d03 Chapter Contents
d03 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_pde_parab_1d_fd_ode_remesh (d03ppc)

## 1  Purpose

nag_pde_parab_1d_fd_ode_remesh (d03ppc) integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable, with scope for coupled ordinary differential equations (ODEs), and automatic adaptive spatial remeshing. The spatial discretization is performed using finite differences, and the method of lines is employed to reduce the PDEs to a system of ODEs. The resulting system is solved using a Backward Differentiation Formula (BDF) method or a Theta method (switching between Newton's method and functional iteration).

## 2  Specification

 #include #include
void  nag_pde_parab_1d_fd_ode_remesh (Integer npde, Integer m, double *ts, double tout,
 void (*pdedef)(Integer npde, double t, double x, const double u[], const double ux[], Integer ncode, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires, Nag_Comm *comm),
 void (*bndary)(Integer npde, double t, const double u[], const double ux[], Integer ncode, const double v[], const double vdot[], Integer ibnd, double beta[], double gamma[], Integer *ires, Nag_Comm *comm),
 void (*uvinit)(Integer npde, Integer npts, Integer nxi, const double x[], const double xi[], double u[], Integer ncode, double v[], Nag_Comm *comm),
double u[], Integer npts, double x[], Integer ncode,
 void (*odedef)(Integer npde, double t, Integer ncode, const double v[], const double vdot[], Integer nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires, Nag_Comm *comm),
Integer nxi, const double xi[], Integer neqn, const double rtol[], const double atol[], Integer itol, Nag_NormType norm, Nag_LinAlgOption laopt, const double algopt[], Nag_Boolean remesh, Integer nxfix, const double xfix[], Integer nrmesh, double dxmesh, double trmesh, Integer ipminf, double xratio, double con,
 void (*monitf)(double t, Integer npts, Integer npde, const double x[], const double u[], const double r[], double fmon[], Nag_Comm *comm),
double rsave[], Integer lrsave, Integer isave[], Integer lisave, Integer itask, Integer itrace, const char *outfile, Integer *ind, Nag_Comm *comm, Nag_D03_Save *saved, NagError *fail)

## 3  Description

nag_pde_parab_1d_fd_ode_remesh (d03ppc) integrates the system of parabolic-elliptic equations and coupled ODEs
 $∑j=1npdePi,j ∂Uj ∂t +Qi=x-m ∂∂x xmRi, i=1,2,…,npde , a≤x≤b,t≥t0,$ (1)
 $Fit,V,V.,ξ,U*,Ux*,R*,Ut*,Uxt*=0, i=1,2,…,ncode,$ (2)
where (1) defines the PDE part and (2) generalizes the coupled ODE part of the problem.
In (1), ${P}_{i,j}$ and ${R}_{i}$ depend on $x$, $t$, $U$, ${U}_{x}$, and $V$; ${Q}_{i}$ depends on $x$, $t$, $U$, ${U}_{x}$, $V$ and linearly on $\stackrel{.}{V}$. The vector $U$ is the set of PDE solution values
 $Ux,t=U1x,t,…,Unpdex,tT,$
and the vector ${U}_{x}$ is the partial derivative with respect to $x$. The vector $V$ is the set of ODE solution values
 $Vt=V1t,…,VncodetT,$
and $\stackrel{.}{V}$ denotes its derivative with respect to time.
In (2), $\xi$ represents a vector of ${n}_{\xi }$ spatial coupling points at which the ODEs are coupled to the PDEs. These points may or may not be equal to some of the PDE spatial mesh points. ${U}^{*}$, ${U}_{x}^{*}$, ${R}^{*}$, ${U}_{t}^{*}$ and ${U}_{xt}^{*}$ are the functions $U$, ${U}_{x}$, $R$, ${U}_{t}$ and ${U}_{xt}$ evaluated at these coupling points. Each ${F}_{i}$ may only depend linearly on time derivatives. Hence the equation (2) may be written more precisely as
 $F=G-AV.-B Ut* Uxt* ,$ (3)
where $F={\left[{F}_{1},\dots ,{F}_{{\mathbf{ncode}}}\right]}^{\mathrm{T}}$, $G$ is a vector of length ncode, $A$ is an ncode by ncode matrix, $B$ is an ncode by $\left({n}_{\xi }×{\mathbf{npde}}\right)$ matrix and the entries in $G$, $A$ and $B$ may depend on $t$, $\xi$, ${U}^{*}$, ${U}_{x}^{*}$ and $V$. In practice you only need to supply a vector of information to define the ODEs and not the matrices $A$ and $B$. (See Section 5 for the specification of odedef.)
The integration in time is from ${t}_{0}$ to ${t}_{\mathrm{out}}$, over the space interval $a\le x\le b$, where $a={x}_{1}$ and $b={x}_{{\mathbf{npts}}}$ are the leftmost and rightmost points of a mesh ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$ defined initially by you and (possibly) adapted automatically during the integration according to user-specified criteria. The coordinate system in space is defined by the following values of $m$; $m=0$ for Cartesian coordinates, $m=1$ for cylindrical polar coordinates and $m=2$ for spherical polar coordinates.
The PDE system which is defined by the functions ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ must be specified in pdedef.
The initial $\left(t={t}_{0}\right)$ values of the functions $U\left(x,t\right)$ and $V\left(t\right)$ must be specified in uvinit. Note that uvinit will be called again following any initial remeshing, and so $U\left(x,{t}_{0}\right)$ should be specified for all values of $x$ in the interval $a\le x\le b$, and not just the initial mesh points.
The functions ${R}_{i}$ which may be thought of as fluxes, are also used in the definition of the boundary conditions. The boundary conditions must have the form
 $βix,tRix,t,U,Ux,V=γix,t,U,Ux,V,V., i=1,2,…,npde,$ (4)
where $x=a$ or $x=b$.
The boundary conditions must be specified in bndary. The function ${\gamma }_{i}$ may depend linearly on $\stackrel{.}{V}$.
The problem is subject to the following restrictions:
 (i) In (1), ${\stackrel{.}{V}}_{\mathit{j}}\left(t\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{ncode}}$, may only appear linearly in the functions ${Q}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, with a similar restriction for $\gamma$; (ii) ${P}_{i,j}$ and the flux ${R}_{i}$ must not depend on any time derivatives; (iii) ${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction; (iv) The evaluation of the terms ${P}_{\mathit{i},j}$, ${Q}_{\mathit{i}}$ and ${R}_{\mathit{i}}$ is done approximately at the mid-points of the mesh ${\mathbf{x}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$, by calling the pdedef for each mid-point in turn. Any discontinuities in these functions must therefore be at one or more of the fixed mesh points specified by xfix; (v) At least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the PDE problem; (vi) If $m>0$ and ${x}_{1}=0.0$, which is the left boundary point, then it must be ensured that the PDE solution is bounded at this point. This can be done by either specifying the solution at $x=0.0$ or by specifying a zero flux there, that is ${\beta }_{i}=1.0$ and ${\gamma }_{i}=0.0$. See also Section 8.
The algebraic-differential equation system which is defined by the functions ${F}_{i}$ must be specified in odedef. You must also specify the coupling points $\xi$ in the array xi.
The parabolic equations are approximated by a system of ODEs in time for the values of ${U}_{i}$ at mesh points. For simple problems in Cartesian coordinates, this system is obtained by replacing the space derivatives by the usual central, three-point finite difference formula. However, for polar and spherical problems, or problems with nonlinear coefficients, the space derivatives are replaced by a modified three-point formula which maintains second order accuracy. In total there are ${\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{ncode}}$ ODEs in time direction. This system is then integrated forwards in time using a Backward Differentiation Formula (BDF) or a Theta method.
The adaptive space remeshing can be used to generate meshes that automatically follow the changing time-dependent nature of the solution, generally resulting in a more efficient and accurate solution using fewer mesh points than may be necessary with a fixed uniform or non-uniform mesh. Problems with travelling wavefronts or variable-width boundary layers for example will benefit from using a moving adaptive mesh. The discrete time-step method used here (developed by Furzeland (1984)) automatically creates a new mesh based on the current solution profile at certain time-steps, and the solution is then interpolated onto the new mesh and the integration continues.
The method requires you to supply a monitf which specifies in an analytical or numerical form the particular aspect of the solution behaviour you wish to track. This so-called monitor function is used to choose a mesh which equally distributes the integral of the monitor function over the domain. A typical choice of monitor function is the second space derivative of the solution value at each point (or some combination of the second space derivatives if there is more than one solution component), which results in refinement in regions where the solution gradient is changing most rapidly.
You must specify the frequency of mesh updates together with certain other criteria such as adjacent mesh ratios. Remeshing can be expensive and you are encouraged to experiment with the different options in order to achieve an efficient solution which adequately tracks the desired features of the solution.
Note that unless the monitor function for the initial solution values is zero at all user-specified initial mesh points, a new initial mesh is calculated and adopted according to the user-specified remeshing criteria. uvinit will then be called again to determine the initial solution values at the new mesh points (there is no interpolation at this stage) and the integration proceeds.

