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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_pde_2d_gen_order2_rectangle (d03ra)

## Purpose

nag_pde_2d_gen_order2_rectangle (d03ra) integrates a system of linear or nonlinear, time-dependent partial differential equations (PDEs) in two space dimensions on a rectangular domain. The method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs) which are solved using a backward differentiation formula (BDF) method. The resulting system of nonlinear equations is solved using a modified Newton method and a Bi-CGSTAB iterative linear solver with ILU preconditioning. Local uniform grid refinement is used to improve the accuracy of the solution. nag_pde_2d_gen_order2_rectangle (d03ra) originates from the VLUGR2 package (see Blom and Verwer (1993) and Blom et al. (1996)).

## Syntax

[ts, dt, rwk, iwk, ind, ifail] = d03ra(ts, tout, dt, xmin, xmax, ymin, ymax, nx, ny, tols, tolt, pdedef, bndary, pdeiv, monitr, opti, optr, rwk, iwk, itrace, ind, 'npde', npde, 'leniwk', leniwk, 'lenlwk', lenlwk)
[ts, dt, rwk, iwk, ind, ifail] = nag_pde_2d_gen_order2_rectangle(ts, tout, dt, xmin, xmax, ymin, ymax, nx, ny, tols, tolt, pdedef, bndary, pdeiv, monitr, opti, optr, rwk, iwk, itrace, ind, 'npde', npde, 'leniwk', leniwk, 'lenlwk', lenlwk)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 22: lenrwk has been removed from the interface; lenlwk has been made optional
.

## Description

nag_pde_2d_gen_order2_rectangle (d03ra) integrates the system of PDEs:
 Fj(t,x,y,u,ut,ux,uy,uxx,uxy,uyy) = 0,  j = 1,2, … ,npde, $Fj(t,x,y,u,ut,ux,uy,uxx,uxy,uyy)=0, j=1,2,…,npde,$ (1)
for x$x$ and y$y$ in the rectangular domain xminxxmax${x}_{\mathrm{min}}\le x\le {x}_{\mathrm{max}}$, yminyymax${y}_{\mathrm{min}}\le y\le {y}_{\mathrm{max}}$, and time interval t0ttout${t}_{0}\le t\le {t}_{\mathrm{out}}$, where the vector u$u$ is the set of solution values
 u(x,y,t) = [u1(x,y,t), … ,unpde(x,y,t)]T, $u(x,y,t)=[u1(x,y,t),…,unpde(x,y,t)]T,$
and ut${u}_{t}$ denotes partial differentiation with respect to t$t$, and similarly for ux${u}_{x}$ etc.
The functions Fj${F}_{j}$ must be supplied by you in pdedef. Similarly the initial values of the functions u(x,y,t)$u\left(x,y,t\right)$ must be specified at t = t0$t={t}_{0}$ in pdeiv.
Note that whilst complete generality is offered by the master equations (1), nag_pde_2d_gen_order2_rectangle (d03ra) is not appropriate for all PDEs. In particular, hyperbolic systems should not be solved using this function. Also, at least one component of ut${u}_{t}$ must appear in the system of PDEs.
The boundary conditions must be supplied by you in bndary in the form
 Gj (t,x,y,u,ut,ux,uy) = 0 , $Gj (t,x,y,u,ut,ux,uy) = 0 ,$ (2)
for all y$y$ when xmin ${x}_{\mathrm{min}}$ or xmax ${x}_{\mathrm{max}}$ and for all x$x$ when y = ymin $y={y}_{\mathrm{min}}$ or y = ymax $y={y}_{\mathrm{max}}$ and j = 1,2,,npde $j=1,2,\dots ,{\mathbf{npde}}$
The domain is covered by a uniform coarse base grid of size nx × ny${n}_{x}×{n}_{y}$ specified by you, and nested finer uniform subgrids are subsequently created in regions with high spatial activity. The refinement is controlled using a space monitor which is computed from the current solution and a user-supplied space tolerance tols. A number of optional parameters, e.g., the maximum number of grid levels at any time, and some weighting factors, can be specified in the arrays opti and optr. Further details of the refinement strategy can be found in Section [Further Comments].
The system of PDEs and the boundary conditions are discretized in space on each grid using a standard second-order finite difference scheme (centred on the internal domain and one-sided at the boundaries), and the resulting system of ODEs is integrated in time using a second-order, two-step, implicit BDF method with variable step size. The time integration is controlled using a time monitor computed at each grid level from the current solution and a user-supplied time tolerance tolt, and some further optional user-specified weighting factors held in optr (see Section [Further Comments] for details). The time monitor is used to compute a new step size, subject to restrictions on the size of the change between steps, and (optional) user-specified maximum and minimum step sizes held in dt. The step size is adjusted so that the remaining integration interval is an integer number times Δt$\Delta t$. In this way a solution is obtained at t = tout$t={t}_{\mathrm{out}}$.
A modified Newton method is used to solve the nonlinear equations arising from the time integration. You may specify (in opti) the maximum number of Newton iterations to be attempted. A Jacobian matrix is calculated at the beginning of each time step. If the Newton process diverges or the maximum number of iterations is exceeded, a new Jacobian is calculated using the most recent iterates and the Newton process is restarted. If convergence is not achieved after the (optional) user-specified maximum number of new Jacobian evaluations, the time step is retried with Δt = Δt / 4$\Delta t=\Delta t/4$. The linear systems arising from the Newton iteration are solved using a Bi-CGSTAB iterative method, in combination with ILU preconditioning. The maximum number of iterations can be specified by you in opti.
The solution at all grid levels is stored in the workspace arrays, along with other information needed for a restart (i.e., a continuation call). It is not intended that you extract the solution from these arrays, indeed the necessary information regarding these arrays is not included. The user-supplied monitor monitr should be used to obtain the solution at particular levels and times. monitr is called at the end of every time step, with the last step being identified via the input parameter tlast.
Within pdeiv, pdedef, bndary and monitr the data structure is as follows. Each point on a particular grid is given an index (ranging from 1$1$ to the total number of points on the grid) and all coordinate or solution information is stored in arrays according to this index, e.g., x(i)${\mathbf{x}}\left(i\right)$ and y(i)${\mathbf{y}}\left(i\right)$ contain the x$x$- and y$y$ coordinate of point i$i$, and u(i,j)${\mathbf{u}}\left(i,j\right)$ contains the j$j$th solution component uj${u}_{j}$ at point i$i$.
Further details of the underlying algorithm can be found in Section [Further Comments] and in Blom and Verwer (1993) and Blom et al. (1996) and the references therein.

## References

Adjerid S and Flaherty J E (1988) A local refinement finite element method for two-dimensional parabolic systems SIAM J. Sci. Statist. Comput. 9 792–811
Blom J G, Trompert R A and Verwer J G (1996) Algorithm 758. VLUGR2: A vectorizable adaptive grid solver for PDEs in 2D Trans. Math. Software 22 302–328
Blom J G and Verwer J G (1993) VLUGR2: A vectorized local uniform grid refinement code for PDEs in 2D Report NM-R9306 CWI, Amsterdam
Brown P N, Hindmarsh A C and Petzold L R (1994) Using Krylov methods in the solution of large scale differential-algebraic systems SIAM J. Sci. Statist. Comput. 15 1467–1488
Trompert R A (1993) Local uniform grid refinement and systems of coupled partial differential equations Appl. Numer. Maths 12 331–355
Trompert R A and Verwer J G (1993) Analysis of the implicit Euler local uniform grid refinement method SIAM J. Sci. Comput. 14 259–278

## Parameters

### Compulsory Input Parameters

1:     ts – double scalar
The initial value of the independent variable t$t$.
Constraint: ${\mathbf{ts}}<{\mathbf{tout}}$.
2:     tout – double scalar
The final value of t$t$ to which the integration is to be carried out.
3:     dt(3$3$) – double array
The initial, minimum and maximum time step sizes respectively.
dt(1)${\mathbf{dt}}\left(1\right)$
Specifies the initial time step size to be used on the first entry, i.e., when ind = 0${\mathbf{ind}}=0$. If dt(1) = 0.0${\mathbf{dt}}\left(1\right)=0.0$ then the default value dt(1) = 0.01 × (toutts)${\mathbf{dt}}\left(1\right)=0.01×\left({\mathbf{tout}}-{\mathbf{ts}}\right)$ is used. On subsequent entries (ind = 1${\mathbf{ind}}=1$), the value of dt(1)${\mathbf{dt}}\left(1\right)$ is not referenced.
dt(2)${\mathbf{dt}}\left(2\right)$
Specifies the minimum time step size to be attempted by the integrator. If dt(2) = 0.0${\mathbf{dt}}\left(2\right)=0.0$ the default value dt(2) = 10.0 × machine precision is used.
dt(3)${\mathbf{dt}}\left(3\right)$
Specifies the maximum time step size to be attempted by the integrator. If dt(3) = 0.0${\mathbf{dt}}\left(3\right)=0.0$ the default value dt(3) = toutts${\mathbf{dt}}\left(3\right)={\mathbf{tout}}-{\mathbf{ts}}$ is used.
Constraints:
• if ind = 0${\mathbf{ind}}=0$, dt(1)0.0${\mathbf{dt}}\left(1\right)\ge 0.0$;
• if ind = 0${\mathbf{ind}}=0$ and dt(1) > 0.0${\mathbf{dt}}\left(1\right)>0.0$,
10.0 × machine precision × max (|ts|,|tout|)dt(1)toutts and
dt(2)dt(1)dt(3)${\mathbf{dt}}\left(2\right)\le {\mathbf{dt}}\left(1\right)\le {\mathbf{dt}}\left(3\right)$, where the values of dt(2)${\mathbf{dt}}\left(2\right)$ and dt(3)${\mathbf{dt}}\left(3\right)$ will have been reset to their default values if zero on entry;
• 0 dt(2) dt(3) $0\le {\mathbf{dt}}\left(2\right)\le {\mathbf{dt}}\left(3\right)$.
4:     xmin – double scalar
5:     xmax – double scalar
The extents of the rectangular domain in the x$x$-direction, i.e., the x$x$ coordinates of the left and right boundaries respectively.
Constraint: ${\mathbf{xmin}}<{\mathbf{xmax}}$ and xmax must be sufficiently distinguishable from xmin for the precision of the machine being used.
6:     ymin – double scalar
7:     ymax – double scalar
The extents of the rectangular domain in the y$y$-direction, i.e., the y$y$ coordinates of the lower and upper boundaries respectively.
Constraint: ${\mathbf{ymin}}<{\mathbf{ymax}}$ and ymax must be sufficiently distinguishable from ymin for the precision of the machine being used.
8:     nx – int64int32nag_int scalar
The number of grid points in the x$x$-direction (including the boundary points).
Constraint: nx4${\mathbf{nx}}\ge 4$.
9:     ny – int64int32nag_int scalar
The number of grid points in the y$y$-direction (including the boundary points).
Constraint: ny4${\mathbf{ny}}\ge 4$.
10:   tols – double scalar
The space tolerance used in the grid refinement strategy (σ$\sigma$ in equation (4)). See Section [Refinement Strategy].
Constraint: tols > 0.0${\mathbf{tols}}>0.0$.
11:   tolt – double scalar
The time tolerance used to determine the time step size (τ$\tau$ in equation (7)). See Section [Time Integration ].
Constraint: tolt > 0.0${\mathbf{tolt}}>0.0$.
12:   pdedef – function handle or string containing name of m-file
pdedef must evaluate the functions Fj${F}_{j}$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, in equation (1) which define the system of PDEs (i.e., the residuals of the resulting ODE system) at all interior points of the domain. Values at points on the boundaries of the domain are ignored and will be overwritten by bndary. pdedef is called for each subgrid in turn.
[res] = pdedef(npts, npde, t, x, y, u, ut, ux, uy, uxx, uxy, uyy)

Input Parameters

1:     npts – int64int32nag_int scalar
The number of grid points in the current grid.
2:     npde – int64int32nag_int scalar
The number of PDEs in the system.
3:     t – double scalar
The current value of the independent variable t$t$.
4:     x(npts) – double array
x(i)${\mathbf{x}}\left(\mathit{i}\right)$ contains the x$x$ coordinate of the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
5:     y(npts) – double array
y(i)${\mathbf{y}}\left(\mathit{i}\right)$ contains the y$y$ coordinate of the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
6:     u(npts,npde) – double array
u(i,j)${\mathbf{u}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
7:     ut(npts,npde) – double array
ut(i,j)${\mathbf{ut}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (u)/(t) $\frac{\partial u}{\partial t}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
8:     ux(npts,npde) – double array
ux(i,j)${\mathbf{ux}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (u)/(x) $\frac{\partial u}{\partial x}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
9:     uy(npts,npde) – double array
uy(i,j)${\mathbf{uy}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (u)/(y) $\frac{\partial u}{\partial y}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
10:   uxx(npts,npde) – double array
uxx(i,j)${\mathbf{uxx}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (2u)/(x2) $\frac{{\partial }^{2}u}{\partial {x}^{2}}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
11:   uxy(npts,npde) – double array
uxy(i,j)${\mathbf{uxy}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (2u)/(xy) $\frac{{\partial }^{2}u}{\partial x\partial y}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
12:   uyy(npts,npde) – double array
uyy(i,j)${\mathbf{uyy}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (2u)/(y2) $\frac{{\partial }^{2}u}{\partial {y}^{2}}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.

Output Parameters

1:     res(npts,npde) – double array
res(i,j)${\mathbf{res}}\left(\mathit{i},\mathit{j}\right)$ must contain the value of Fj${F}_{\mathit{j}}$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$, although the residuals at boundary points will be ignored (and overwritten later on) and so they need not be specified here.
13:   bndary – function handle or string containing name of m-file
bndary must evaluate the functions Gj${G}_{\mathit{j}}$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, in equation (2) which define the boundary conditions at all boundary points of the domain. Residuals at interior points must not be altered by this function.
[res] = bndary(npts, npde, t, x, y, u, ut, ux, uy, nbpts, lbnd, res)

Input Parameters

1:     npts – int64int32nag_int scalar
The number of grid points in the current grid.
2:     npde – int64int32nag_int scalar
The number of PDEs in the system.
3:     t – double scalar
The current value of the independent variable t$t$.
4:     x(npts) – double array
x(i)${\mathbf{x}}\left(\mathit{i}\right)$ contains the x$x$ coordinate of the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
5:     y(npts) – double array
y(i)${\mathbf{y}}\left(\mathit{i}\right)$ contains the y$y$ coordinate of the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
6:     u(npts,npde) – double array
u(i,j)${\mathbf{u}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
7:     ut(npts,npde) – double array
ut(i,j)${\mathbf{ut}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (u)/(t) $\frac{\partial u}{\partial t}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
8:     ux(npts,npde) – double array
ux(i,j)${\mathbf{ux}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (u)/(x) $\frac{\partial u}{\partial x}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
9:     uy(npts,npde) – double array
uy(i,j)${\mathbf{uy}}\left(\mathit{i},\mathit{j}\right)$ contains the value of (u)/(y) $\frac{\partial u}{\partial y}$ for the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
10:   nbpts – int64int32nag_int scalar
The number of boundary points in the grid.
11:   lbnd(nbpts) – int64int32nag_int array
lbnd(i)${\mathbf{lbnd}}\left(\mathit{i}\right)$ contains the grid index for the i$\mathit{i}$th boundary point, for i = 1,2,,nbpts$\mathit{i}=1,2,\dots ,{\mathbf{nbpts}}$. Hence the i$\mathit{i}$th boundary point has coordinates x(lbnd(i))${\mathbf{x}}\left({\mathbf{lbnd}}\left(\mathit{i}\right)\right)$ and y(lbnd(i))${\mathbf{y}}\left({\mathbf{lbnd}}\left(\mathit{i}\right)\right)$, and the corresponding solution values are u(lbnd(i),npde)${\mathbf{u}}\left({\mathbf{lbnd}}\left(\mathit{i}\right),{\mathbf{npde}}\right)$, etc.
12:   res(npts,npde) – double array
res(i,j)${\mathbf{res}}\left(\mathit{i},\mathit{j}\right)$ contains the value of Fj${F}_{\mathit{j}}$, for i = 1,2,,npde$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$, as returned by pdedef. The residuals at the boundary points will be overwritten and so need not have been set by pdedef.

Output Parameters

1:     res(npts,npde) – double array
res(lbnd(i),j)${\mathbf{res}}\left({\mathbf{lbnd}}\left(\mathit{i}\right),\mathit{j}\right)$ must contain the value of Gj${G}_{\mathit{j}}$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, at the i$\mathit{i}$th boundary point, for i = 1,2,,nbpts$\mathit{i}=1,2,\dots ,{\mathbf{nbpts}}$.
Note: elements of res corresponding to interior points must not be altered.
14:   pdeiv – function handle or string containing name of m-file
pdeiv must specify the initial values of the PDE components u$u$ at all points in the grid. pdeiv is not referenced if, on entry, ind = 1${\mathbf{ind}}=1$.
[u] = pdeiv(npts, npde, t, x, y)

Input Parameters

1:     npts – int64int32nag_int scalar
The number of grid points in the grid.
2:     npde – int64int32nag_int scalar
The number of PDEs in the system.
3:     t – double scalar
The (initial) value of the independent variable t$t$.
4:     x(npts) – double array
x(i)${\mathbf{x}}\left(\mathit{i}\right)$ contains the x$x$ coordinate of the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
5:     y(npts) – double array
y(i)${\mathbf{y}}\left(\mathit{i}\right)$ contains the y$y$ coordinate of the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.

Output Parameters

1:     u(npts,npde) – double array
u(i,j)${\mathbf{u}}\left(\mathit{i},\mathit{j}\right)$ must contain the value of the j$\mathit{j}$th PDE component at the i$\mathit{i}$th grid point, for i = 1,2,,npts$\mathit{i}=1,2,\dots ,{\mathbf{npts}}$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
15:   monitr – function handle or string containing name of m-file
monitr is called by nag_pde_2d_gen_order2_rectangle (d03ra) at the end of every successful time step, and may be used to examine or print the solution or perform other tasks such as error calculations, particularly at the final time step, indicated by the parameter tlast. The input arguments contain information about the grid and solution at all grid levels used.
monitr can also be used to force an immediate tidy termination of the solution process and return to the calling program.
[ierr] = monitr(npde, t, dt, dtnew, tlast, nlev, ngpts, xpts, ypts, lsol, sol, ierr)

Input Parameters

1:     npde – int64int32nag_int scalar
The number of PDEs in the system.
2:     t – double scalar
The current value of the independent variable t$t$, i.e., the time at the end of the integration step just completed.
3:     dt – double scalar
The current time step size Δt$\Delta t$, i.e., the time step size used for the integration step just completed.
4:     dtnew – double scalar
The step size that will be used for the next time step.
5:     tlast – logical scalar
Indicates if intermediate or final time step. tlast = false${\mathbf{tlast}}=\mathbf{false}$ for an intermediate step, tlast = true${\mathbf{tlast}}=\mathbf{true}$ for the last call to monitr before returning to your program.
6:     nlev – int64int32nag_int scalar
The number of grid levels used at time t.
7:     ngpts(nlev) – int64int32nag_int array
ngpts(l)${\mathbf{ngpts}}\left(\mathit{l}\right)$ contains the number of grid points at level l$\mathit{l}$, for l = 1,2,,nlev$\mathit{l}=1,2,\dots ,{\mathbf{nlev}}$.
8:     xpts(lpts$\mathit{lpts}$) – double array
Contains the x$x$ coordinates of the grid points in each level in turn, i.e., x(i)${\mathbf{x}}\left(\mathit{i}\right)$, for i = 1,2,,ngpts(l)$\mathit{i}=1,2,\dots ,{\mathbf{ngpts}}\left(\mathit{l}\right)$ and l = 1,2,,nlev$\mathit{l}=1,2,\dots ,{\mathbf{nlev}}$.
So for level l$\mathit{l}$, x(i) = xpts(k + i)${\mathbf{x}}\left(\mathit{i}\right)={\mathbf{xpts}}\left(\mathit{k}+\mathit{i}\right)$, where k = ngpts(1) + ngpts(2) + + ngpts(l1)$\mathit{k}={\mathbf{ngpts}}\left(1\right)+{\mathbf{ngpts}}\left(2\right)+\cdots +{\mathbf{ngpts}}\left(\mathit{l}-1\right)$, for i = 1,2,,ngpts(l)$\mathit{i}=1,2,\dots ,{\mathbf{ngpts}}\left(\mathit{l}\right)$ and l = 1,2,,nlev$\mathit{l}=1,2,\dots ,{\mathbf{nlev}}$.
9:     ypts(lpts$\mathit{lpts}$) – double array
Contains the y$y$ coordinates of the grid points in each level in turn, i.e., y(i)${\mathbf{y}}\left(\mathit{i}\right)$, for i = 1,2,,ngpts(l)$\mathit{i}=1,2,\dots ,{\mathbf{ngpts}}\left(\mathit{l}\right)$ and l = 1,2,,nlev$\mathit{l}=1,2,\dots ,{\mathbf{nlev}}$.
So for level l$\mathit{l}$, y(i) = ypts(k + i)${\mathbf{y}}\left(\mathit{i}\right)={\mathbf{ypts}}\left(\mathit{k}+\mathit{i}\right)$, where k = ngpts(1) + ngpts(2) + + ngpts(l1)$\mathit{k}={\mathbf{ngpts}}\left(1\right)+{\mathbf{ngpts}}\left(2\right)+\cdots +{\mathbf{ngpts}}\left(\mathit{l}-1\right)$, for i = 1,2,,ngpts(l)$\mathit{i}=1,2,\dots ,{\mathbf{ngpts}}\left(\mathit{l}\right)$ and l = 1,2,,nlev$\mathit{l}=1,2,\dots ,{\mathbf{nlev}}$.
10:   lsol(nlev) – int64int32nag_int array
lsol(l)${\mathbf{lsol}}\left(\mathit{l}\right)$ contains the pointer to the solution in sol at grid level l$\mathit{l}$ and time t. (lsol(l)${\mathbf{lsol}}\left(\mathit{l}\right)$ actually contains the array index immediately preceding the start of the solution in sol.)
11:   sol( : $:$) – double array
Contains the solution u(ngpts(l),npde)${\mathbf{u}}\left({\mathbf{ngpts}}\left(\mathit{l}\right),{\mathbf{npde}}\right)$ at time t for each grid level l$\mathit{l}$ in turn, positioned according to lsol, i.e., for level l$\mathit{l}$, u(i,j) = sol(lsol(l) + (j1) × ngpts(l) + i)${\mathbf{u}}\left(\mathit{i},\mathit{j}\right)={\mathbf{sol}}\left({\mathbf{lsol}}\left(\mathit{l}\right)+\left(\mathit{j}-1\right)×{\mathbf{ngpts}}\left(\mathit{l}\right)+\mathit{i}\right)$, for i = 1,2,,ngpts(l)$\mathit{i}=1,2,\dots ,{\mathbf{ngpts}}\left(\mathit{l}\right)$, j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$ and l = 1,2,,nlev$\mathit{l}=1,2,\dots ,{\mathbf{nlev}}$.
Default: lenrwk6 × npde$\mathit{lenrwk}-6×{\mathbf{npde}}$
12:   ierr – int64int32nag_int scalar
Will be set to 0$0$.

Output Parameters

1:     ierr – int64int32nag_int scalar
Should be set to 1$1$ to force a tidy termination and an immediate return to the calling program with ${\mathbf{ifail}}={\mathbf{4}}$. ierr should remain unchanged otherwise.
16:   opti(4$4$) – int64int32nag_int array
May be set to control various options available in the integrator.
opti(1) = 0${\mathbf{opti}}\left(1\right)=0$
All the default options are employed.
opti(1) > 0${\mathbf{opti}}\left(1\right)>0$
The default value of opti(i)${\mathbf{opti}}\left(\mathit{i}\right)$, for i = 2,3,4$\mathit{i}=2,3,4$, can be obtained by setting opti(i) = 0${\mathbf{opti}}\left(\mathit{i}\right)=0$.
opti(1)${\mathbf{opti}}\left(1\right)$
Specifies the maximum number of grid levels allowed (including the base grid). opti(1)0${\mathbf{opti}}\left(1\right)\ge 0$. The default value is opti(1) = 3${\mathbf{opti}}\left(1\right)=3$.
opti(2)${\mathbf{opti}}\left(2\right)$
Specifies the maximum number of Jacobian evaluations allowed during each nonlinear equations solution. opti(2)0${\mathbf{opti}}\left(2\right)\ge 0$. The default value is opti(2) = 2${\mathbf{opti}}\left(2\right)=2$.
opti(3)${\mathbf{opti}}\left(3\right)$
Specifies the maximum number of Newton iterations in each nonlinear equations solution. opti(3)0${\mathbf{opti}}\left(3\right)\ge 0$. The default value is opti(3) = 10${\mathbf{opti}}\left(3\right)=10$.
opti(4)${\mathbf{opti}}\left(4\right)$
Specifies the maximum number of iterations in each linear equations solution. opti(4)0${\mathbf{opti}}\left(4\right)\ge 0$. The default value is opti(4) = 100${\mathbf{opti}}\left(4\right)=100$.
Constraint: opti(1)0${\mathbf{opti}}\left(1\right)\ge 0$ and if opti(1) > 0${\mathbf{opti}}\left(1\right)>0$, opti(i)0${\mathbf{opti}}\left(\mathit{i}\right)\ge 0$, for i = 2,3,4$\mathit{i}=2,3,4$.
17:   optr(3$3$,npde) – double array
May be used to specify the optional vectors umax${u}^{\mathrm{max}}$, ws${w}^{\mathrm{s}}$ and wt${w}^{\mathrm{t}}$ in the space and time monitors (see Section [Further Comments]).
If an optional vector is not required then all its components should be set to 1.0$1.0$.
optr(1,j)${\mathbf{optr}}\left(1,\mathit{j}\right)$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, specifies ujmax${u}_{\mathit{j}}^{\mathrm{max}}$, the approximate maximum absolute value of the j$\mathit{j}$th component of u$u$, as used in (4) and (7). optr(1,j) > 0.0${\mathbf{optr}}\left(1,\mathit{j}\right)>0.0$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
optr(2,j)${\mathbf{optr}}\left(2,\mathit{j}\right)$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, specifies wjs${w}_{\mathit{j}}^{\mathrm{s}}$, the weighting factors used in the space monitor (see (4)) to indicate the relative importance of the j$\mathit{j}$th component of u$u$ on the space monitor. optr(2,j)0.0${\mathbf{optr}}\left(2,\mathit{j}\right)\ge 0.0$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
optr(3,j)${\mathbf{optr}}\left(3,\mathit{j}\right)$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, specifies wjt${w}_{\mathit{j}}^{\mathrm{t}}$, the weighting factors used in the time monitor (see (6)) to indicate the relative importance of the j$\mathit{j}$th component of u$u$ on the time monitor. optr(3,j)0.0${\mathbf{optr}}\left(3,\mathit{j}\right)\ge 0.0$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
Constraints:
• optr(1,j) > 0.0${\mathbf{optr}}\left(1,\mathit{j}\right)>0.0$, for j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$;
• optr(i,j)0.0${\mathbf{optr}}\left(\mathit{i},\mathit{j}\right)\ge 0.0$, for i = 2,3$\mathit{i}=2,3$ and j = 1,2,,npde$\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
18:   rwk(lenrwk) – double array
lenrwk, the dimension of the array, must satisfy the constraint lenrwknx × ny × npde × (14 + 18 × npde) + 2 × nx × ny$\mathit{lenrwk}\ge {\mathbf{nx}}×{\mathbf{ny}}×{\mathbf{npde}}×\left(14+18×{\mathbf{npde}}\right)+2×{\mathbf{nx}}×{\mathbf{ny}}$ (the required size for the initial grid).
The required value of lenrwk cannot be determined exactly in advance, but a suggested value is
 lenrwk = maxpts × npde × (5 × l + 18 × npde + 9) + 2 × maxpts, $lenrwk=maxpts×npde×(5×l+18×npde+9)+2×maxpts,$
where l = opti(1)$\mathit{l}={\mathbf{opti}}\left(1\right)$ if opti(1)0${\mathbf{opti}}\left(1\right)\ne 0$ and l = 3$\mathit{l}=3$ otherwise, and maxpts$\mathit{maxpts}$ is the expected maximum number of grid points at any one level. If during the execution the supplied value is found to be too small then the function returns with ${\mathbf{ifail}}={\mathbf{3}}$ and an estimated required size is printed on the current error message unit (see nag_file_set_unit_error (x04aa)).
Constraint: lenrwknx × ny × npde × (14 + 18 × npde) + 2 × nx × ny$\mathit{lenrwk}\ge {\mathbf{nx}}×{\mathbf{ny}}×{\mathbf{npde}}×\left(14+18×{\mathbf{npde}}\right)+2×{\mathbf{nx}}×{\mathbf{ny}}$ (the required size for the initial grid).
19:   iwk(leniwk) – int64int32nag_int array
leniwk, the dimension of the array, must satisfy the constraint leniwk19 × nx × ny + 9${\mathbf{leniwk}}\ge 19×{\mathbf{nx}}×{\mathbf{ny}}+9$ (the required size for the initial grid).
If ind = 0${\mathbf{ind}}=0$, iwk need not be set. Otherwise iwk must remain unchanged from a previous call to nag_pde_2d_gen_order2_rectangle (d03ra).
20:   itrace – int64int32nag_int scalar
The level of trace information required from nag_pde_2d_gen_order2_rectangle (d03ra). itrace may take the value 1$-1$, 0$0$, 1$1$, 2$2$ or 3$3$.
itrace = 1${\mathbf{itrace}}=-1$
No output is generated.
itrace = 0${\mathbf{itrace}}=0$
Only warning messages are printed.
itrace > 0${\mathbf{itrace}}>0$
Output from the underlying solver is printed on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)). This output contains details of the time integration, the nonlinear iteration and the linear solver.
If itrace < 1${\mathbf{itrace}}<-1$, then 1$-1$ is assumed and similarly if itrace > 3${\mathbf{itrace}}>3$, then 3$3$ is assumed.
The advisory messages are given in greater detail as itrace increases. Setting itrace = 1${\mathbf{itrace}}=1$ allows you to monitor the progress of the integration without possibly excessive information.
21:   ind – int64int32nag_int scalar
Must be set to 0$0$ or 1$1$.
ind = 0${\mathbf{ind}}=0$
Starts the integration in time.
ind = 1${\mathbf{ind}}=1$
Continues the integration after an earlier exit from the function. In this case, only the following parameters may be reset between calls to nag_pde_2d_gen_order2_rectangle (d03ra): tout, dt(2)${\mathbf{dt}}\left(2\right)$, dt(3)${\mathbf{dt}}\left(3\right)$, tols, tolt, opti, optr, itrace and ifail.
Constraint: 0ind1$0\le {\mathbf{ind}}\le 1$.
Note:  for users of the NAG Toolbox only, additional values of ind are allowed. See Section [PDEs ()] in the (smpin) for further details.