## 4  References

Berzins M (1990) Developments in the NAG Library software for parabolic equations Scientific Software Systems (eds J C Mason and M G Cox) 59–72 Chapman and Hall
Berzins M, Dew P M and Furzeland R M (1989) Developing software for time-dependent problems using the method of lines and differential-algebraic integrators Appl. Numer. Math. 5 375–397
Berzins M and Furzeland R M (1992) An adaptive theta method for the solution of stiff and nonstiff differential equations Appl. Numer. Math. 9 1–19
Furzeland R M (1984) The construction of adaptive space meshes TNER.85.022 Thornton Research Centre, Chester
Skeel R D and Berzins M (1990) A method for the spatial discretization of parabolic equations in one space variable SIAM J. Sci. Statist. Comput. 11(1) 1–32

## 5  Arguments

1:     npdeIntegerInput
On entry: the number of PDEs to be solved.
Constraint: ${\mathbf{npde}}\ge 1$.
2:     mIntegerInput
On entry: the coordinate system used:
${\mathbf{m}}=0$
Indicates Cartesian coordinates.
${\mathbf{m}}=1$
Indicates cylindrical polar coordinates.
${\mathbf{m}}=2$
Indicates spherical polar coordinates.
Constraint: ${\mathbf{m}}=0$, $1$ or $2$.
3:     tsdouble *Input/Output
On entry: the initial value of the independent variable $t$.
On exit: the value of $t$ corresponding to the solution values in u. Normally ${\mathbf{ts}}={\mathbf{tout}}$.
Constraint: ${\mathbf{ts}}<{\mathbf{tout}}$.
4:     toutdoubleInput
On entry: the final value of $t$ to which the integration is to be carried out.
5:     pdedeffunction, supplied by the userExternal Function
pdedef must evaluate the functions ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$, ${U}_{x}$ and $V$. ${Q}_{i}$ may depend linearly on $\stackrel{.}{V}$. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_parab_1d_fd_ode_remesh (d03ppc).
The specification of pdedef is:
 void pdedef (Integer npde, double t, double x, const double u[], const double ux[], Integer ncode, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires, Nag_Comm *comm)
1:     npdeIntegerInput
On entry: the number of PDEs in the system.
2:     tdoubleInput
On entry: the current value of the independent variable $t$.
3:     xdoubleInput
On entry: the current value of the space variable $x$.
4:     u[npde]const doubleInput
On entry: ${\mathbf{u}}\left[\mathit{i}-1\right]$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
5:     ux[npde]const doubleInput
On entry: ${\mathbf{ux}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
6:     ncodeIntegerInput
On entry: the number of coupled ODEs in the system.
7:     v[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{v}}\left[\mathit{i}-1\right]$ contains the value of component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
8:     vdot[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{vdot}}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
Note:  ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$, may only appear linearly in ${Q}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
9:     p[${\mathbf{npde}}×{\mathbf{npde}}$]doubleOutput
On exit: ${\mathbf{p}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}\left(x,t,U,{U}_{x},V\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
10:   q[npde]doubleOutput
On exit: ${\mathbf{q}}\left[\mathit{i}-1\right]$ must be set to the value of ${Q}_{\mathit{i}}\left(x,t,U,{U}_{x},V,\stackrel{.}{V}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
11:   r[npde]doubleOutput
On exit: ${\mathbf{r}}\left[\mathit{i}-1\right]$ must be set to the value of ${R}_{\mathit{i}}\left(x,t,U,{U}_{x},V\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
12:   iresInteger *Input/Output
On entry: set to $-1\text{​ or ​}1$.
On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to NE_USER_STOP.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, then nag_pde_parab_1d_fd_ode_remesh (d03ppc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
13:   commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to pdedef.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode_remesh (d03ppc) you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from nag_pde_parab_1d_fd_ode_remesh (d03ppc) (see Section 3.2.1 in the Essential Introduction).
6:     bndaryfunction, supplied by the userExternal Function
bndary must evaluate the functions ${\beta }_{i}$ and ${\gamma }_{i}$ which describe the boundary conditions, as given in (4).
The specification of bndary is:
 void bndary (Integer npde, double t, const double u[], const double ux[], Integer ncode, const double v[], const double vdot[], Integer ibnd, double beta[], double gamma[], Integer *ires, Nag_Comm *comm)
1:     npdeIntegerInput
On entry: the number of PDEs in the system.
2:     tdoubleInput
On entry: the current value of the independent variable $t$.
3:     u[npde]const doubleInput
On entry: ${\mathbf{u}}\left[\mathit{i}-1\right]$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
4:     ux[npde]const doubleInput
On entry: ${\mathbf{ux}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
5:     ncodeIntegerInput
On entry: the number of coupled ODEs in the system.
6:     v[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{v}}\left[\mathit{i}-1\right]$ contains the value of component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
7:     vdot[ncode]const doubleInput
On entry: ${\mathbf{vdot}}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$, may only appear linearly in ${\gamma }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
8:     ibndIntegerInput
On entry: specifies which boundary conditions are to be evaluated.
${\mathbf{ibnd}}=0$
bndary must set up the coefficients of the left-hand boundary, $x=a$.
${\mathbf{ibnd}}\ne 0$
bndary must set up the coefficients of the right-hand boundary, $x=b$.
9:     beta[npde]doubleOutput
On exit: ${\mathbf{beta}}\left[\mathit{i}-1\right]$ must be set to the value of ${\beta }_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
10:   gamma[npde]doubleOutput
On exit: ${\mathbf{gamma}}\left[\mathit{i}-1\right]$ must be set to the value of ${\gamma }_{\mathit{i}}\left(x,t,U,{U}_{x},V,\stackrel{.}{V}\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
11:   iresInteger *Input/Output
On entry: set to $-1\text{​ or ​}1$.
On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to NE_USER_STOP.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, then nag_pde_parab_1d_fd_ode_remesh (d03ppc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
12:   commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to bndary.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode_remesh (d03ppc) you may allocate memory and initialize these pointers with various quantities for use by bndary when called from nag_pde_parab_1d_fd_ode_remesh (d03ppc) (see Section 3.2.1 in the Essential Introduction).