### Optional Input Parameters

1:     npde – int64int32nag_int scalar
Default: The dimension of the array optr.
The number of PDEs in the system.
Constraint: npde1${\mathbf{npde}}\ge 1$.
2:     leniwk – int64int32nag_int scalar
Default: The dimension of the array iwk.
The dimension of the array iwk as declared in the (sub)program from which nag_pde_2d_gen_order2_rectangle (d03ra) is called.
The required value of leniwk cannot be determined exactly in advance, but a suggested value is
 leniwk = maxpts × (14 + 5 × m) + 7 × m + 2,$leniwk=maxpts×(14+5×m)+7×m+2,$
where maxpts$\mathit{maxpts}$ is the expected maximum number of grid points at any one level and m = opti(1)$m={\mathbf{opti}}\left(1\right)$ if opti(1) > 0${\mathbf{opti}}\left(1\right)>0$ and m = 3$m=3$ otherwise. If during the execution the supplied value is found to be too small then the function returns with ${\mathbf{ifail}}={\mathbf{3}}$ and an estimated required size is printed on the current error message unit (see nag_file_set_unit_error (x04aa)).
Constraint: leniwk19 × nx × ny + 9${\mathbf{leniwk}}\ge 19×{\mathbf{nx}}×{\mathbf{ny}}+9$ (the required size for the initial grid).
3:     lenlwk – int64int32nag_int scalar
The dimension of the array lwk as declared in the (sub)program from which nag_pde_2d_gen_order2_rectangle (d03ra) is called.
The required value of lenlwk cannot be determined exactly in advanced, but a suggested value is
 lenlwk = maxpts + 1,$lenlwk=maxpts+1,$
where maxpts$\mathit{maxpts}$ is the expected maximum number of grid points at any one level. If during the execution the supplied value is found to be too small then the function returns with ${\mathbf{ifail}}={\mathbf{3}}$ and an estimated required size is printed on the current error message unit (see nag_file_set_unit_error (x04aa)).
Default: maxpts + 1$\mathit{maxpts}+1$
Constraint: lenlwknx × ny + 1${\mathbf{lenlwk}}\ge {\mathbf{nx}}×{\mathbf{ny}}+1$ (the required size for the initial grid).

lenrwk lwk

### Output Parameters

1:     ts – double scalar
The value of t$t$ which has been reached. Normally ${\mathbf{ts}}={\mathbf{tout}}$.
2:     dt(3$3$) – double array
dt(1)${\mathbf{dt}}\left(1\right)$ contains the time step size for the next time step. dt(2)${\mathbf{dt}}\left(2\right)$ and dt(3)${\mathbf{dt}}\left(3\right)$ are unchanged or set to their default values if zero on entry.
3:     rwk(lenrwk) – double array
lenrwknx × ny × npde × (14 + 18 × npde) + 2 × nx × ny$\mathit{lenrwk}\ge {\mathbf{nx}}×{\mathbf{ny}}×{\mathbf{npde}}×\left(14+18×{\mathbf{npde}}\right)+2×{\mathbf{nx}}×{\mathbf{ny}}$ (the required size for the initial grid).
Communication array, used to store information between calls to nag_pde_2d_gen_order2_rectangle (d03ra).
4:     iwk(leniwk) – int64int32nag_int array
The following components of the array iwk concern the efficiency of the integration. Here, m$m$ is the maximum number of grid levels allowed (m = opti(1)$m={\mathbf{opti}}\left(1\right)$ if opti(1) > 1${\mathbf{opti}}\left(1\right)>1$ and m = 3$m=3$ otherwise), and l$\mathit{l}$ is a grid level taking the values l = 1,2,,nl$\mathit{l}=1,2,\dots ,\mathit{nl}$, where nl$\mathit{nl}$ is the number of levels used.
iwk(1)${\mathbf{iwk}}\left(1\right)$
Contains the number of steps taken in time.
iwk(2)${\mathbf{iwk}}\left(2\right)$
Contains the number of rejected time steps.
iwk(2 + l)${\mathbf{iwk}}\left(2+\mathit{l}\right)$
Contains the total number of residual evaluations performed (i.e., the number of times pdedef was called) at grid level l$\mathit{l}$.
iwk(2 + m + l)${\mathbf{iwk}}\left(2+m+\mathit{l}\right)$
Contains the total number of Jacobian evaluations performed at grid level l$\mathit{l}$.
iwk(2 + 2 × m + l)${\mathbf{iwk}}\left(2+2×m+\mathit{l}\right)$
Contains the total number of Newton iterations performed at grid level l$\mathit{l}$.
iwk(2 + 3 × m + l)${\mathbf{iwk}}\left(2+3×m+\mathit{l}\right)$
Contains the total number of linear solver iterations performed at grid level l$\mathit{l}$.
iwk(2 + 4 × m + l)${\mathbf{iwk}}\left(2+4×m+\mathit{l}\right)$
Contains the maximum number of Newton iterations performed at any one time step at grid level l$\mathit{l}$.
iwk(2 + 5 × m + l)${\mathbf{iwk}}\left(2+5×m+\mathit{l}\right)$
Contains the maximum number of linear solver iterations performed at any one time step at grid level l$\mathit{l}$.
Note: the total and maximum numbers are cumulative over all calls to nag_pde_2d_gen_order2_rectangle (d03ra). If the specified maximum number of Newton or linear solver iterations is exceeded at any stage, then the maximums above are set to the specified maximum plus one.
5:     ind – int64int32nag_int scalar
ind = 1${\mathbf{ind}}=1$.
Note:  for users of the NAG Toolbox only, additional values of ind are allowed. See Section [PDEs ()] in the (smpin) for further details.
6:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail = 1${\mathbf{ifail}}=1$
 On entry, npde < 1${\mathbf{npde}}<1$, or ${\mathbf{tout}}\le {\mathbf{ts}}$, or tout is too close to ts, or ind = 0${\mathbf{ind}}=0$ and dt(1) < 0.0${\mathbf{dt}}\left(1\right)<0.0$, or dt(i) < 0.0${\mathbf{dt}}\left(i\right)<0.0$, for i = 2​ or ​3$\mathit{i}=2\text{​ or ​}3$, or dt(2) > dt(3)${\mathbf{dt}}\left(2\right)>{\mathbf{dt}}\left(3\right)$, or ind = 0${\mathbf{ind}}=0$ and 0.0 < dt(1) < 10 × machine precision × max (|ts|,|tout|), or ind = 0${\mathbf{ind}}=0$ and dt(1) > tout − ts${\mathbf{dt}}\left(1\right)>{\mathbf{tout}}-{\mathbf{ts}}$, or ind = 0${\mathbf{ind}}=0$ and dt(1) < dt(2)${\mathbf{dt}}\left(1\right)<{\mathbf{dt}}\left(2\right)$ or dt(1) > dt(3)${\mathbf{dt}}\left(1\right)>{\mathbf{dt}}\left(3\right)$, or ${\mathbf{xmin}}\ge {\mathbf{xmax}}$, or xmax too close to xmin, or ${\mathbf{ymin}}\ge {\mathbf{ymax}}$, or ymax too close to ymin, or nx or ny < 4${\mathbf{ny}}<4$, or tols or tolt ≤ 0.0${\mathbf{tolt}}\le 0.0$, or opti(1) < 0${\mathbf{opti}}\left(1\right)<0$, or opti(1) > 0${\mathbf{opti}}\left(1\right)>0$ and opti(j) < 0${\mathbf{opti}}\left(j\right)<0$, for j = 2$\mathit{j}=2$, 3$3$ or 4$4$, or optr(1,j) ≤ 0.0${\mathbf{optr}}\left(1,j\right)\le 0.0$, for some j = 1,2, … ,npde$j=1,2,\dots ,{\mathbf{npde}}$, or optr(2,j) < 0.0${\mathbf{optr}}\left(2,j\right)<0.0$, for some j = 1,2, … ,npde$j=1,2,\dots ,{\mathbf{npde}}$, or optr(3,j) < 0.0${\mathbf{optr}}\left(3,j\right)<0.0$, for some j = 1,2, … ,npde$j=1,2,\dots ,{\mathbf{npde}}$, or lenrwk, leniwk or lenlwk too small for initial grid level, or ind ≠ 0${\mathbf{ind}}\ne 0$ or 1$1$, or ind = 1${\mathbf{ind}}=1$ on initial entry to nag_pde_2d_gen_order2_rectangle (d03ra).
ifail = 2${\mathbf{ifail}}=2$
The time step size to be attempted is less than the specified minimum size. This may occur following time step failures and subsequent step size reductions caused by one or more of the following:
• the requested accuracy could not be achieved, i.e., tolt is too small,
• the maximum number of linear solver iterations, Newton iterations or Jacobian evaluations is too small,
• ILU decomposition of the Jacobian matrix could not be performed, possibly due to singularity of the Jacobian.
Setting itrace to a higher value may provide further information.
In the latter two cases you are advised to check their problem formulation in pdedef and/or bndary, and the initial values in pdeiv if appropriate.
ifail = 3${\mathbf{ifail}}=3$
One or more of the workspace arrays is too small for the required number of grid points. An estimate of the required sizes for the current stage is output, but more space may be required at a later stage.
W ifail = 4${\mathbf{ifail}}=4$
ierr was set to 1$1$ in monitr, forcing control to be passed back to calling program. Integration was successful as far as t = ts${\mathbf{t}}={\mathbf{ts}}$.
W ifail = 5${\mathbf{ifail}}=5$
The integration has been completed but the maximum number of levels specified in opti(1)${\mathbf{opti}}\left(1\right)$ was insufficient at one or more time steps, meaning that the requested space accuracy could not be achieved. To avoid this warning either increase the value of opti(1)${\mathbf{opti}}\left(1\right)$ or decrease the value of tols.

## Accuracy

There are three sources of error in the algorithm: space and time discretization, and interpolation (linear) between grid levels. The space and time discretization errors are controlled separately using the parameters tols and tolt described in the following section, and you should test the effects of varying these parameters. Interpolation errors are generally implicitly controlled by the refinement criterion since in areas where interpolation errors are potentially large, the space monitor will also be large. It can be shown that the global spatial accuracy is comparable to that which would be obtained on a uniform grid of the finest grid size. A full error analysis can be found in Trompert and Verwer (1993).

### Algorithm Outline

The local uniform grid refinement method is summarised as follows:
1. Initialize the course base grid, an initial solution and an initial time step.
2. Solve the system of PDEs on the current grid with the current time step.
3. If the required accuracy in space and the maximum number of grid levels have not yet been reached:
 (a) Determine new finer grid at forward time level. (b) Get solution values at previous time level(s) on new grid. (c) Interpolate internal boundary values from old grid at forward time. (d) Get initial values for the Newton process at forward time. (e) Go to 2$2$.
4. Update the coarser grid solution using the finer grid values.
5. Estimate error in time integration. If time error is acceptable advance time level.
6. Determine new step size then go to 2$2$ with coarse base as current grid.

### Refinement Strategy

For each grid point i$i$ a space monitor μis${\mu }_{i}^{\mathrm{s}}$ is determined by
 μis = max {γj(|Δx2(∂2)/( ∂ x2 )uj(xi,yi,t)| + |Δy2(∂2)/( ∂ y2 )uj(xi,yi,t)|)}, j = 1,npde
$μ i s = max j=1,npde { γ j ( | Δ x 2 ∂ 2 ∂ x 2 u j ( x i , y i ,t) | + | Δ y 2 ∂ 2 ∂ y 2 u j ( x i , y i ,t) | ) } ,$
(3)
where Δx$\Delta x$ and Δy$\Delta y$ are the grid widths in the x$x$ and y$y$ directions; and xi${x}_{i}$, yi${y}_{i}$ are the x$x$ and y$y$ coordinates at grid point i$i$. The parameter γj${\gamma }_{j}$ is obtained from
 γj = (wjs)/(ujmax σ), $γj=wjs ujmax σ ,$ (4)
where σ$\sigma$ is the user-supplied space tolerance; wjs${w}_{j}^{s}$ is a weighting factor for the relative importance of the j$j$th PDE component on the space monitor; and ujmax${u}_{j}^{\mathrm{max}}$ is the approximate maximum absolute value of the j$j$th component. A value for σ$\sigma$ must be supplied by you. Values for wjs${w}_{j}^{\mathrm{s}}$ and ujmax${u}_{j}^{\mathrm{max}}$ must also be supplied but may be set to the value 1.0$1.0$ if little information about the solution is known.
A new level of refinement is created if
 max {μis} > 0.9  or  1.0, i
$maxi{μis}>0.9 or 1.0,$
(5)
depending on the grid level at the previous step in order to avoid fluctuations in the number of grid levels between time steps. If (5) is satisfied then all grid points for which μis > 0.25${\mu }_{i}^{s}>0.25$ are flagged and surrounding cells are quartered in size.
No derefinement takes place as such, since at each time step the solution on the base grid is computed first and new finer grids are then created based on the new solution. Hence derefinement occurs implicitly. See Section [Algorithm Outline].

### Time Integration

The time integration is controlled using a time monitor calculated at each level l$\mathit{l}$ up to the maximum level used, given by
 μlt = sqrt(1/N ∑ j = 1npdewjt ∑ i = 1ngpts(l) ((Δt)/(αij)ut(xi,yi,t))2) $μlt=1N∑j=1npdewjt∑i=1 ngptsl (Δtαij ut(xi,yi,t)) 2$ (6)
where ngpts(l)${\mathbf{ngpts}}\left(\mathit{l}\right)$ is the total number of points on grid level l$\mathit{l}$; N = ngpts(l) × npde$N={\mathbf{ngpts}}\left(\mathit{l}\right)×{\mathbf{npde}}$; Δt$\Delta t$ is the current time step; ut${u}_{t}$ is the time derivative of u$u$ which is approximated by first-order finite differences; wjt${w}_{j}^{\mathrm{t}}$ is the time equivalent of the space weighting factor wjs${w}_{j}^{\mathrm{s}}$; and αij${\alpha }_{ij}$ is given by
 αij = τ ((ujmax)/100 + |u(xi,yi,t)|) $αij=τ (ujmax100+|u(xi,yi,t)|)$ (7)
where ujmax${u}_{j}^{\mathrm{max}}$ is as before, and τ$\tau$ is the user-specified time tolerance.
An integration step is rejected and retried at all levels if
 max {μlt} > 1.0. l
$maxl{μlt}>1.0.$
(8)