7:     uvinitfunction, supplied by the userExternal Function
uvinit must supply the initial $\left(t={t}_{0}\right)$ values of $U\left(x,t\right)$ and $V\left(t\right)$ for all values of $x$ in the interval $a\le x\le b$.
The specification of uvinit is:
 void uvinit (Integer npde, Integer npts, Integer nxi, const double x[], const double xi[], double u[], Integer ncode, double v[], Nag_Comm *comm)
1:     npdeIntegerInput
On entry: the number of PDEs in the system.
2:     nptsIntegerInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
3:     nxiIntegerInput
On entry: the number of ODE/PDE coupling points.
4:     x[npts]const doubleInput
On entry: the current mesh. ${\mathbf{x}}\left[\mathit{i}-1\right]$ contains the value of ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
5:     xi[nxi]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left[\mathit{i}-1\right]$ contains the value of the ODE/PDE coupling point, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$.
6:     u[${\mathbf{npde}}×{\mathbf{npts}}$]doubleOutput
On exit: if ${\mathbf{nxi}}>0$, ${\mathbf{u}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of the component ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
7:     ncodeIntegerInput
On entry: the number of coupled ODEs in the system.
8:     v[ncode]doubleOutput
On exit: ${\mathbf{v}}\left[\mathit{i}-1\right]$ contains the value of component ${V}_{\mathit{i}}\left({t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
9:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to uvinit.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode_remesh (d03ppc) you may allocate memory and initialize these pointers with various quantities for use by uvinit when called from nag_pde_parab_1d_fd_ode_remesh (d03ppc) (see Section 3.2.1 in the Essential Introduction).
8:     u[neqn]doubleInput/Output
On entry: if ${\mathbf{ind}}=1$, the value of u must be unchanged from the previous call.
On exit: ${\mathbf{u}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the computed solution ${U}_{\mathit{i}}\left({x}_{\mathit{j}},t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and ${\mathbf{u}}\left[{\mathbf{npts}}×{\mathbf{npde}}+\mathit{k}-1\right]$ contains ${V}_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,{\mathbf{ncode}}$, evaluated at $t={\mathbf{ts}}$.
9:     nptsIntegerInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
Constraint: ${\mathbf{npts}}\ge 3$.
10:   x[npts]doubleInput/Output
On entry: the initial mesh points in the space direction. ${\mathbf{x}}\left[0\right]$ must specify the left-hand boundary, $a$ and ${\mathbf{x}}\left[{\mathbf{npts}}-1\right]$ must specify the right-hand boundary, $b$.
Constraint: ${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{npts}}-1\right]$.
On exit: the final values of the mesh points.
11:   ncodeIntegerInput
On entry: the number of coupled ODE in the system.
Constraint: ${\mathbf{ncode}}\ge 0$.
12:   odedeffunction, supplied by the userExternal Function
odedef must evaluate the functions $F$, which define the system of ODEs, as given in (3).
odedef will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to nag_pde_parab_1d_fd_ode_remesh (d03ppc).
The specification of odedef is:
 void odedef (Integer npde, double t, Integer ncode, const double v[], const double vdot[], Integer nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires, Nag_Comm *comm)
1:     npdeIntegerInput
On entry: the number of PDEs in the system.
2:     tdoubleInput
On entry: the current value of the independent variable $t$.
3:     ncodeIntegerInput
On entry: the number of coupled ODEs in the system.
4:     v[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{v}}\left[\mathit{i}-1\right]$ contains the value of component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
5:     vdot[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{vdot}}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
6:     nxiIntegerInput
On entry: the number of ODE/PDE coupling points.
7:     xi[nxi]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left[\mathit{i}-1\right]$ contains the ODE/PDE coupling points, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$.
8:     ucp[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucp}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of ${U}_{\mathit{i}}\left(x,t\right)$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
9:     ucpx[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpx}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
10:   rcp[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: ${\mathbf{rcp}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of the flux ${R}_{\mathit{i}}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
11:   ucpt[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpt}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of $\frac{\partial {U}_{\mathit{i}}}{\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
12:   ucptx[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: ${\mathbf{ucptx}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of $\frac{{\partial }^{2}{U}_{\mathit{i}}}{\partial x\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
13:   f[ncode]doubleOutput
On exit: ${\mathbf{f}}\left[\mathit{i}-1\right]$ must contain the $\mathit{i}$th component of $F$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$, where $F$ is defined as
 $F=G-AV.-B Ut* Uxt* ,$ (5)
or
 $F=-AV.-B Ut* Uxt* .$ (6)
The definition of $F$ is determined by the input value of ires.
14:   iresInteger *Input/Output
On entry: the form of $F$ that must be returned in the array f.
${\mathbf{ires}}=1$
Equation (5) must be used.
${\mathbf{ires}}=-1$
Equation (6) must be used.
On exit: should usually remain unchanged. However, you may reset ires to force the integration function to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to NE_USER_STOP.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, then nag_pde_parab_1d_fd_ode_remesh (d03ppc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
15:   commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to odedef.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode_remesh (d03ppc) you may allocate memory and initialize these pointers with various quantities for use by odedef when called from nag_pde_parab_1d_fd_ode_remesh (d03ppc) (see Section 3.2.1 in the Essential Introduction).
13:   nxiIntegerInput
On entry: the number of ODE/PDE coupling points.
Constraints:
• if ${\mathbf{ncode}}=0$, ${\mathbf{nxi}}=0$;
• if ${\mathbf{ncode}}>0$, ${\mathbf{nxi}}\ge 0$.
14:   xi[nxi]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$, must be set to the ODE/PDE coupling points.