## Example

```function nag_pde_2d_gen_order2_rectangle_example
global ssav1 ssav2;
ex1;
% Plot results.
fig1 = figure('Number', 'off');
ex1_plot;
ex2;
% Plot results.
fig2 = figure('Number', 'off');
ex2_plot(1);
fig3 = figure('Number', 'off');
ex2_plot(2);

function ex1
% First nag_pde_2d_gen_order2_rectangle example.  Temperature distribution
% for a single, one-step reaction of a mixture of two chemicals.
global heat_release activ_energy diffusion damkohler;
global iout isav;

% Set values for problem parameters.
mxlev = 3;
npde = 1;
npts = 2000;
xmin = 0.0;
xmax = 1.0;
ymin = 0.0;
ymax = 1.0;
tols = 0.5;
tolt = 0.01;
nx = 21;
ny = 21;
reaction_rate = 5.0;
activ_energy = 20.0;
heat_release = 1.0;
diffusion = 0.1;
damkohler = reaction_rate*exp(activ_energy)/(heat_release*activ_energy);

leniwk = npts*(14 + 5*mxlev) + 7*mxlev + 2;
lenlwk = npts + 1;
lenrwk = npts*npde*(5*mxlev + 18*npde + 9) + 2*npts;

% The above are the recommended values from the documentation, but
% nag_pde_2d_gen_order2_rectangle returns with ifail = 3 and the
% minumum workspace size requirements if these are used.  So we beef them
% up here with some fudge factors.
leniwk = 2.0*leniwk;
lenlwk = 2.0*lenlwk;
lenrwk = 2.0*lenrwk;

% Create some work arrays - see NAG documentation.
rwk = zeros(lenrwk, 1);
iwk = zeros(leniwk, 1);

% Initialize some problem variables.
ind = 0;
itrace = 0;
ts = 0.0;
dt = [0.1e-2; 0.0; 0.0];
twant = [0.24; 0.25];
opti = zeros(4, 1);
optr = ones(3, npde);

% What's the index of the next set of saved results?
isav = 1;

% We run through the problem twice - once from t=0 up to t=twant(1), and
% then onwards to t=twant(2).

fprintf('\n\nnag_pde_2d_gen_order2_rectangle example program #1 results\n');
for iout = 1:2
tout = twant(iout);
[ts, dt, rwk, iwk, ind, ifail] = ...
nag_pde_2d_gen_order2_rectangle(ts, tout, dt, xmin, xmax, ymin, ymax, ...
int64(nx), int64(ny), tols, tolt, @ex1_pdedef, @ex1_bndary, ...
@ex1_pdeiv, @ex1_monitr, int64(opti), optr, rwk, ...
int64(iwk), int64(itrace), int64(ind), 'lenlwk', int64(lenlwk));
if ifail ~= 0
% Parameters out of range, or convergence problems.
% Print message and exit.
error('Warning: nag_pde_2d_gen_order2_rectangle returned with ifail = %1d ',ifail);
end

% Output some statistics.
fprintf('\nStatistics\n');
fprintf('\nTime =%8.4f \n', ts);
fprintf('Total number of accepted timesteps %5d \n', iwk(1));
fprintf('Total number of rejected timesteps %5d \n', iwk(2));
fprintf('\n                    Total number of\n');
fprintf('          Residual  Jacobian    Newton   Lin sys\n');
fprintf('             evals     evals     iters     iters\n');
fprintf('At level\n');
maxlev = 3;
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%10d%10d%10d%10d\n', j, iwk(j+2), ...
iwk(j+2+maxlev), iwk(j+2+2*maxlev), iwk(j+2+3*maxlev));
end;
end;
fprintf('\n                 Maximum number of\n');
fprintf('          Newton iters Lin sys iters\n');
fprintf('At level\n');
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%14d%14d\n', j, iwk(j+2+4*maxlev), ...
iwk(j+2+5*maxlev));
end
end
end

function [res] = ex1_pdedef(npts, npde, t, x, y, u, ut, ux, uy, ...
uxx, uxy, uyy)
% Evaluate the system of PDEs for this problem.

global heat_release activ_energy diffusion damkohler;

res = zeros(npts, npde);
for i = 1:int32(npts) % can't use int64 in loop range.
res(i,1) = ut(i,1) - diffusion*(uxx(i,1) + uyy(i,1)) - ...
damkohler*(1.0 + heat_release - u(i,1))*exp(-activ_energy/u(i,1));
end

function [u] = ex1_pdeiv(npts, npde, t, x, y)
% Evaluate initial conditions for PDEs for this system.

u = ones(npts, npde);

function [res] = ex1_bndary(npts, npde, t, x, y, u, ut, ux, uy, ...
nbpts, lbnd, res)
% Implement boundary conditions for the domain.

tol = 10.0*nag_machine_precision;
for i = 1:int32(nbpts) % can't use int64 in loop range.
j = lbnd(i);
if (abs(x(j)) <= tol)
res(j,1) = ux(j,1);
elseif (abs(x(j)-1.0) <= tol)
res(j,1) = u(j,1) - 1.0;
elseif (abs(y(j)) <= tol)
res(j,1) = uy(j,1);
elseif (abs(y(j)-1.0) <= tol)
res(j,1) = u(j,1) - 1.0;
end
end

function [ierr] = ex1_monitr(npde, t, dt, dtnew, tlast, nlev, ...
ngpts, xpts, ypts, lsol, sol, ierr)
% Save the results at specified times.
times = [0.001; 0.228; 0.240; 0.25];

global iout isav;
global xsav ysav ssav1 tsav;

% If it's the last time this routine gets called on the second time round,
% then output the results to screen.
if (tlast && iout == 2)
fprintf(['\nSolution at every 20th grid point in level %d ', ...
'at time %8.4f : \n\n      x          y        ', ...
'approx u\n\n'], nlev, t);
npts = ngpts(nlev);
ipsol = lsol(nlev);
for i = 1:20:int32(npts) % can't use int64 in loop range.
j = i+int32(ipsol); % can't combine integers of different classes.
fprintf('%9.4e %9.4e %9.4e\n', xpts(i), ypts(i), sol(j));
end;
end

% Do we want to save this time step?  Only if the current time is equal to
% or just past the next time we've specified.
if (t < times(isav))
return;
end
fprintf('Saving set %d at time = %f\n', isav, t);

% Specify the grid level at which we're going to extract the solution.
level = 1;
npts = ngpts(level);
ipsol = lsol(level);

% Allocate space for saves the first time through.
nside = round(sqrt(double(npts)));
if isav == 1
nsav = length(times);
xsav = zeros(nside, 1);
ysav = zeros(nside, 1);
ssav1 = zeros(nside, nside, nsav);
tsav = zeros(nsav, 1);
end

% Save this time step.
tsav(isav) = t;
for i = 1:nside
for j = 1:nside
k = (i-1)*nside + j;
xsav(j) = xpts(k);
ysav(i) = ypts(k);
ssav1(j, i, isav) = sol(ipsol+k);
end
end

% Look for the next time step, unless we're being called for the last time
% on the second time round.
if ~(tlast && iout == 2)
isav = isav+1;
end

function ex1_plot
% Plot the results.
global xsav ysav ssav1 tsav isav;
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% Set the colours for each surface, and the array of handles for the plots.
colours = [[1 0 0]; [0 1 0]; [1 0 1]; [0 0 1]];
h = zeros(isav,1);
% Check the allocation of colours against the number we're going to plot.
if (length(colours(:,1)) < isav)
fprintf(['Not enough colours allocated in example1_plot: ', ...
'%d needed\n'], isav);
end
% Plot all the surfaces, and colour each one.
for i = 1:isav
h(i) = mesh(xsav, ysav, ssav1(:,:,i));
hold on;
set(h(i), 'EdgeColor', colours(i,:));
end
hold off;
% Label the axes, and set the title.
xlabel('x', labFmt{:});
ylabel('y', labFmt{:});
zlabel('U(x,y,t)', labFmt{:});
title({'Model for a Single, One-step Reaction', ...
'of a Mixture of Two Chemicals'}, titleFmt{:});
% Add a legend, using the plot handles and the array of time values.
legend(h, num2str(tsav(1:isav)));
% Set the view to something nice (determined empirically).
view(15, 30);

function ex2
% Second nag_pde_2d_gen_order2_rectangle example.  Concentration distribution
% in a multispecies (predator-prey) food web model.
global alpha beta;
global iout icount;

% Set values for problem parameters.
npde = 2;
npts = 2000;
xmin = 0.0;
xmax = 1.0;
ymin = 0.0;
ymax = 1.0;
tols = 0.075;
tolt = 0.1;
nx = 11;
ny = 11;
alpha = 50.0;
beta = 300.0;

maxlev = 4;
leniwk = npts*(14 + 5*maxlev) + 7*maxlev + 2;
lenlwk = npts + 1;
lenrwk = npts*npde*(5*maxlev + 18*npde + 9) + 2*npts;

% The above are the recommended values from the documentation, but
% nag_pde_2d_gen_order2_rectangle returns with ifail = 3 and the minumum
% workspace size requirements if these are used.  So we beef them up here
% with some fudge factors.
leniwk = 2.0*leniwk;
lenlwk = 2.5*lenlwk;
lenrwk = 3.0*lenrwk;

% Create some work arrays - see NAG documentation.
rwk = zeros(lenrwk, 1);
iwk = zeros(leniwk, 1);

% Initialize some problem variables.
ind = 0;
itrace = 0;
ts = 0.0;
dt = [0.5e-3; 1.0e-6; 0.0];
twant = [0.01; 0.025];
opti = zeros(4, 1);
opti(1) = 4;
optr = ones(3, npde);
optr(1,1) = 250.0;
optr(1,2) = 1.5e+6;

% Counts how many times monitor routine is called.
icount = 0;
fprintf('\n\nnag_pde_2d_gen_order2_rectangle example program #2 results\n');

% We run through the problem twice - once from t=0 up to t=twant(1), and
% then onwards to t=twant(2).
for iout = 1:2
tout = twant(iout);
[ts, dt, rwk, iwk, ind, ifail] = ...
nag_pde_2d_gen_order2_rectangle(ts, tout, dt, xmin, xmax, ymin, ymax, ...
int64(nx), int64(ny), tols, tolt, @ex2_pdedef, @ex2_bndary, ...
@ex2_pdeiv, @ex2_monitr, int64(opti), optr, rwk, ...
int64(iwk), int64(itrace), int64(ind), 'lenlwk', int64(lenlwk));
if ifail ~= 0
% Parameters out of range, or convergence problems.
% Print message and exit.
error('Warning: nag_pde_2d_gen_order2_rectangle returned with ifail = %1d ',ifail);
end

% Output some statistics.
fprintf('\nStatistics\n');
fprintf('\nTime =%8.4f \n', ts);
fprintf('Total number of accepted timesteps %5d \n', iwk(1));
fprintf('Total number of rejected timesteps %5d \n', iwk(2));
fprintf('\n                    Total number of\n');
fprintf('          Residual  Jacobian    Newton   Lin sys\n');
fprintf('             evals     evals     iters     iters\n');
fprintf('At level\n');
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%10d%10d%10d%10d\n', j, iwk(j+2), ...
iwk(j+2+maxlev), iwk(j+2+2*maxlev), iwk(j+2+3*maxlev));
end;
end;
fprintf('\n                 Maximum number of\n');
fprintf('          Newton iters Lin sys iters\n');
fprintf('At level\n');
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%14d%14d\n', j, iwk(j+2+4*maxlev), ...
iwk(j+2+5*maxlev));
end
end
end

function [res] = ex2_pdedef(npts, npde, t, x, y, u, ut, ux, uy, ...
uxx, uxy, uyy)
% Evaluate the system of PDEs for this problem.

global alpha beta;

fp = 4.0*pi;
res = zeros(npts, npde);
for i = 1:int32(npts) % can't use int64 in loop range.
b1 = 1.0 + alpha*x(i)*y(i) + beta*sin(fp*x(i))*sin(fp*y(i));
res(i,1) = ut(i,1) - uxx(i,1) - uyy(i,1) - ...
u(i,1)*(b1 - u(i,1) - 0.5e-6*u(i,2));
res(i,2) = -0.05*(uxx(i,2) + uyy(i,2)) - ...
u(i,2)*(-b1 + 1.0e4*u(i,1) - u(i,2));
end

function [u] = ex2_pdeiv(npts, npde, t, x, y)
% Evaluate initial conditions for PDEs for this system.

global alpha beta;

fp = 4.0*pi;
u = zeros(npts, npde);
for i = 1:int32(npts) % can't use int64 in loop range.
u(i,1) = 10.0 + (16.0*x(i)*(1.0 - x(i))*y(i)*(1.0 - y(i)))^2;
u(i,2) = -1.0 - alpha*x(i)*y(i) - beta*sin(fp*x(i))*sin(fp*y(i)) + ...
1.0e4*u(i,1);
end

function [res] = ex2_bndary(npts, npde, t, x, y, u, ut, ux, uy, ...
nbpts, lbnd, res)
% Implement boundary conditions for the domain.

tol = 10.0*nag_machine_precision;
for i = 1:int32(nbpts) % can't use int64 in loop range.
j = lbnd(i);
if (abs(x(j)) <= tol || abs(x(j)-1.0) <= tol)
res(j,1) = ux(j,1);
res(j,2) = ux(j,2);
elseif (abs(y(j)) <= tol || abs(y(j)-1.0) <= tol)
res(j,1) = uy(j,1);
res(j,2) = uy(j,2);
end
end

function [ierr] = ex2_monitr(npde, t, dt, dtnew, tlast, nlev, ...
ngpts, xpts, ypts, lsol, sol, ierr)
% Monitor the results at specified intervals.

global iout icount;
global xsav ysav ssav2 tsav isav;

% If it's the last time this routine gets called on the second time round,
% then output the results to screen.
if (tlast && iout == 2)

% Specify the grid level at which we're going to print the solution.
level = nlev;
npts = ngpts(level);
ipsol = lsol(level);
fprintf(['\nSolution at every 2nd grid point in level %d ', ...
'at time %8.4f : \n\n       x           y     ', ...
'approx c1   approx c2\n\n'], level, t);

disp(npts);
% Print the solution.
nside = round(sqrt(double(npts)));
for i = 1:2:double(npts)
fprintf('%9.4e %9.4e %9.4e %9.4e\n', xpts(i), ...
ypts(i), sol(ipsol+i), sol(ipsol+npts+i));
end
end
% Specify the maximum number of times the results get saved, and the
% interval for saves.
nsav = 5;
nint = 10;

% Do we want to save this time?
icount = icount+1;
if mod(icount,nint) ~= 1 && ~tlast
return;
end

isav = 1 + round(icount/nint);
if isav > nsav
fprintf('Not enough save space allocated in monitr2: %d\n', ...
nsav);
return;
end
fprintf('Saving set %d at time = %f\n', isav, t);

% Specify the grid level at which we're going to extract the solution.
level = 1;
npts = ngpts(level);
ipsol = lsol(level);

% Allocate space for saves the first time through.
nside = round(sqrt(double(npts)));
if icount == 1
xsav = zeros(nside, 1);
ysav = zeros(nside, 1);
ssav2 = zeros(nside, nside, 2, nsav);
tsav = zeros(nsav, 1);
end

% Save this time step.  For this problem, we're calculating two
% concentrations as a function of x and y (and time).
tsav(isav) = t;
for i = 1:nside
for j = 1:nside
k = (i-1)*nside + j;
xsav(j) = xpts(k);
ysav(i) = ypts(k);
ssav2(j, i, 1, isav) = sol(ipsol+k);
ssav2(j, i, 2, isav) = sol(ipsol+npts+k);
end
end

function ex2_plot(id)
% Plot the results.
global xsav ysav ssav2 tsav isav;
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% Check the value of the array identifier.
if (id ~= 1 && id ~= 2)
error('Illegal value for array identifier in plot: %d', id);
end
% Set the colours for each surface, and the array of handles for the plots.
colours = [[1 0 0]; [0 1 0]; [1 0 1]; [0 0 1]];
h = zeros(isav,1);
% Check the allocation of colours against the number we're going to plot.
if (length(colours(:,1)) < isav)
fprintf(['Not enough colours allocated in example2_plot: ', ...
'%d needed\n'], isav);
end
% Plot all the surfaces, and colour each one.
for i = 1:isav
h(i) = mesh(xsav, ysav, ssav2(:,:,id,i));
hold on;
set(h(i), 'EdgeColor', colours(i,:));
end
hold off;
% Label the axes, and set the title according to the array identifier.
xlabel('x', labFmt{:});
ylabel('y', labFmt{:});
if (id == 1)
title({'Multispecies Food Web Model:', ...
'Time-dependent Predator Concentration'}, titleFmt{:});
zlabel('Predator concentration', labFmt{:});
else
title({'Multispecies Food Web Model:', ...
'Time-dependent Prey Concentration'}, titleFmt{:});
zlabel('Prey concentration', labFmt{:});
end
% Add a legend, using the plot handles and the array of time values.
legend(h, num2str(tsav(1:isav)));
% Set the view to something nice (determined empirically).
view(60, 40);
```
```

nag_pde_2d_gen_order2_rectangle example program #1 results
Saving set 1 at time = 0.001000
Saving set 2 at time = 0.228996
Saving set 3 at time = 0.240000

Statistics

Time =  0.2400
Total number of accepted timesteps    75
Total number of rejected timesteps     0

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1       600        75       150       159

Maximum number of
Newton iters Lin sys iters
At level
1             2             2

Solution at every 20th grid point in level 3 at time   0.2500 :

x          y        approx u

0.0000e+00 0.0000e+00 2.0000e+00
1.0000e+00 0.0000e+00 1.4759e+00
9.5000e-01 5.0000e-02 1.9999e+00
9.0000e-01 1.0000e-01 1.3413e+00
8.5000e-01 1.5000e-01 1.9995e+00
8.0000e-01 2.0000e-01 1.2833e+00
7.5000e-01 2.5000e-01 1.9966e+00
7.0000e-01 3.0000e-01 1.2399e+00
6.5000e-01 3.5000e-01 1.9826e+00
6.0000e-01 4.0000e-01 1.2132e+00
5.5000e-01 4.5000e-01 1.8646e+00
5.0000e-01 5.0000e-01 1.1895e+00
4.5000e-01 5.5000e-01 1.6382e+00
4.0000e-01 6.0000e-01 2.0000e+00
3.5000e-01 6.5000e-01 1.4678e+00
3.0000e-01 7.0000e-01 1.9999e+00
2.5000e-01 7.5000e-01 1.3355e+00
2.0000e-01 8.0000e-01 1.9993e+00
1.5000e-01 8.5000e-01 1.2780e+00
1.0000e-01 9.0000e-01 1.9950e+00
5.0000e-02 9.5000e-01 1.2356e+00
0.0000e+00 1.0000e+00 1.9731e+00
1.0000e+00 1.0000e+00 1.2087e+00
7.5000e-01 2.5000e-02 1.7809e+00
6.0000e-01 7.5000e-02 1.1850e+00
7.7500e-01 1.0000e-01 1.9237e+00
6.2500e-01 1.5000e-01 1.2548e+00
8.0000e-01 1.7500e-01 1.9999e+00
6.5000e-01 2.2500e-01 1.3982e+00
8.2500e-01 2.5000e-01 2.0000e+00
6.7500e-01 3.0000e-01 1.8013e+00
8.5000e-01 3.2500e-01 1.2524e+00
7.0000e-01 3.7500e-01 1.9999e+00
8.7500e-01 4.0000e-01 1.9850e+00
7.2500e-01 4.5000e-01 1.9118e+00
9.0000e-01 4.7500e-01 1.6189e+00
6.5000e-01 5.2500e-01 1.9929e+00
7.2500e-01 5.5000e-01 1.9994e+00
8.0000e-01 5.7500e-01 2.0000e+00
3.7500e-01 6.0000e-01 2.0000e+00
8.7500e-01 6.0000e-01 1.9996e+00
4.5000e-01 6.2500e-01 1.2027e+00
7.5000e-02 6.5000e-01 1.9999e+00
5.7500e-01 6.5000e-01 1.9989e+00
2.0000e-01 6.7500e-01 1.2408e+00
7.0000e-01 6.7500e-01 1.9994e+00
3.2500e-01 7.0000e-01 1.9961e+00
8.2500e-01 7.0000e-01 1.2441e+00
4.5000e-01 7.2500e-01 1.9964e+00
1.2500e-01 7.5000e-01 1.9807e+00
6.2500e-01 7.5000e-01 1.2462e+00
3.0000e-01 7.7500e-01 1.9822e+00
2.5000e-02 8.0000e-01 1.9237e+00
5.2500e-01 8.0000e-01 1.2195e+00
2.5000e-01 8.2500e-01 1.8620e+00
2.5000e-02 8.5000e-01 1.6830e+00
5.2500e-01 8.5000e-01 1.6454e+00
3.0000e-01 8.7500e-01 1.6299e+00
1.7500e-01 9.0000e-01 1.3982e+00
7.0000e-01 0.0000e+00 1.4758e+00
7.8750e-01 1.2500e-02 1.4591e+00
7.1250e-01 3.7500e-02 1.2960e+00
8.0000e-01 5.0000e-02 1.3410e+00
7.2500e-01 7.5000e-02 1.3257e+00
8.1250e-01 8.7500e-02 1.2281e+00
7.3750e-01 1.1250e-01 1.2828e+00
8.2500e-01 1.2500e-01 1.2657e+00
7.5000e-01 1.5000e-01 1.2400e+00
6.7500e-01 1.7500e-01 1.2378e+00
7.6250e-01 1.8750e-01 1.2068e+00
6.8750e-01 2.1250e-01 1.2130e+00
7.7500e-01 2.2500e-01 1.2053e+00
7.0000e-01 2.5000e-01 1.1895e+00
7.8750e-01 2.6250e-01 1.1850e+00
Saving set 4 at time = 0.250000

Statistics

Time =  0.2500
Total number of accepted timesteps   180
Total number of rejected timesteps     1

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1      1468       181       382       391
2       662        82       170       170
3       177        22        45        45

Maximum number of
Newton iters Lin sys iters
At level
1             4             2
2             4             1
3             3             1

nag_pde_2d_gen_order2_rectangle example program #2 results
Saving set 1 at time = 0.000500
Saving set 2 at time = 0.007377
Saving set 2 at time = 0.010000

Statistics

Time =  0.0100
Total number of accepted timesteps    14
Total number of rejected timesteps     0

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1       196        14        28        42
2       168        12        24        34
3        70         5        10        16

Maximum number of
Newton iters Lin sys iters
At level
1             2             2
2             2             2
3             2             3
Saving set 3 at time = 0.016563

Solution at every 2nd grid point in level 4 at time   0.0250 :

x           y     approx c1   approx c2

4343

0.0000e+00 0.0000e+00 6.6146e+01 6.6146e+05
2.0000e-01 0.0000e+00 6.6963e+01 6.6962e+05
4.0000e-01 0.0000e+00 6.8880e+01 6.8879e+05
6.0000e-01 0.0000e+00 7.0689e+01 7.0687e+05
8.0000e-01 0.0000e+00 7.1226e+01 7.1223e+05
1.0000e+00 0.0000e+00 6.9633e+01 6.9630e+05
1.0000e-01 1.0000e-01 6.5564e+01 6.5561e+05
3.0000e-01 1.0000e-01 5.9242e+01 5.9240e+05
5.0000e-01 1.0000e-01 5.1376e+01 5.1374e+05
7.0000e-01 1.0000e-01 4.2883e+01 4.2882e+05
9.0000e-01 1.0000e-01 3.9626e+01 3.9625e+05
0.0000e+00 2.0000e-01 4.5742e+01 4.5740e+05
2.0000e-01 2.0000e-01 5.0545e+01 5.0542e+05
4.0000e-01 2.0000e-01 5.3180e+01 5.3177e+05
6.0000e-01 2.0000e-01 5.3162e+01 5.3159e+05
8.0000e-01 2.0000e-01 5.0482e+01 5.0479e+05
1.0000e+00 2.0000e-01 4.5607e+01 4.5605e+05
1.0000e-01 3.0000e-01 3.9376e+01 3.9375e+05
3.0000e-01 3.0000e-01 6.6350e+01 6.6350e+05
5.0000e-01 3.0000e-01 6.7224e+01 6.7223e+05
7.0000e-01 3.0000e-01 6.9263e+01 6.9261e+05
9.0000e-01 3.0000e-01 7.1200e+01 7.1196e+05
0.0000e+00 4.0000e-01 7.1824e+01 7.1820e+05
2.0000e-01 4.0000e-01 7.0253e+01 7.0248e+05
4.0000e-01 4.0000e-01 6.6131e+01 6.6127e+05
6.0000e-01 4.0000e-01 5.9694e+01 5.9691e+05
8.0000e-01 4.0000e-01 5.1675e+01 5.1673e+05
1.0000e+00 4.0000e-01 4.3136e+01 4.3134e+05
1.0000e-01 5.0000e-01 3.9812e+01 3.9810e+05
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1.0000e+00 5.0000e-01 3.8059e+01 3.8066e+05
8.7500e-02 5.1250e-01 5.0818e+01 5.0818e+05
1.1250e-01 5.1250e-01 6.6407e+01 6.6401e+05
1.3750e-01 5.1250e-01 8.3734e+01 8.3721e+05
1.6250e-01 5.1250e-01 1.0097e+02 1.0096e+06
5.5000e-01 5.1250e-01 1.1588e+02 1.1586e+06
5.7500e-01 5.1250e-01 1.2637e+02 1.2634e+06
6.0000e-01 5.1250e-01 1.3117e+02 1.3115e+06
6.2500e-01 5.1250e-01 1.3035e+02 1.3032e+06
6.5000e-01 5.1250e-01 1.2533e+02 1.2531e+06
6.7500e-01 5.1250e-01 1.1869e+02 1.1867e+06
7.0000e-01 5.1250e-01 1.1372e+02 1.1371e+06
7.8750e-01 5.1250e-01 3.5415e+01 3.5414e+05
8.1250e-01 5.1250e-01 4.5481e+01 4.5474e+05
8.3750e-01 5.1250e-01 5.5781e+01 5.5769e+05
8.6250e-01 5.1250e-01 6.4933e+01 6.4916e+05
8.8750e-01 5.1250e-01 7.1438e+01 7.1419e+05
9.1250e-01 5.1250e-01 7.4114e+01 7.4094e+05
9.3750e-01 5.1250e-01 7.2477e+01 7.2458e+05
0.0000e+00 5.2500e-01 6.6910e+01 6.6893e+05
2.5000e-02 5.2500e-01 5.8533e+01 5.8521e+05
5.0000e-02 5.2500e-01 4.8839e+01 4.8831e+05
7.5000e-02 5.2500e-01 3.9250e+01 3.9248e+05
1.0000e-01 5.2500e-01 3.0865e+01 3.0868e+05
1.2500e-01 5.2500e-01 2.4377e+01 2.4385e+05
1.5000e-01 5.2500e-01 2.0090e+01 2.0102e+05
1.7500e-01 5.2500e-01 1.8003e+01 1.8018e+05
2.0000e-01 5.2500e-01 1.8143e+01 1.8159e+05
2.2500e-01 5.2500e-01 2.0613e+01 2.0627e+05
5.3750e-01 5.2500e-01 2.5634e+01 2.5646e+05
5.6250e-01 5.2500e-01 3.3481e+01 3.3489e+05
5.8750e-01 5.2500e-01 4.4321e+01 4.4324e+05
6.1250e-01 5.2500e-01 5.7982e+01 5.7978e+05
6.3750e-01 5.2500e-01 7.3731e+01 7.3722e+05
6.6250e-01 5.2500e-01 9.0179e+01 9.0165e+05
6.8750e-01 5.2500e-01 1.0543e+02 1.0541e+06
7.1250e-01 5.2500e-01 1.1748e+02 1.1746e+06
7.5000e-01 5.2500e-01 1.2483e+02 1.2480e+06
7.7500e-01 5.2500e-01 1.2691e+02 1.2689e+06
8.0000e-01 5.2500e-01 1.2443e+02 1.2441e+06
8.2500e-01 5.2500e-01 1.1921e+02 1.1920e+06
8.5000e-01 5.2500e-01 1.1389e+02 1.1388e+06
8.7500e-01 5.2500e-01 1.1156e+02 1.1155e+06
9.0000e-01 5.2500e-01 3.9781e+01 3.9777e+05
9.2500e-01 5.2500e-01 4.9454e+01 4.9446e+05
9.5000e-01 5.2500e-01 5.8643e+01 5.8631e+05
1.2500e-02 5.3750e-01 6.6010e+01 6.5996e+05
3.7500e-02 5.3750e-01 7.0329e+01 7.0314e+05
6.2500e-02 5.3750e-01 7.0842e+01 7.0827e+05
8.7500e-02 5.3750e-01 6.7503e+01 6.7488e+05
1.1250e-01 5.3750e-01 6.0993e+01 6.0981e+05
1.3750e-01 5.3750e-01 5.2506e+01 5.2498e+05
1.6250e-01 5.3750e-01 4.3399e+01 4.3395e+05
1.8750e-01 5.3750e-01 2.4748e+01 2.4754e+05
2.1250e-01 5.3750e-01 2.0642e+01 2.0651e+05
5.2500e-01 5.3750e-01 1.8654e+01 1.8665e+05
5.5000e-01 5.3750e-01 1.8855e+01 1.8866e+05
5.7500e-01 5.3750e-01 2.1361e+01 2.1371e+05
6.0000e-01 5.3750e-01 2.6365e+01 2.6373e+05
6.2500e-01 5.3750e-01 3.4078e+01 3.4083e+05
6.5000e-01 5.3750e-01 4.4583e+01 4.4584e+05
6.7500e-01 5.3750e-01 5.7637e+01 5.7634e+05
7.0000e-01 5.3750e-01 7.2498e+01 7.2490e+05
7.2500e-01 5.3750e-01 8.7861e+01 8.7849e+05
7.6250e-01 5.3750e-01 1.0202e+02 1.0201e+06
7.8750e-01 5.3750e-01 1.1323e+02 1.1321e+06
8.1250e-01 5.3750e-01 1.2019e+02 1.2017e+06
8.3750e-01 5.3750e-01 1.2245e+02 1.2243e+06
8.6250e-01 5.3750e-01 1.2064e+02 1.2063e+06
8.8750e-01 5.3750e-01 1.1638e+02 1.1636e+06
9.1250e-01 5.3750e-01 1.1196e+02 1.1195e+06
9.3750e-01 5.3750e-01 1.1002e+02 1.1002e+06
0.0000e+00 5.5000e-01 3.9294e+01 3.9291e+05
2.5000e-02 5.5000e-01 4.8402e+01 4.8396e+05
5.0000e-02 5.5000e-01 5.6964e+01 5.6956e+05
7.5000e-02 5.5000e-01 6.3783e+01 6.3773e+05
1.0000e-01 5.5000e-01 6.7775e+01 6.7764e+05
1.2500e-01 5.5000e-01 6.8276e+01 6.8265e+05
1.5000e-01 5.5000e-01 6.5247e+01 6.5236e+05
1.7500e-01 5.5000e-01 5.9280e+01 5.9271e+05
2.0000e-01 5.5000e-01 5.1432e+01 5.1426e+05
2.2500e-01 5.5000e-01 4.2922e+01 4.2918e+05
3.8750e-01 5.5000e-01 2.5120e+01 2.5123e+05
4.1250e-01 5.5000e-01 2.1138e+01 2.1143e+05
4.3750e-01 5.5000e-01 1.9240e+01 1.9247e+05
4.6250e-01 5.5000e-01 1.9496e+01 1.9503e+05
5.0000e-01 5.5000e-01 2.2033e+01 2.2039e+05
5.2500e-01 5.5000e-01 2.7020e+01 2.7024e+05
5.5000e-01 5.5000e-01 3.4611e+01 3.4613e+05
5.7500e-01 5.5000e-01 4.4817e+01 4.4817e+05
6.0000e-01 5.5000e-01 5.7338e+01 5.7335e+05
6.2500e-01 5.5000e-01 7.1422e+01 7.1415e+05
6.5000e-01 5.5000e-01 8.5844e+01 8.5835e+05
6.7500e-01 5.5000e-01 9.9062e+01 9.9051e+05
7.0000e-01 5.5000e-01 1.0954e+02 1.0953e+06
7.2500e-01 5.5000e-01 1.1616e+02 1.1615e+06
7.5000e-01 5.5000e-01 1.1856e+02 1.1855e+06
7.7500e-01 5.5000e-01 1.1733e+02 1.1732e+06
8.0000e-01 5.5000e-01 1.1389e+02 1.1388e+06
8.2500e-01 5.5000e-01 1.1026e+02 1.1025e+06
8.5000e-01 5.5000e-01 1.0868e+02 1.0868e+06
8.7500e-01 5.5000e-01 3.8985e+01 3.8983e+05
9.0000e-01 5.5000e-01 4.7678e+01 4.7675e+05
9.2500e-01 5.5000e-01 5.5804e+01 5.5799e+05
9.5000e-01 5.5000e-01 6.2242e+01 6.2236e+05
9.7500e-01 5.5000e-01 6.6007e+01 6.6000e+05
1.0000e+00 5.5000e-01 6.6500e+01 6.6493e+05
1.2500e-02 5.6250e-01 6.3684e+01 6.3678e+05
3.7500e-02 5.6250e-01 5.8095e+01 5.8089e+05
6.2500e-02 5.6250e-01 5.0693e+01 5.0689e+05
8.7500e-02 5.6250e-01 4.2616e+01 4.2613e+05
1.1250e-01 5.6250e-01 2.5244e+01 2.5246e+05
1.3750e-01 5.6250e-01 2.1491e+01 2.1492e+05
1.6250e-01 5.6250e-01 1.9664e+01 1.9665e+05
1.8750e-01 5.6250e-01 1.9959e+01 1.9961e+05
2.1250e-01 5.6250e-01 2.2517e+01 2.2519e+05
3.7500e-01 5.6250e-01 2.7491e+01 2.7492e+05
4.0000e-01 5.6250e-01 3.4996e+01 3.4996e+05
4.2500e-01 5.6250e-01 4.4992e+01 4.4990e+05
4.5000e-01 5.6250e-01 5.7137e+01 5.7133e+05
4.7500e-01 5.6250e-01 7.0675e+01 7.0670e+05
5.1250e-01 5.6250e-01 8.4438e+01 8.4432e+05
5.3750e-01 5.6250e-01 9.6997e+01 9.6990e+05
5.6250e-01 5.6250e-01 1.0696e+02 1.0695e+06
5.8750e-01 5.6250e-01 1.1334e+02 1.1334e+06
6.1250e-01 5.6250e-01 1.1584e+02 1.1584e+06
6.3750e-01 5.6250e-01 1.1501e+02 1.1500e+06
6.6250e-01 5.6250e-01 1.1214e+02 1.1213e+06
6.8750e-01 5.6250e-01 1.0908e+02 1.0907e+06
7.1250e-01 5.6250e-01 1.0776e+02 1.0776e+06
7.3750e-01 5.6250e-01 3.8882e+01 3.8880e+05
7.6250e-01 5.6250e-01 4.7437e+01 4.7434e+05
7.8750e-01 5.6250e-01 5.5417e+01 5.5413e+05
8.1250e-01 5.6250e-01 6.1728e+01 6.1724e+05
8.3750e-01 5.6250e-01 6.5417e+01 6.5412e+05
8.6250e-01 5.6250e-01 6.5907e+01 6.5902e+05
8.8750e-01 5.6250e-01 6.3163e+01 6.3158e+05
9.1250e-01 5.6250e-01 5.7699e+01 5.7695e+05
9.3750e-01 5.6250e-01 5.0446e+01 5.0443e+05
9.6250e-01 5.6250e-01 4.2514e+01 4.2511e+05
9.8750e-01 5.6250e-01 2.5368e+01 2.5369e+05
0.0000e+00 5.7500e-01 2.1608e+01 2.1608e+05
2.5000e-02 5.7500e-01 1.9804e+01 1.9805e+05
5.0000e-02 5.7500e-01 2.0112e+01 2.0113e+05
7.5000e-02 5.7500e-01 2.2678e+01 2.2678e+05
1.0000e-01 5.7500e-01 2.7648e+01 2.7648e+05
1.2500e-01 5.7500e-01 3.5124e+01 3.5123e+05
1.5000e-01 5.7500e-01 4.5049e+01 4.5047e+05
1.7500e-01 5.7500e-01 5.7069e+01 5.7065e+05
2.0000e-01 5.7500e-01 7.0426e+01 7.0421e+05
2.2500e-01 5.7500e-01 8.3969e+01 8.3963e+05
2.5000e-01 5.7500e-01 9.6308e+01 9.6302e+05
3.8750e-01 5.7500e-01 1.0610e+02 1.0610e+06
4.1250e-01 5.7500e-01 1.1241e+02 1.1240e+06
4.3750e-01 5.7500e-01 1.1494e+02 1.1493e+06
4.6250e-01 5.7500e-01 1.1423e+02 1.1422e+06
5.0000e-01 5.7500e-01 1.1156e+02 1.1155e+06
5.2500e-01 5.7500e-01 1.0868e+02 1.0868e+06
5.5000e-01 5.7500e-01 1.0746e+02 1.0745e+06
Saving set 4 at time = 0.025000

Statistics

Time =  0.0250
Total number of accepted timesteps    29
Total number of rejected timesteps     0

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1       406        29        58        87
2       378        27        54        79
3       280        20        40        61
4        98         7        14        27

Maximum number of
Newton iters Lin sys iters
At level
1             2             2
2             2             2
3             2             3
4             2             3