Constraint: ${\mathbf{x}}\left[0\right]\le {\mathbf{xi}}\left[0\right]<{\mathbf{xi}}\left[1\right]<\cdots <{\mathbf{xi}}\left[{\mathbf{nxi}}-1\right]\le {\mathbf{x}}\left[{\mathbf{npts}}-1\right]$.
15:   neqnIntegerInput
On entry: the number of ODEs in the time direction.
Constraint: ${\mathbf{neqn}}={\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{ncode}}$.
16:   rtol[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array rtol must be at least
• $1$ when ${\mathbf{itol}}=1$ or $2$;
• ${\mathbf{neqn}}$ when ${\mathbf{itol}}=3$ or $4$.
On entry: the relative local error tolerance.
Constraint: ${\mathbf{rtol}}\left[i-1\right]\ge 0.0$ for all relevant $i$.
17:   atol[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array atol must be at least
• $1$ when ${\mathbf{itol}}=1$ or $3$;
• ${\mathbf{neqn}}$ when ${\mathbf{itol}}=2$ or $4$.
On entry: the absolute local error tolerance.
Constraints:
• ${\mathbf{atol}}\left[i-1\right]\ge 0.0$ for all relevant $i$;
• Corresponding elements of atol and rtol cannot both be $0.0$.
18:   itolIntegerInput
On entry: a value to indicate the form of the local error test. itol indicates to nag_pde_parab_1d_fd_ode_remesh (d03ppc) whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $‖{e}_{i}/{w}_{i}‖<1.0$, where ${w}_{i}$ is defined as follows:
 itol rtol atol ${w}_{i}$ 1 scalar scalar ${\mathbf{rtol}}\left[0\right]×\left|{U}_{i}\right|+{\mathbf{atol}}\left[0\right]$ 2 scalar vector ${\mathbf{rtol}}\left[0\right]×\left|{U}_{i}\right|+{\mathbf{atol}}\left[i-1\right]$ 3 vector scalar ${\mathbf{rtol}}\left[i-1\right]×\left|{U}_{i}\right|+{\mathbf{atol}}\left[0\right]$ 4 vector vector ${\mathbf{rtol}}\left[i-1\right]×\left|{U}_{i}\right|+{\mathbf{atol}}\left[i-1\right]$
In the above, ${e}_{\mathit{i}}$ denotes the estimated local error for the $\mathit{i}$th component of the coupled PDE/ODE system in time, ${\mathbf{u}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{neqn}}$.
The choice of norm used is defined by the argument norm.
Constraint: $1\le {\mathbf{itol}}\le 4$.
19:   normNag_NormTypeInput
On entry: the type of norm to be used.
${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$
Maximum norm.
${\mathbf{norm}}=\mathrm{Nag_TwoNorm}$
Averaged ${L}_{2}$ norm.
If ${{\mathbf{u}}}_{\mathrm{norm}}$ denotes the norm of the vector u of length neqn, then for the averaged ${L}_{2}$ norm
 $unorm=1neqn∑i=1neqnu[i-1]/wi2,$
while for the maximum norm
 $u norm = maxi u[i-1] / wi .$
See the description of itol for the formulation of the weight vector $w$.
Constraint: ${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$ or $\mathrm{Nag_TwoNorm}$.
20:   laoptNag_LinAlgOptionInput
On entry: the type of matrix algebra required.
${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$
Full matrix methods to be used.
${\mathbf{laopt}}=\mathrm{Nag_LinAlgBand}$
Banded matrix methods to be used.
${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$
Sparse matrix methods to be used.
Constraint: ${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$, $\mathrm{Nag_LinAlgBand}$ or $\mathrm{Nag_LinAlgSparse}$.
Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., ${\mathbf{ncode}}=0$).
21:   algopt[$30$]const doubleInput
On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, then ${\mathbf{algopt}}\left[0\right]$ should be set to $0.0$. Default values will also be used for any other elements of algopt set to zero. The permissible values, default values, and meanings are as follows:
${\mathbf{algopt}}\left[0\right]$
Selects the ODE integration method to be used. If ${\mathbf{algopt}}\left[0\right]=1.0$, a BDF method is used and if ${\mathbf{algopt}}\left[0\right]=2.0$, a Theta method is used. The default value is ${\mathbf{algopt}}\left[0\right]=1.0$.
If ${\mathbf{algopt}}\left[0\right]=2.0$, then ${\mathbf{algopt}}\left[\mathit{i}-1\right]$, for $\mathit{i}=2,3,4$ are not used.
${\mathbf{algopt}}\left[1\right]$
Specifies the maximum order of the BDF integration formula to be used. ${\mathbf{algopt}}\left[1\right]$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is ${\mathbf{algopt}}\left[1\right]=5.0$.
${\mathbf{algopt}}\left[2\right]$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If ${\mathbf{algopt}}\left[2\right]=1.0$ a modified Newton iteration is used and if ${\mathbf{algopt}}\left[2\right]=2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, then there is an automatic switch to the modified Newton iteration. The default value is ${\mathbf{algopt}}\left[2\right]=1.0$.
${\mathbf{algopt}}\left[3\right]$
Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as ${P}_{i,\mathit{j}}=0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, for some $i$ or when there is no ${\stackrel{.}{V}}_{i}\left(t\right)$ dependence in the coupled ODE system. If ${\mathbf{algopt}}\left[3\right]=1.0$, then the Petzold test is used. If ${\mathbf{algopt}}\left[3\right]=2.0$, then the Petzold test is not used. The default value is ${\mathbf{algopt}}\left[3\right]=1.0$.
If ${\mathbf{algopt}}\left[0\right]=1.0$, then ${\mathbf{algopt}}\left[\mathit{i}-1\right]$, for $\mathit{i}=5,6,7$, are not used.
${\mathbf{algopt}}\left[4\right]$
Specifies the value of Theta to be used in the Theta integration method. $0.51\le {\mathbf{algopt}}\left[4\right]\le 0.99$. The default value is ${\mathbf{algopt}}\left[4\right]=0.55$.
${\mathbf{algopt}}\left[5\right]$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If ${\mathbf{algopt}}\left[5\right]=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left[5\right]=2.0$, a functional iteration method is used. The default value is ${\mathbf{algopt}}\left[5\right]=1.0$.
${\mathbf{algopt}}\left[6\right]$
Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If ${\mathbf{algopt}}\left[6\right]=1.0$, then switching is allowed and if ${\mathbf{algopt}}\left[6\right]=2.0$, then switching is not allowed. The default value is ${\mathbf{algopt}}\left[6\right]=1.0$.
${\mathbf{algopt}}\left[10\right]$
Specifies a point in the time direction, ${t}_{\mathrm{crit}}$, beyond which integration must not be attempted. The use of ${t}_{\mathrm{crit}}$ is described under the argument itask. If ${\mathbf{algopt}}\left[0\right]\ne 0.0$, a value of $0.0$ for ${\mathbf{algopt}}\left[10\right]$, say, should be specified even if itask subsequently specifies that ${t}_{\mathrm{crit}}$ will not be used.
${\mathbf{algopt}}\left[11\right]$
Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left[11\right]$ should be set to $0.0$.
${\mathbf{algopt}}\left[12\right]$
Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left[12\right]$ should be set to $0.0$.
${\mathbf{algopt}}\left[13\right]$
Specifies the initial step size to be attempted by the integrator. If ${\mathbf{algopt}}\left[13\right]=0.0$, then the initial step size is calculated internally.
${\mathbf{algopt}}\left[14\right]$