```
```function d03ra_example
global ssav1 ssav2;
ex1;
% Plot results.
fig1 = figure('Number', 'off');
ex1_plot;
ex2;
% Plot results.
fig2 = figure('Number', 'off');
ex2_plot(1);
fig3 = figure('Number', 'off');
ex2_plot(2);

function ex1
% First d03ra example.  Temperature distribution for a single, one-step
% reaction of a mixture of two chemicals.
global heat_release activ_energy diffusion damkohler;
global iout isav;

% Set values for problem parameters.
mxlev = 3;
npde = 1;
npts = 2000;
xmin = 0.0;
xmax = 1.0;
ymin = 0.0;
ymax = 1.0;
tols = 0.5;
tolt = 0.01;
nx = 21;
ny = 21;
reaction_rate = 5.0;
activ_energy = 20.0;
heat_release = 1.0;
diffusion = 0.1;
damkohler = reaction_rate*exp(activ_energy)/(heat_release*activ_energy);

leniwk = npts*(14 + 5*mxlev) + 7*mxlev + 2;
lenlwk = npts + 1;
lenrwk = npts*npde*(5*mxlev + 18*npde + 9) + 2*npts;

% The above are the recommended values from the documentation, but d03ra
% returns with ifail = 3 and the minumum workspace size requirements if
% these are used.  So we beef them up here with some fudge factors.
leniwk = 2.0*leniwk;
lenlwk = 2.0*lenlwk;
lenrwk = 2.0*lenrwk;

% Create some work arrays - see NAG documentation.
rwk = zeros(lenrwk, 1);
iwk = zeros(leniwk, 1);

% Initialize some problem variables.
ind = 0;
itrace = 0;
ts = 0.0;
dt = [0.1e-2; 0.0; 0.0];
twant = [0.24; 0.25];
opti = zeros(4, 1);
optr = ones(3, npde);

% What's the index of the next set of saved results?
isav = 1;

% We run through the problem twice - once from t=0 up to t=twant(1), and
% then onwards to t=twant(2).

fprintf('\n\nd03ra example program #1 results\n');
for iout = 1:2
tout = twant(iout);
[ts, dt, rwk, iwk, ind, ifail] = ...
d03ra(ts, tout, dt, xmin, xmax, ymin, ymax, int64(nx), ...
int64(ny), tols, tolt, @ex1_pdedef, @ex1_bndary, ...
@ex1_pdeiv, @ex1_monitr, int64(opti), optr, rwk, ...
int64(iwk), int64(itrace), int64(ind), 'lenlwk', int64(lenlwk));
if ifail ~= 0
% Parameters out of range, or convergence problems.
% Print message and exit.
error('Warning: d03ra returned with ifail = %1d ',ifail);
end

% Output some statistics.
fprintf('\nStatistics\n');
fprintf('\nTime =%8.4f \n', ts);
fprintf('Total number of accepted timesteps %5d \n', iwk(1));
fprintf('Total number of rejected timesteps %5d \n', iwk(2));
fprintf('\n                    Total number of\n');
fprintf('          Residual  Jacobian    Newton   Lin sys\n');
fprintf('             evals     evals     iters     iters\n');
fprintf('At level\n');
maxlev = 3;
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%10d%10d%10d%10d\n', j, iwk(j+2), ...
iwk(j+2+maxlev), iwk(j+2+2*maxlev), iwk(j+2+3*maxlev));
end;
end;
fprintf('\n                 Maximum number of\n');
fprintf('          Newton iters Lin sys iters\n');
fprintf('At level\n');
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%14d%14d\n', j, iwk(j+2+4*maxlev), ...
iwk(j+2+5*maxlev));
end
end
end

function [res] = ex1_pdedef(npts, npde, t, x, y, u, ut, ux, uy, ...
uxx, uxy, uyy)
% Evaluate the system of PDEs for this problem.

global heat_release activ_energy diffusion damkohler;

res = zeros(npts, npde);
for i = 1:int32(npts) % can't use int64 in loop range.
res(i,1) = ut(i,1) - diffusion*(uxx(i,1) + uyy(i,1)) - ...
damkohler*(1.0 + heat_release - u(i,1))*exp(-activ_energy/u(i,1));
end

function [u] = ex1_pdeiv(npts, npde, t, x, y)
% Evaluate initial conditions for PDEs for this system.

u = ones(npts, npde);

function [res] = ex1_bndary(npts, npde, t, x, y, u, ut, ux, uy, ...
nbpts, lbnd, res)
% Implement boundary conditions for the domain.

tol = 10.0*x02aj();
for i = 1:int32(nbpts) % can't use int64 in loop range.
j = lbnd(i);
if (abs(x(j)) <= tol)
res(j,1) = ux(j,1);
elseif (abs(x(j)-1.0) <= tol)
res(j,1) = u(j,1) - 1.0;
elseif (abs(y(j)) <= tol)
res(j,1) = uy(j,1);
elseif (abs(y(j)-1.0) <= tol)
res(j,1) = u(j,1) - 1.0;
end
end

function [ierr] = ex1_monitr(npde, t, dt, dtnew, tlast, nlev, ...
ngpts, xpts, ypts, lsol, sol, ierr)
% Save the results at specified times.
times = [0.001; 0.228; 0.240; 0.25];

global iout isav;
global xsav ysav ssav1 tsav;

% If it's the last time this routine gets called on the second time round,
% then output the results to screen.
if (tlast && iout == 2)
fprintf(['\nSolution at every 20th grid point in level %d ', ...
'at time %8.4f : \n\n      x          y        ', ...
'approx u\n\n'], nlev, t);
npts = ngpts(nlev);
ipsol = lsol(nlev);
for i = 1:20:int32(npts) % can't use int64 in loop range.
j = i+int32(ipsol); % can't combine integers of different classes.
fprintf('%9.4e %9.4e %9.4e\n', xpts(i), ypts(i), sol(j));
end;
end

% Do we want to save this time step?  Only if the current time is equal to
% or just past the next time we've specified.
if (t < times(isav))
return;
end
fprintf('Saving set %d at time = %f\n', isav, t);

% Specify the grid level at which we're going to extract the solution.
level = 1;
npts = ngpts(level);
ipsol = lsol(level);

% Allocate space for saves the first time through.
nside = round(sqrt(double(npts)));
if isav == 1
nsav = length(times);
xsav = zeros(nside, 1);
ysav = zeros(nside, 1);
ssav1 = zeros(nside, nside, nsav);
tsav = zeros(nsav, 1);
end

% Save this time step.
tsav(isav) = t;
for i = 1:nside
for j = 1:nside
k = (i-1)*nside + j;
xsav(j) = xpts(k);
ysav(i) = ypts(k);
ssav1(j, i, isav) = sol(ipsol+k);
end
end

% Look for the next time step, unless we're being called for the last time
% on the second time round.
if ~(tlast && iout == 2)
isav = isav+1;
end

function ex1_plot
% Plot the results.
global xsav ysav ssav1 tsav isav;
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% Set the colours for each surface, and the array of handles for the plots.
colours = [[1 0 0]; [0 1 0]; [1 0 1]; [0 0 1]];
h = zeros(isav,1);
% Check the allocation of colours against the number we're going to plot.
if (length(colours(:,1)) < isav)
fprintf(['Not enough colours allocated in example1_plot: ', ...
'%d needed\n'], isav);
end
% Plot all the surfaces, and colour each one.
for i = 1:isav
h(i) = mesh(xsav, ysav, ssav1(:,:,i));
hold on;
set(h(i), 'EdgeColor', colours(i,:));
end
hold off;
% Label the axes, and set the title.
xlabel('x', labFmt{:});
ylabel('y', labFmt{:});
zlabel('U(x,y,t)', labFmt{:});
title({'Model for a Single, One-step Reaction', ...
'of a Mixture of Two Chemicals'}, titleFmt{:});
% Add a legend, using the plot handles and the array of time values.
legend(h, num2str(tsav(1:isav)));
% Set the view to something nice (determined empirically).
view(15, 30);

function ex2
% Second d03ra example.  Concentration distribution in a multispecies
% (predator-prey) food web model.
global alpha beta;
global iout icount;

% Set values for problem parameters.
npde = 2;
npts = 2000;
xmin = 0.0;
xmax = 1.0;
ymin = 0.0;
ymax = 1.0;
tols = 0.075;
tolt = 0.1;
nx = 11;
ny = 11;
alpha = 50.0;
beta = 300.0;

maxlev = 4;
leniwk = npts*(14 + 5*maxlev) + 7*maxlev + 2;
lenlwk = npts + 1;
lenrwk = npts*npde*(5*maxlev + 18*npde + 9) + 2*npts;

% The above are the recommended values from the documentation, but d03ra
% returns with ifail = 3 and the minumum workspace size requirements if
% these are used.  So we beef them up here with some fudge factors.
leniwk = 2.0*leniwk;
lenlwk = 2.5*lenlwk;
lenrwk = 3.0*lenrwk;

% Create some work arrays - see NAG documentation.
rwk = zeros(lenrwk, 1);
iwk = zeros(leniwk, 1);

% Initialize some problem variables.
ind = 0;
itrace = 0;
ts = 0.0;
dt = [0.5e-3; 1.0e-6; 0.0];
twant = [0.01; 0.025];
opti = zeros(4, 1);
opti(1) = 4;
optr = ones(3, npde);
optr(1,1) = 250.0;
optr(1,2) = 1.5e+6;

% Counts how many times monitor routine is called.
icount = 0;
fprintf('\n\nd03ra example program #2 results\n');

% We run through the problem twice - once from t=0 up to t=twant(1), and
% then onwards to t=twant(2).
for iout = 1:2
tout = twant(iout);
[ts, dt, rwk, iwk, ind, ifail] = ...
d03ra(ts, tout, dt, xmin, xmax, ymin, ymax, int64(nx), ...
int64(ny), tols, tolt, @ex2_pdedef, @ex2_bndary, ...
@ex2_pdeiv, @ex2_monitr, int64(opti), optr, rwk, ...
int64(iwk), int64(itrace), int64(ind), 'lenlwk', int64(lenlwk));
if ifail ~= 0
% Parameters out of range, or convergence problems.
% Print message and exit.
error('Warning: d03ra returned with ifail = %1d ',ifail);
end

% Output some statistics.
fprintf('\nStatistics\n');
fprintf('\nTime =%8.4f \n', ts);
fprintf('Total number of accepted timesteps %5d \n', iwk(1));
fprintf('Total number of rejected timesteps %5d \n', iwk(2));
fprintf('\n                    Total number of\n');
fprintf('          Residual  Jacobian    Newton   Lin sys\n');
fprintf('             evals     evals     iters     iters\n');
fprintf('At level\n');
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%10d%10d%10d%10d\n', j, iwk(j+2), ...
iwk(j+2+maxlev), iwk(j+2+2*maxlev), iwk(j+2+3*maxlev));
end;
end;
fprintf('\n                 Maximum number of\n');
fprintf('          Newton iters Lin sys iters\n');
fprintf('At level\n');
for j = 1:maxlev
if iwk(j+2) ~= 0
fprintf('%8d%14d%14d\n', j, iwk(j+2+4*maxlev), ...
iwk(j+2+5*maxlev));
end
end
end

function [res] = ex2_pdedef(npts, npde, t, x, y, u, ut, ux, uy, ...
uxx, uxy, uyy)
% Evaluate the system of PDEs for this problem.

global alpha beta;

fp = 4.0*pi;
res = zeros(npts, npde);
for i = 1:int32(npts) % can't use int64 in loop range.
b1 = 1.0 + alpha*x(i)*y(i) + beta*sin(fp*x(i))*sin(fp*y(i));
res(i,1) = ut(i,1) - uxx(i,1) - uyy(i,1) - ...
u(i,1)*(b1 - u(i,1) - 0.5e-6*u(i,2));
res(i,2) = -0.05*(uxx(i,2) + uyy(i,2)) - ...
u(i,2)*(-b1 + 1.0e4*u(i,1) - u(i,2));
end

function [u] = ex2_pdeiv(npts, npde, t, x, y)
% Evaluate initial conditions for PDEs for this system.

global alpha beta;

fp = 4.0*pi;
u = zeros(npts, npde);
for i = 1:int32(npts) % can't use int64 in loop range.
u(i,1) = 10.0 + (16.0*x(i)*(1.0 - x(i))*y(i)*(1.0 - y(i)))^2;
u(i,2) = -1.0 - alpha*x(i)*y(i) - beta*sin(fp*x(i))*sin(fp*y(i)) + ...
1.0e4*u(i,1);
end

function [res] = ex2_bndary(npts, npde, t, x, y, u, ut, ux, uy, ...
nbpts, lbnd, res)
% Implement boundary conditions for the domain.

tol = 10.0*x02aj;
for i = 1:int32(nbpts) % can't use int64 in loop range.
j = lbnd(i);
if (abs(x(j)) <= tol || abs(x(j)-1.0) <= tol)
res(j,1) = ux(j,1);
res(j,2) = ux(j,2);
elseif (abs(y(j)) <= tol || abs(y(j)-1.0) <= tol)
res(j,1) = uy(j,1);
res(j,2) = uy(j,2);
end
end

function [ierr] = ex2_monitr(npde, t, dt, dtnew, tlast, nlev, ...
ngpts, xpts, ypts, lsol, sol, ierr)
% Monitor the results at specified intervals.

global iout icount;
global xsav ysav ssav2 tsav isav;

% If it's the last time this routine gets called on the second time round,
% then output the results to screen.
if (tlast && iout == 2)

% Specify the grid level at which we're going to print the solution.
level = nlev;
npts = ngpts(level);
ipsol = lsol(level);
fprintf(['\nSolution at every 2nd grid point in level %d ', ...
'at time %8.4f : \n\n       x           y     ', ...
'approx c1   approx c2\n\n'], level, t);

% Print the solution.
nside = round(sqrt(double(npts)));
for i = 1:2:double(npts)
fprintf('%9.4e %9.4e %9.4e %9.4e\n', xpts(i), ...
ypts(i), sol(ipsol+i), sol(ipsol+npts+i));
end
end
% Specify the maximum number of times the results get saved, and the
% interval for saves.
nsav = 5;
nint = 10;

% Do we want to save this time?
icount = icount+1;
if mod(icount,nint) ~= 1 && ~tlast
return;
end

isav = 1 + round(icount/nint);
if isav > nsav
fprintf('Not enough save space allocated in monitr2: %d\n', ...
nsav);
return;
end
fprintf('Saving set %d at time = %f\n', isav, t);

% Specify the grid level at which we're going to extract the solution.
level = 1;
npts = ngpts(level);
ipsol = lsol(level);

% Allocate space for saves the first time through.
nside = round(sqrt(double(npts)));
if icount == 1
xsav = zeros(nside, 1);
ysav = zeros(nside, 1);
ssav2 = zeros(nside, nside, 2, nsav);
tsav = zeros(nsav, 1);
end

% Save this time step.  For this problem, we're calculating two
% concentrations as a function of x and y (and time).
tsav(isav) = t;
for i = 1:nside
for j = 1:nside
k = (i-1)*nside + j;
xsav(j) = xpts(k);
ysav(i) = ypts(k);
ssav2(j, i, 1, isav) = sol(ipsol+k);
ssav2(j, i, 2, isav) = sol(ipsol+npts+k);
end
end

function ex2_plot(id)
% Plot the results.
global xsav ysav ssav2 tsav isav;
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% Check the value of the array identifier.
if (id ~= 1 && id ~= 2)
fprintf('Illegal value for array identifier in plot: %d\n', id);
return;
end
% Set the colours for each surface, and the array of handles for the plots.
colours = [[1 0 0]; [0 1 0]; [1 0 1]; [0 0 1]];
h = zeros(isav,1);
% Check the allocation of colours against the number we're going to plot.
if (length(colours(:,1)) < isav)
fprintf(['Not enough colours allocated in example2_plot: ', ...
'%d needed\n'], isav);
end
% Plot all the surfaces, and colour each one.
for i = 1:isav
h(i) = mesh(xsav, ysav, ssav2(:,:,id,i));
hold on;
set(h(i), 'EdgeColor', colours(i,:));
end
hold off;
% Label the axes, and set the title according to the array identifier.
xlabel('x', labFmt{:});
ylabel('y', labFmt{:});
if (id == 1)
title({'Multispecies Food Web Model:', ...
'Time-dependent Predator Concentration'}, titleFmt{:});
zlabel('Predator concentration', labFmt{:});
else
title({'Multispecies Food Web Model:', ...
'Time-dependent Prey Concentration'}, titleFmt{:});
zlabel('Prey concentration', labFmt{:});
end
% Add a legend, using the plot handles and the array of time values.
legend(h, num2str(tsav(1:isav)));
% Set the view to something nice (determined empirically).
view(60, 40);
```
```

d03ra example program #1 results
Saving set 1 at time = 0.001000
Saving set 2 at time = 0.228996
Saving set 3 at time = 0.240000

Statistics

Time =  0.2400
Total number of accepted timesteps    75
Total number of rejected timesteps     0

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1       600        75       150       159

Maximum number of
Newton iters Lin sys iters
At level
1             2             2

Solution at every 20th grid point in level 3 at time   0.2500 :

x          y        approx u

0.0000e+00 0.0000e+00 2.0000e+00
1.0000e+00 0.0000e+00 1.4759e+00
9.5000e-01 5.0000e-02 1.9999e+00
9.0000e-01 1.0000e-01 1.3413e+00
8.5000e-01 1.5000e-01 1.9995e+00
8.0000e-01 2.0000e-01 1.2833e+00
7.5000e-01 2.5000e-01 1.9966e+00
7.0000e-01 3.0000e-01 1.2399e+00
6.5000e-01 3.5000e-01 1.9826e+00
6.0000e-01 4.0000e-01 1.2132e+00
5.5000e-01 4.5000e-01 1.8646e+00
5.0000e-01 5.0000e-01 1.1895e+00
4.5000e-01 5.5000e-01 1.6382e+00
4.0000e-01 6.0000e-01 2.0000e+00
3.5000e-01 6.5000e-01 1.4678e+00
3.0000e-01 7.0000e-01 1.9999e+00
2.5000e-01 7.5000e-01 1.3355e+00
2.0000e-01 8.0000e-01 1.9993e+00
1.5000e-01 8.5000e-01 1.2780e+00
1.0000e-01 9.0000e-01 1.9950e+00
5.0000e-02 9.5000e-01 1.2356e+00
0.0000e+00 1.0000e+00 1.9731e+00
1.0000e+00 1.0000e+00 1.2087e+00
7.5000e-01 2.5000e-02 1.7809e+00
6.0000e-01 7.5000e-02 1.1850e+00
7.7500e-01 1.0000e-01 1.9237e+00
6.2500e-01 1.5000e-01 1.2548e+00
8.0000e-01 1.7500e-01 1.9999e+00
6.5000e-01 2.2500e-01 1.3982e+00
8.2500e-01 2.5000e-01 2.0000e+00
6.7500e-01 3.0000e-01 1.8013e+00
8.5000e-01 3.2500e-01 1.2524e+00
7.0000e-01 3.7500e-01 1.9999e+00
8.7500e-01 4.0000e-01 1.9850e+00
7.2500e-01 4.5000e-01 1.9118e+00
9.0000e-01 4.7500e-01 1.6189e+00
6.5000e-01 5.2500e-01 1.9929e+00
7.2500e-01 5.5000e-01 1.9994e+00
8.0000e-01 5.7500e-01 2.0000e+00
3.7500e-01 6.0000e-01 2.0000e+00
8.7500e-01 6.0000e-01 1.9996e+00
4.5000e-01 6.2500e-01 1.2027e+00
7.5000e-02 6.5000e-01 1.9999e+00
5.7500e-01 6.5000e-01 1.9989e+00
2.0000e-01 6.7500e-01 1.2408e+00
7.0000e-01 6.7500e-01 1.9994e+00
3.2500e-01 7.0000e-01 1.9961e+00
8.2500e-01 7.0000e-01 1.2441e+00
4.5000e-01 7.2500e-01 1.9964e+00
1.2500e-01 7.5000e-01 1.9807e+00
6.2500e-01 7.5000e-01 1.2462e+00
3.0000e-01 7.7500e-01 1.9822e+00
2.5000e-02 8.0000e-01 1.9237e+00
5.2500e-01 8.0000e-01 1.2195e+00
2.5000e-01 8.2500e-01 1.8620e+00
2.5000e-02 8.5000e-01 1.6830e+00
5.2500e-01 8.5000e-01 1.6454e+00
3.0000e-01 8.7500e-01 1.6299e+00
1.7500e-01 9.0000e-01 1.3982e+00
7.0000e-01 0.0000e+00 1.4758e+00
7.8750e-01 1.2500e-02 1.4591e+00
7.1250e-01 3.7500e-02 1.2960e+00
8.0000e-01 5.0000e-02 1.3410e+00
7.2500e-01 7.5000e-02 1.3257e+00
8.1250e-01 8.7500e-02 1.2281e+00
7.3750e-01 1.1250e-01 1.2828e+00
8.2500e-01 1.2500e-01 1.2657e+00
7.5000e-01 1.5000e-01 1.2400e+00
6.7500e-01 1.7500e-01 1.2378e+00
7.6250e-01 1.8750e-01 1.2068e+00
6.8750e-01 2.1250e-01 1.2130e+00
7.7500e-01 2.2500e-01 1.2053e+00
7.0000e-01 2.5000e-01 1.1895e+00
7.8750e-01 2.6250e-01 1.1850e+00
Saving set 4 at time = 0.250000

Statistics

Time =  0.2500
Total number of accepted timesteps   180
Total number of rejected timesteps     1

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1      1468       181       382       391
2       662        82       170       170
3       177        22        45        45

Maximum number of
Newton iters Lin sys iters
At level
1             4             2
2             4             1
3             3             1

d03ra example program #2 results
Saving set 1 at time = 0.000500
Saving set 2 at time = 0.007377
Saving set 2 at time = 0.010000

Statistics

Time =  0.0100
Total number of accepted timesteps    14
Total number of rejected timesteps     0

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1       196        14        28        42
2       168        12        24        34
3        70         5        10        16

Maximum number of
Newton iters Lin sys iters
At level
1             2             2
2             2             2
3             2             3
Saving set 3 at time = 0.016563

Solution at every 2nd grid point in level 4 at time   0.0250 :

x           y     approx c1   approx c2

0.0000e+00 0.0000e+00 6.6146e+01 6.6146e+05
2.0000e-01 0.0000e+00 6.6963e+01 6.6962e+05
4.0000e-01 0.0000e+00 6.8880e+01 6.8879e+05
6.0000e-01 0.0000e+00 7.0689e+01 7.0687e+05
8.0000e-01 0.0000e+00 7.1226e+01 7.1223e+05
1.0000e+00 0.0000e+00 6.9633e+01 6.9630e+05
1.0000e-01 1.0000e-01 6.5564e+01 6.5561e+05
3.0000e-01 1.0000e-01 5.9242e+01 5.9240e+05
5.0000e-01 1.0000e-01 5.1376e+01 5.1374e+05
7.0000e-01 1.0000e-01 4.2883e+01 4.2882e+05
9.0000e-01 1.0000e-01 3.9626e+01 3.9625e+05
0.0000e+00 2.0000e-01 4.5742e+01 4.5740e+05
2.0000e-01 2.0000e-01 5.0545e+01 5.0542e+05
4.0000e-01 2.0000e-01 5.3180e+01 5.3177e+05
6.0000e-01 2.0000e-01 5.3162e+01 5.3159e+05
8.0000e-01 2.0000e-01 5.0482e+01 5.0479e+05
1.0000e+00 2.0000e-01 4.5607e+01 4.5605e+05
1.0000e-01 3.0000e-01 3.9376e+01 3.9375e+05
3.0000e-01 3.0000e-01 6.6350e+01 6.6350e+05
5.0000e-01 3.0000e-01 6.7224e+01 6.7223e+05
7.0000e-01 3.0000e-01 6.9263e+01 6.9261e+05
9.0000e-01 3.0000e-01 7.1200e+01 7.1196e+05
0.0000e+00 4.0000e-01 7.1824e+01 7.1820e+05
2.0000e-01 4.0000e-01 7.0253e+01 7.0248e+05
4.0000e-01 4.0000e-01 6.6131e+01 6.6127e+05
6.0000e-01 4.0000e-01 5.9694e+01 5.9691e+05
8.0000e-01 4.0000e-01 5.1675e+01 5.1673e+05
1.0000e+00 4.0000e-01 4.3136e+01 4.3134e+05
1.0000e-01 5.0000e-01 3.9812e+01 3.9810e+05
3.0000e-01 5.0000e-01 4.6073e+01 4.6069e+05
5.0000e-01 5.0000e-01 5.0986e+01 5.0982e+05
7.0000e-01 5.0000e-01 5.3685e+01 5.3680e+05
9.0000e-01 5.0000e-01 5.3668e+01 5.3663e+05
0.0000e+00 6.0000e-01 5.0926e+01 5.0921e+05
2.0000e-01 6.0000e-01 4.5942e+01 4.5939e+05
4.0000e-01 6.0000e-01 3.9568e+01 3.9566e+05
6.0000e-01 6.0000e-01 6.6963e+01 6.6962e+05
8.0000e-01 6.0000e-01 6.8006e+01 6.8004e+05
1.0000e+00 6.0000e-01 7.0413e+01 7.0408e+05
1.0000e-01 7.0000e-01 7.2731e+01 7.2723e+05
3.0000e-01 7.0000e-01 7.3620e+01 7.3611e+05
5.0000e-01 7.0000e-01 7.2112e+01 7.2103e+05
7.0000e-01 7.0000e-01 6.7831e+01 6.7822e+05
9.0000e-01 7.0000e-01 6.1051e+01 6.1043e+05
0.0000e+00 8.0000e-01 5.2573e+01 5.2567e+05
2.0000e-01 8.0000e-01 4.3388e+01 4.3386e+05
4.0000e-01 8.0000e-01 4.0372e+01 4.0367e+05
6.0000e-01 8.0000e-01 4.7064e+01 4.7058e+05
8.0000e-01 8.0000e-01 5.2309e+01 5.2301e+05
1.0000e+00 8.0000e-01 5.5199e+01 5.5190e+05
1.0000e-01 9.0000e-01 5.5185e+01 5.5175e+05
3.0000e-01 9.0000e-01 5.2257e+01 5.2249e+05
5.0000e-01 9.0000e-01 4.6947e+01 4.6941e+05
7.0000e-01 9.0000e-01 4.0144e+01 4.0140e+05
9.0000e-01 9.0000e-01 6.7858e+01 6.7856e+05
0.0000e+00 1.0000e+00 6.9130e+01 6.9126e+05
2.0000e-01 1.0000e+00 7.2059e+01 7.2051e+05
4.0000e-01 1.0000e+00 7.4925e+01 7.4914e+05
6.0000e-01 1.0000e+00 7.6199e+01 7.6185e+05
8.0000e-01 1.0000e+00 7.4789e+01 7.4775e+05
1.0000e+00 1.0000e+00 7.0282e+01 7.0268e+05
5.0000e-02 0.0000e+00 6.3009e+01 6.2998e+05
1.5000e-01 0.0000e+00 5.3893e+01 5.3885e+05
2.5000e-01 0.0000e+00 4.4146e+01 4.4142e+05
3.5000e-01 0.0000e+00 4.1273e+01 4.1266e+05
4.5000e-01 0.0000e+00 4.8513e+01 4.8503e+05
5.5000e-01 0.0000e+00 5.4232e+01 5.4220e+05
6.5000e-01 0.0000e+00 5.7399e+01 5.7385e+05
7.5000e-01 0.0000e+00 5.7390e+01 5.7376e+05
8.5000e-01 0.0000e+00 5.4194e+01 5.4182e+05
9.5000e-01 0.0000e+00 4.8419e+01 4.8409e+05
0.0000e+00 5.0000e-02 4.1077e+01 4.1071e+05
1.0000e-01 5.0000e-02 6.8880e+01 6.8879e+05
2.0000e-01 5.0000e-02 7.0413e+01 7.0408e+05
3.0000e-01 5.0000e-02 7.3943e+01 7.3932e+05
4.0000e-01 5.0000e-02 7.7450e+01 7.7436e+05
5.0000e-01 5.0000e-02 7.9179e+01 7.9162e+05
6.0000e-01 5.0000e-02 7.7890e+01 7.7872e+05
7.0000e-01 5.0000e-02 7.3125e+01 7.3108e+05
8.0000e-01 5.0000e-02 6.5284e+01 6.5269e+05
9.0000e-01 5.0000e-02 5.5431e+01 5.5420e+05
1.0000e+00 5.0000e-02 4.4904e+01 4.4898e+05
5.0000e-02 1.0000e-01 4.2323e+01 4.2315e+05
1.5000e-01 1.0000e-01 5.0216e+01 5.0203e+05
2.5000e-01 1.0000e-01 5.6491e+01 5.6474e+05
3.5000e-01 1.0000e-01 5.9983e+01 5.9965e+05
4.5000e-01 1.0000e-01 5.9981e+01 5.9963e+05
5.5000e-01 1.0000e-01 5.6473e+01 5.6456e+05
6.5000e-01 1.0000e-01 5.0154e+01 5.0141e+05
7.5000e-01 1.0000e-01 4.2170e+01 4.2161e+05
8.5000e-01 1.0000e-01 6.9876e+01 6.9874e+05
9.5000e-01 1.0000e-01 7.1672e+01 7.1666e+05
0.0000e+00 1.5000e-01 7.5818e+01 7.5805e+05
1.0000e-01 1.5000e-01 7.9991e+01 7.9974e+05
2.0000e-01 1.5000e-01 8.2201e+01 8.2180e+05
3.0000e-01 1.5000e-01 8.1048e+01 8.1026e+05
4.0000e-01 1.5000e-01 7.6024e+01 7.6003e+05
5.0000e-01 1.5000e-01 6.7601e+01 6.7584e+05
6.0000e-01 1.5000e-01 5.7001e+01 5.6988e+05
7.0000e-01 1.5000e-01 4.5711e+01 4.5704e+05
8.0000e-01 1.5000e-01 4.3428e+01 4.3418e+05
9.0000e-01 1.5000e-01 5.1981e+01 5.1965e+05
1.0000e+00 1.5000e-01 5.8830e+01 5.8810e+05
5.0000e-02 2.0000e-01 6.2663e+01 6.2641e+05
1.5000e-01 2.0000e-01 6.2669e+01 6.2648e+05
2.5000e-01 2.0000e-01 5.8838e+01 5.8818e+05
3.5000e-01 2.0000e-01 5.1961e+01 5.1945e+05
4.5000e-01 2.0000e-01 4.3331e+01 4.3321e+05
5.5000e-01 2.0000e-01 7.0690e+01 7.0687e+05
6.5000e-01 2.0000e-01 7.2731e+01 7.2724e+05
7.5000e-01 2.0000e-01 7.7451e+01 7.7436e+05
8.5000e-01 2.0000e-01 8.2255e+01 8.2234e+05
9.5000e-01 2.0000e-01 8.4927e+01 8.4903e+05
0.0000e+00 2.5000e-01 8.3913e+01 8.3888e+05
1.0000e-01 2.5000e-01 7.8659e+01 7.8635e+05
2.0000e-01 2.5000e-01 6.9705e+01 6.9685e+05
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4.3750e-01 5.5000e-01 1.9240e+01 1.9247e+05
4.6250e-01 5.5000e-01 1.9496e+01 1.9503e+05
5.0000e-01 5.5000e-01 2.2033e+01 2.2039e+05
5.2500e-01 5.5000e-01 2.7020e+01 2.7024e+05
5.5000e-01 5.5000e-01 3.4611e+01 3.4613e+05
5.7500e-01 5.5000e-01 4.4817e+01 4.4817e+05
6.0000e-01 5.5000e-01 5.7338e+01 5.7335e+05
6.2500e-01 5.5000e-01 7.1422e+01 7.1415e+05
6.5000e-01 5.5000e-01 8.5844e+01 8.5835e+05
6.7500e-01 5.5000e-01 9.9062e+01 9.9051e+05
7.0000e-01 5.5000e-01 1.0954e+02 1.0953e+06
7.2500e-01 5.5000e-01 1.1616e+02 1.1615e+06
7.5000e-01 5.5000e-01 1.1856e+02 1.1855e+06
7.7500e-01 5.5000e-01 1.1733e+02 1.1732e+06
8.0000e-01 5.5000e-01 1.1389e+02 1.1388e+06
8.2500e-01 5.5000e-01 1.1026e+02 1.1025e+06
8.5000e-01 5.5000e-01 1.0868e+02 1.0868e+06
8.7500e-01 5.5000e-01 3.8985e+01 3.8983e+05
9.0000e-01 5.5000e-01 4.7678e+01 4.7675e+05
9.2500e-01 5.5000e-01 5.5804e+01 5.5799e+05
9.5000e-01 5.5000e-01 6.2242e+01 6.2236e+05
9.7500e-01 5.5000e-01 6.6007e+01 6.6000e+05
1.0000e+00 5.5000e-01 6.6500e+01 6.6493e+05
1.2500e-02 5.6250e-01 6.3684e+01 6.3678e+05
3.7500e-02 5.6250e-01 5.8095e+01 5.8089e+05
6.2500e-02 5.6250e-01 5.0693e+01 5.0689e+05
8.7500e-02 5.6250e-01 4.2616e+01 4.2613e+05
1.1250e-01 5.6250e-01 2.5244e+01 2.5246e+05
1.3750e-01 5.6250e-01 2.1491e+01 2.1492e+05
1.6250e-01 5.6250e-01 1.9664e+01 1.9665e+05
1.8750e-01 5.6250e-01 1.9959e+01 1.9961e+05
2.1250e-01 5.6250e-01 2.2517e+01 2.2519e+05
3.7500e-01 5.6250e-01 2.7491e+01 2.7492e+05
4.0000e-01 5.6250e-01 3.4996e+01 3.4996e+05
4.2500e-01 5.6250e-01 4.4992e+01 4.4990e+05
4.5000e-01 5.6250e-01 5.7137e+01 5.7133e+05
4.7500e-01 5.6250e-01 7.0675e+01 7.0670e+05
5.1250e-01 5.6250e-01 8.4438e+01 8.4432e+05
5.3750e-01 5.6250e-01 9.6997e+01 9.6990e+05
5.6250e-01 5.6250e-01 1.0696e+02 1.0695e+06
5.8750e-01 5.6250e-01 1.1334e+02 1.1334e+06
6.1250e-01 5.6250e-01 1.1584e+02 1.1584e+06
6.3750e-01 5.6250e-01 1.1501e+02 1.1500e+06
6.6250e-01 5.6250e-01 1.1214e+02 1.1213e+06
6.8750e-01 5.6250e-01 1.0908e+02 1.0907e+06
7.1250e-01 5.6250e-01 1.0776e+02 1.0776e+06
7.3750e-01 5.6250e-01 3.8882e+01 3.8880e+05
7.6250e-01 5.6250e-01 4.7437e+01 4.7434e+05
7.8750e-01 5.6250e-01 5.5417e+01 5.5413e+05
8.1250e-01 5.6250e-01 6.1728e+01 6.1724e+05
8.3750e-01 5.6250e-01 6.5417e+01 6.5412e+05
8.6250e-01 5.6250e-01 6.5907e+01 6.5902e+05
8.8750e-01 5.6250e-01 6.3163e+01 6.3158e+05
9.1250e-01 5.6250e-01 5.7699e+01 5.7695e+05
9.3750e-01 5.6250e-01 5.0446e+01 5.0443e+05
9.6250e-01 5.6250e-01 4.2514e+01 4.2511e+05
9.8750e-01 5.6250e-01 2.5368e+01 2.5369e+05
0.0000e+00 5.7500e-01 2.1608e+01 2.1608e+05
2.5000e-02 5.7500e-01 1.9804e+01 1.9805e+05
5.0000e-02 5.7500e-01 2.0112e+01 2.0113e+05
7.5000e-02 5.7500e-01 2.2678e+01 2.2678e+05
1.0000e-01 5.7500e-01 2.7648e+01 2.7648e+05
1.2500e-01 5.7500e-01 3.5124e+01 3.5123e+05
1.5000e-01 5.7500e-01 4.5049e+01 4.5047e+05
1.7500e-01 5.7500e-01 5.7069e+01 5.7065e+05
2.0000e-01 5.7500e-01 7.0426e+01 7.0421e+05
2.2500e-01 5.7500e-01 8.3969e+01 8.3963e+05
2.5000e-01 5.7500e-01 9.6308e+01 9.6302e+05
3.8750e-01 5.7500e-01 1.0610e+02 1.0610e+06
4.1250e-01 5.7500e-01 1.1241e+02 1.1240e+06
4.3750e-01 5.7500e-01 1.1494e+02 1.1493e+06
4.6250e-01 5.7500e-01 1.1423e+02 1.1422e+06
5.0000e-01 5.7500e-01 1.1156e+02 1.1155e+06
5.2500e-01 5.7500e-01 1.0868e+02 1.0868e+06
5.5000e-01 5.7500e-01 1.0746e+02 1.0745e+06
Saving set 4 at time = 0.025000

Statistics

Time =  0.0250
Total number of accepted timesteps    29
Total number of rejected timesteps     0

Total number of
Residual  Jacobian    Newton   Lin sys
evals     evals     iters     iters
At level
1       406        29        58        87
2       378        27        54        79
3       280        20        40        61
4        98         7        14        27

Maximum number of
Newton iters Lin sys iters
At level
1             2             2
2             2             2
3             2             3
4             2             3

```