Specifies the maximum number of steps to be attempted by the integrator in any one call. If ${\mathbf{algopt}}\left[14\right]=0.0$, then no limit is imposed.
${\mathbf{algopt}}\left[22\right]$
Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of $U$, ${U}_{t}$, $V$ and $\stackrel{.}{V}$. If ${\mathbf{algopt}}\left[22\right]=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left[22\right]=2.0$, functional iteration is used. The default value is ${\mathbf{algopt}}\left[22\right]=1.0$.
${\mathbf{algopt}}\left[28\right]$ and ${\mathbf{algopt}}\left[29\right]$ are used only for the sparse matrix algebra option, ${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$.
${\mathbf{algopt}}\left[28\right]$
Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<{\mathbf{algopt}}\left[28\right]<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If ${\mathbf{algopt}}\left[28\right]$ lies outside this range then the default value is used. If the functions regard the Jacobian matrix as numerically singular then increasing ${\mathbf{algopt}}\left[28\right]$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is ${\mathbf{algopt}}\left[28\right]=0.1$.
${\mathbf{algopt}}\left[29\right]$
Is used as a relative pivot threshold during subsequent Jacobian decompositions (see ${\mathbf{algopt}}\left[28\right]$) below which an internal error is invoked. If ${\mathbf{algopt}}\left[29\right]$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian is found to be numerically singular (see ${\mathbf{algopt}}\left[28\right]$). The default value is ${\mathbf{algopt}}\left[29\right]=0.0001$.
22:   remeshNag_BooleanInput
On entry: indicates whether or not spatial remeshing should be performed.
${\mathbf{remesh}}=\mathrm{Nag_TRUE}$
Indicates that spatial remeshing should be performed as specified.
${\mathbf{remesh}}=\mathrm{Nag_FALSE}$
Indicates that spatial remeshing should be suppressed.
Note:  remesh should not be changed between consecutive calls to nag_pde_parab_1d_fd_ode_remesh (d03ppc). Remeshing can be switched off or on at specified times by using appropriate values for the arguments nrmesh and trmesh at each call.
23:   nxfixIntegerInput
On entry: the number of fixed mesh points.
Constraint: $0\le {\mathbf{nxfix}}\le {\mathbf{npts}}-2$
Note: the end points ${\mathbf{x}}\left[0\right]$ and ${\mathbf{x}}\left[{\mathbf{npts}}-1\right]$ are fixed automatically and hence should not be specified as fixed points.
24:   xfix[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array xfix must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nxfix}}\right)$.
On entry: ${\mathbf{xfix}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxfix}}$, must contain the value of the $x$ coordinate at the $\mathit{i}$th fixed mesh point.
Constraints:
• ${\mathbf{xfix}}\left[\mathit{i}-1\right]<{\mathbf{xfix}}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxfix}}-1$;
• each fixed mesh point must coincide with a user-supplied initial mesh point, that is ${\mathbf{xfix}}\left[\mathit{i}-1\right]={\mathbf{x}}\left[\mathit{j}-1\right]$ for some $j$, $2\le j\le {\mathbf{npts}}-1$
Note: the positions of the fixed mesh points in the array x remain fixed during remeshing, and so the number of mesh points between adjacent fixed points (or between fixed points and end points) does not change. You should take this into account when choosing the initial mesh distribution.
.
25:   nrmeshIntegerInput
On entry: specifies the spatial remeshing frequency and criteria for the calculation and adoption of a new mesh.
${\mathbf{nrmesh}}<0$
Indicates that a new mesh is adopted according to the argument dxmesh. The mesh is tested every $\left|{\mathbf{nrmesh}}\right|$ timesteps.
${\mathbf{nrmesh}}=0$
Indicates that remeshing should take place just once at the end of the first time step reached when $t>{\mathbf{trmesh}}$.
${\mathbf{nrmesh}}>0$
Indicates that remeshing will take place every nrmesh time steps, with no testing using dxmesh.
Note: nrmesh may be changed between consecutive calls to nag_pde_parab_1d_fd_ode_remesh (d03ppc) to give greater flexibility over the times of remeshing.
26:   dxmeshdoubleInput
On entry: determines whether a new mesh is adopted when nrmesh is set less than zero. A possible new mesh is calculated at the end of every $\left|{\mathbf{nrmesh}}\right|$ time steps, but is adopted only if
 $xinew>xi old +dxmesh×xi+1 old -xi old$
or
 $xinew
dxmesh thus imposes a lower limit on the difference between one mesh and the next.
Constraint: ${\mathbf{dxmesh}}\ge 0.0$.
27:   trmeshdoubleInput
On entry: specifies when remeshing will take place when nrmesh is set to zero. Remeshing will occur just once at the end of the first time step reached when $t$ is greater than trmesh.
Note: trmesh may be changed between consecutive calls to nag_pde_parab_1d_fd_ode_remesh (d03ppc) to force remeshing at several specified times.
28:   ipminfIntegerInput
On entry: the level of trace information regarding the adaptive remeshing.
${\mathbf{ipminf}}=0$
No trace information.
${\mathbf{ipminf}}=1$
Brief summary of mesh characteristics.
${\mathbf{ipminf}}=2$
More detailed information, including old and new mesh points, mesh sizes and monitor function values.
Constraint: ${\mathbf{ipminf}}=0$, $1$ or $2$.
29:   xratiodoubleInput
On entry: an input bound on the adjacent mesh ratio (greater than $1.0$ and typically in the range $1.5$ to $3.0$). The remeshing functions will attempt to ensure that
 $xi-xi-1/xratio
Suggested value: ${\mathbf{xratio}}=1.5$.
Constraint: ${\mathbf{xratio}}>1.0$.
30:   condoubleInput
On entry: an input bound on the sub-integral of the monitor function ${F}^{\mathrm{mon}}\left(x\right)$ over each space step. The remeshing functions will attempt to ensure that
 $∫xixi+1Fmonxdx≤con∫x1xnptsFmonxdx,$
(see Furzeland (1984)). con gives you more control over the mesh distribution e.g., decreasing con allows more clustering. A typical value is $2/\left({\mathbf{npts}}-1\right)$, but you are encouraged to experiment with different values. Its value is not critical and the mesh should be qualitatively correct for all values in the range given below.
Suggested value: ${\mathbf{con}}=2.0/\left({\mathbf{npts}}-1\right)$.
Constraint: $0.1/\left({\mathbf{npts}}-1\right)\le {\mathbf{con}}\le 10.0/\left({\mathbf{npts}}-1\right)$.
31:   monitffunction, supplied by the userExternal Function
monitf must supply and evaluate a remesh monitor function to indicate the solution behaviour of interest.
If you specify ${\mathbf{remesh}}=\mathrm{Nag_FALSE}$, i.e., no remeshing, then monitf will not be called and may be specified as NULLFN.
The specification of monitf is:
 void monitf (double t, Integer npts, Integer npde, const double x[], const double u[], const double r[], double fmon[], Nag_Comm *comm)
1:     tdoubleInput
On entry: the current value of the independent variable $t$.
2:     nptsIntegerInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
3:     npdeIntegerInput
On entry: the number of PDEs in the system.
4:     x[npts]const doubleInput
On entry: the current mesh. ${\mathbf{x}}\left[\mathit{i}-1\right]$ contains the value of ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
5:     u[${\mathbf{npde}}×{\mathbf{npts}}$]const doubleInput
On entry: ${\mathbf{u}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of ${U}_{\mathit{i}}\left(x,t\right)$ at $x={\mathbf{x}}\left[\mathit{j}-1\right]$ and time $t$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
6:     r[${\mathbf{npde}}×{\mathbf{npts}}$]const doubleInput
On entry: ${\mathbf{r}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of ${R}_{\mathit{i}}\left(x,t,U,{U}_{x},V\right)$ at $x={\mathbf{x}}\left[\mathit{j}-1\right]$ and time $t$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
7:     fmon[npts]doubleOutput
On exit: ${\mathbf{fmon}}\left[i-1\right]$ must contain the value of the monitor function ${F}^{\mathrm{mon}}\left(x\right)$ at mesh point $x={\mathbf{x}}\left[i-1\right]$.
Constraint: ${\mathbf{fmon}}\left[i-1\right]\ge 0.0$.
8:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to monitf.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_fd_ode_remesh (d03ppc) you may allocate memory and initialize these pointers with various quantities for use by monitf when called from nag_pde_parab_1d_fd_ode_remesh (d03ppc) (see Section 3.2.1 in the Essential Introduction).
32:   rsave[lrsave]doubleCommunication Array
If ${\mathbf{ind}}=0$, rsave need not be set on entry.
If ${\mathbf{ind}}=1$, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
33:   lrsaveIntegerInput
On entry: the dimension of the array rsave. Its size depends on the type of matrix algebra selected.
If ${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$, ${\mathbf{lrsave}}\ge {\mathbf{neqn}}×{\mathbf{neqn}}+{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\mathrm{Nag_LinAlgBand}$, ${\mathbf{lrsave}}\ge \left(3×\mathit{mlu}+1\right)×{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$, ${\mathbf{lrsave}}\ge 4×{\mathbf{neqn}}+11×{\mathbf{neqn}}/2+1+\mathit{nwkres}+\mathit{lenode}$.
Where
 $\mathit{mlu}$ is the lower or upper half bandwidths such that $\mathit{mlu}=2×{\mathbf{npde}}-1$, for PDE problems only; or $\mathit{mlu}={\mathbf{neqn}}-1$, for coupled PDE/ODE problems. $\mathit{nwkres}=\left\{\begin{array}{ll}{\mathbf{npde}}×\left(3×{\mathbf{npde}}+6×{\mathbf{nxi}}+{\mathbf{npts}}+15\right)+{\mathbf{nxi}}+{\mathbf{ncode}}+7×{\mathbf{npts}}+{\mathbf{nxfix}}+1\text{,}& \text{when ​}{\mathbf{ncode}}>0\text{​ and ​}{\mathbf{nxi}}>0\text{;}\\ {\mathbf{npde}}×\left(3×{\mathbf{npde}}+{\mathbf{npts}}+21\right)+{\mathbf{ncode}}+7×{\mathbf{npts}}+{\mathbf{nxfix}}+2\text{,}& \text{when ​}{\mathbf{ncode}}>0\text{​ and ​}{\mathbf{nxi}}=0\text{; or}\\ {\mathbf{npde}}×\left(3×{\mathbf{npde}}+{\mathbf{npts}}+21\right)+7×{\mathbf{npts}}+{\mathbf{nxfix}}+3\text{,}& \text{when ​}{\mathbf{ncode}}=0\text{.}\end{array}\right\$ $\mathit{lenode}=\left\{\begin{array}{cc}\left(6+\mathrm{int}\left({\mathbf{algopt}}\left[1\right]\right)\right)×{\mathbf{neqn}}+50\text{,}& \text{when the BDF method is used;}\\ 9×{\mathbf{neqn}}+50\text{,}& \text{when the Theta method is used.}\end{array}\right\$
Note: when using the sparse option, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if ${\mathbf{itrace}}>0$ and the function returns with NE_INT_2.
34:   isave[lisave]IntegerCommunication Array
If ${\mathbf{ind}}=0$, isave need not be set on entry.
If ${\mathbf{ind}}=1$, isave must be unchanged from the previous call to the function because it contains required information about the iteration required for subsequent calls. In particular:
${\mathbf{isave}}\left[0\right]$
Contains the number of steps taken in time.
${\mathbf{isave}}\left[1\right]$
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
${\mathbf{isave}}\left[2\right]$
Contains the number of Jacobian evaluations performed by the time integrator.
${\mathbf{isave}}\left[3\right]$
Contains the order of the ODE method last used in the time integration.
${\mathbf{isave}}\left[4\right]$
Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.
The rest of the array is used as workspace.
35:   lisaveIntegerInput
On entry: the dimension of the array isave.
Its size depends on the type of matrix algebra selected:
• if ${\mathbf{laopt}}=\mathrm{Nag_LinAlgBand}$, ${\mathbf{lisave}}\ge {\mathbf{neqn}}+25+{\mathbf{nxfix}}$;
• if ${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$, ${\mathbf{lisave}}\ge 25+{\mathbf{nxfix}}$;
• if ${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$, ${\mathbf{lisave}}\ge 25×{\mathbf{neqn}}+25+{\mathbf{nxfix}}$.
Note:  when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed if ${\mathbf{itrace}}>0$ and the function returns with NE_INT_2.
On entry: specifies the task to be performed by the ODE integrator.
${\mathbf{itask}}=1$
Normal computation of output values u at $t={\mathbf{tout}}$.
${\mathbf{itask}}=2$
One step and return.
${\mathbf{itask}}=3$
Stop at first internal integration point at or beyond $t={\mathbf{tout}}$.
${\mathbf{itask}}=4$
Normal computation of output values u at $t={\mathbf{tout}}$ but without overshooting $t={t}_{\mathrm{crit}}$ where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
${\mathbf{itask}}=5$
Take one step in the time direction and return, without passing ${t}_{\mathrm{crit}}$, where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
Constraint: ${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
37:   itraceIntegerInput
On entry: the level of trace information required from nag_pde_parab_1d_fd_ode_remesh (d03ppc) and the underlying ODE solver:
${\mathbf{itrace}}\le -1$
No output is generated.
${\mathbf{itrace}}=0$
Only warning messages from the PDE solver are printed.
${\mathbf{itrace}}=1$
Output from the underlying ODE solver is printed. This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
${\mathbf{itrace}}=2$
Output from the underlying ODE solver is similar to that produced when ${\mathbf{itrace}}=1$, except that the advisory messages are given in greater detail.
${\mathbf{itrace}}\ge 3$
Output from the underlying ODE solver is similar to that produced when ${\mathbf{itrace}}=2$, except that the advisory messages are given in greater detail.
38:   outfileconst char *Input
On entry: the name of a file to which diagnostic output will be directed. If outfile is NULL the diagnostic output will be directed to standard output.
39:   indInteger *Input/Output
On entry: must be set to $0$ or $1$.
${\mathbf{ind}}=0$
Starts or restarts the integration in time.
${\mathbf{ind}}=1$
Continues the integration after an earlier exit from the function. In this case, only the arguments tout and fail and the remeshing arguments nrmesh, dxmesh, trmesh, xratio and con may be reset between calls to nag_pde_parab_1d_fd_ode_remesh (d03ppc).
Constraint: $0\le {\mathbf{ind}}\le 1$.
On exit: ${\mathbf{ind}}=1$.
40:   commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
41:   savedNag_D03_Save *Communication Structure
saved must remain unchanged following a previous call to a Chapter d03 function and prior to any subsequent call to a Chapter d03 function.
42:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ACC_IN_DOUBT
Integration completed, but small changes in atol or rtol are unlikely to result in a changed solution.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_FAILED_DERIV
In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting ${\mathbf{ires}}=3$ in pdedef or bndary.
NE_FAILED_START
atol and rtol were too small to start integration.
Underlying ODE solver cannot make further progress from the point ts with the supplied values of atol and rtol. ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.
NE_FAILED_STEP
Error during Jacobian formulation for ODE system. Increase itrace for further details.
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts: ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.
NE_INCOMPAT_PARAM
On entry, ${\mathbf{con}}=〈\mathit{\text{value}}〉$, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{con}}\le 10.0/\left({\mathbf{npts}}-1\right)$.
On entry, ${\mathbf{con}}=〈\mathit{\text{value}}〉$, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{con}}\ge 0.1/\left({\mathbf{npts}}-1\right)$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[0\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\le 0$ or ${\mathbf{x}}\left[0\right]\ge 0.0$
On entry, the point ${\mathbf{xfix}}\left[i-1\right]$ does not coincide with any ${\mathbf{x}}\left[j-1\right]$: $i=〈\mathit{\text{value}}〉$ and ${\mathbf{xfix}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
NE_INT
ires set to an invalid value in call to pdedef, bndary, or odedef.
On entry, ${\mathbf{ind}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On entry, ${\mathbf{ipminf}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ipminf}}=0$, $1$ or $2$.
On entry, ${\mathbf{itask}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
On entry, ${\mathbf{itol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{itol}}=1$, $2$, $3$ or $4$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}=0$, $1$ or $2$.
On entry, ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncode}}\ge 0$.
On entry, ${\mathbf{npde}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npde}}\ge 1$.
On entry, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npts}}\ge 3$.
On entry, ${\mathbf{nxfix}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nxfix}}\ge 0$.
NE_INT_2
On entry, corresponding elements ${\mathbf{atol}}\left[i-1\right]$ and ${\mathbf{rtol}}\left[j-1\right]$ are both zero: $i=〈\mathit{\text{value}}〉$ and $j=〈\mathit{\text{value}}〉$.
On entry, lisave is too small: ${\mathbf{lisave}}=〈\mathit{\text{value}}〉$. Minimum possible dimension: $〈\mathit{\text{value}}〉$.
On entry, lrsave is too small: ${\mathbf{lrsave}}=〈\mathit{\text{value}}〉$. Minimum possible dimension: $〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nxi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nxi}}=0$ when ${\mathbf{ncode}}=0$.
On entry, ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nxi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nxi}}\ge 0$ when ${\mathbf{ncode}}>0$.
On entry, ${\mathbf{nxfix}}=〈\mathit{\text{value}}〉$, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nxfix}}\le {\mathbf{npts}}-2$.
When using the sparse option lisave or lrsave is too small: ${\mathbf{lisave}}=〈\mathit{\text{value}}〉$, ${\mathbf{lrsave}}=〈\mathit{\text{value}}〉$.
NE_INT_4
On entry, ${\mathbf{neqn}}=〈\mathit{\text{value}}〉$, ${\mathbf{npde}}=〈\mathit{\text{value}}〉$, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{neqn}}={\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{ncode}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
Serious error in internal call to an auxiliary. Increase itrace for further details.
NE_ITER_FAIL
In solving ODE system, the maximum number of steps ${\mathbf{algopt}}\left[14\right]$ has been exceeded. ${\mathbf{algopt}}\left[14\right]=〈\mathit{\text{value}}〉$.
NE_NOT_CLOSE_FILE
Cannot close file $〈\mathit{\text{value}}〉$.
NE_NOT_STRICTLY_INCREASING
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{xfix}}\left[i\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{xfix}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xfix}}\left[i\right]>{\mathbf{xfix}}\left[i-1\right]$.
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{xi}}\left[i\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{xi}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xi}}\left[i\right]>{\mathbf{xi}}\left[i-1\right]$.
On entry, mesh points x appear to be badly ordered: $i=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left[i-1\right]=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[j-1\right]=〈\mathit{\text{value}}〉$.
NE_NOT_WRITE_FILE
Cannot open file $〈\mathit{\text{value}}〉$ for writing.
NE_REAL
On entry, ${\mathbf{dxmesh}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{dxmesh}}\ge 0.0$.
On entry, ${\mathbf{xratio}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xratio}}>1.0$.
NE_REAL_2
On entry, at least one point in xi lies outside $\left[{\mathbf{x}}\left[0\right],{\mathbf{x}}\left[{\mathbf{npts}}-1\right]\right]$: ${\mathbf{x}}\left[0\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[{\mathbf{npts}}-1\right]=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{tout}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tout}}>{\mathbf{ts}}$.
On entry, ${\mathbf{tout}}-{\mathbf{ts}}$ is too small: ${\mathbf{tout}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.
NE_REAL_ARRAY
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{atol}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{atol}}\left[i-1\right]\ge 0.0$.
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{rtol}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rtol}}\left[i-1\right]\ge 0.0$.
NE_REMESH_CHANGED
remesh has been changed between calls to nag_pde_parab_1d_fd_ode_remesh (d03ppc).
NE_SING_JAC
Singular Jacobian of ODE system. Check problem formulation.
NE_TIME_DERIV_DEP
Flux function appears to depend on time derivatives.
NE_USER_STOP
In evaluating residual of ODE system, ${\mathbf{ires}}=2$ has been set in pdedef, bndary, or odedef. Integration is successful as far as ts: ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.
NE_ZERO_WTS
Zero error weights encountered during time integration.

## 7  Accuracy

nag_pde_parab_1d_fd_ode_remesh (d03ppc) controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the accuracy arguments, atol and rtol.

The argument specification allows you to include equations with only first-order derivatives in the space direction but there is no guarantee that the method of integration will be satisfactory for such systems. The position and nature of the boundary conditions in particular are critical in defining a stable problem. It may be advisable in such cases to reduce the whole system to first-order and to use the Keller box scheme function nag_pde_parab_1d_keller_ode_remesh (d03prc).
The time taken depends on the complexity of the parabolic system, the accuracy requested, and the frequency of the mesh updates. For a given system with fixed accuracy and mesh-update frequency it is approximately proportional to neqn.

## 9  Example

This example uses Burgers Equation, a common test problem for remeshing algorithms, given by
 $∂U ∂t =-U ∂U ∂x +E ∂2U ∂x2 ,$
for $x\in \left[0,1\right]$ and $t\in \left[0,1\right]$, where $E$ is a small constant.
The initial and boundary conditions are given by the exact solution
 $Ux,t=0.1exp-A+0.5exp-B+exp-C exp-A+exp-B+exp-C ,$
where
 $A = 50Ex- 0.5+ 4.95t, B = 250Ex- 0.5+ 0.75t, C = 500Ex- 0.375.$

### 9.1  Program Text

Program Text (d03ppce.c)

None.

### 9.3  Program Results

Program Results (d03ppce.r)