NAG CL Interface
d03prc (dim1_​parab_​remesh_​keller)

1 Purpose

d03prc integrates a system of linear or nonlinear, first-order, time-dependent 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 the Keller box scheme (see Keller (1970)) 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 <nag.h>
void  d03prc (Integer npde, double *ts, double tout,
void (*pdedef)(Integer npde, double t, double x, const double u[], const double ut[], const double ux[], Integer nv, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm),
void (*bndary)(Integer npde, double t, Integer ibnd, Integer nobc, const double u[], const double ut[], Integer nv, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm),
void (*uvinit)(Integer npde, Integer npts, Integer nxi, const double x[], const double xi[], double u[], Integer nv, double v[], Nag_Comm *comm),
double u[], Integer npts, double x[], Integer nleft, Integer nv,
void (*odedef)(Integer npde, double t, Integer nv, const double v[], const double vdot[], Integer nxi, const double xi[], const double ucp[], const double ucpx[], const double ucpt[], double r[], 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[], 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)
The function may be called by the names: d03prc, nag_pde_dim1_parab_remesh_keller or nag_pde_parab_1d_keller_ode_remesh.

3 Description

d03prc integrates the system of first-order PDEs and coupled ODEs given by the master equations:
Gi x,t,U,Ux,Ut,V,V. = 0 ,   i=1,2,,npde ,   axb,tt0, (1)
Rit,V,V.,ξ,U*,Ux*,Ut*=0,  i=1,2,,nv. (2)
In the PDE part of the problem given by (1), the functions Gi must have the general form
Gi=j=1npdePi,j Uj t +j=1nvQi,jV.j+Si=0,  i=1,2,,npde, (3)
where Pi,j, Qi,j and Si depend on x, t, U, Ux and V.
The vector U is the set of PDE solution values
Ux,t=U1x,t,,Unpdex,tT,  
and the vector Ux is the partial derivative with respect to x. The vector V is the set of ODE solution values
Vt=V1t,,VnvtT,  
and V. denotes its derivative with respect to time.
In the ODE part given by (2), ξ represents a vector of nξ 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*, Ux* and Ut* are the functions U, Ux and Ut evaluated at these coupling points. Each Ri may only depend linearly on time derivatives. Hence equation (2) may be written more precisely as
R=A-BV.-CUt*, (4)
where R=R1,,RnvT, A is a vector of length nv, B is an nv by nv matrix, C is an nv by nξ×npde matrix and the entries in A, B and C may depend on t, ξ, U*, Ux* and V. In practice you only need to supply a vector of information to define the ODEs and not the matrices B and C. (See Section 5 for the specification of odedef.)
The integration in time is from t0 to tout, over the space interval axb, where a=x1 and b=xnpts are the leftmost and rightmost points of a mesh x1,x2,,xnpts defined initially by you and (possibly) adapted automatically during the integration according to user-specified criteria.
The PDE system which is defined by the functions Gi must be specified in pdedef.
The initial t=t0 values of the functions Ux,t and Vt must be specified in uvinit. Note that uvinit will be called again following any remeshing, and so Ux,t0 should be specified for all values of x in the interval axb, and not just the initial mesh points.
For a first-order system of PDEs, only one boundary condition is required for each PDE component Ui. The npde boundary conditions are separated into na at the left-hand boundary x=a, and nb at the right-hand boundary x=b, such that na+nb=npde. The position of the boundary condition for each component should be chosen with care; the general rule is that if the characteristic direction of Ui at the left-hand boundary (say) points into the interior of the solution domain, then the boundary condition for Ui should be specified at the left-hand boundary. Incorrect positioning of boundary conditions generally results in initialization or integration difficulties in the underlying time integration functions.
The boundary conditions have the master equation form:
GiL x,t,U,Ut,V,V. = 0   at ​ x = a ,   i=1,2,,na , (5)
at the left-hand boundary, and
GiR x,t,U,Ut,V,V. = 0   at ​ x = b ,   i=1,2,,nb , (6)
at the right-hand boundary.
Note that the functions GiL and GiR must not depend on Ux, since spatial derivatives are not determined explicitly in the Keller box scheme functions. If the problem involves derivative (Neumann) boundary conditions then it is generally possible to restate such boundary conditions in terms of permissible variables. Also note that GiL and GiR must be linear with respect to time derivatives, so that the boundary conditions have the general form:
j=1 npde E i,j L Uj t + j=1 nv H i,j L V.j + KiL = 0 ,   i=1,2,,na , (7)
at the left-hand boundary, and
j=1 npde E i,j R Uj t + j=1 nv H i,j R V.j + KiR = 0 ,   i=1,2,,nb , (8)
at the right-hand boundary, where Ei,jL, Ei,jR, Hi,jL, Hi,jR, KiL and KiR depend on x,t,U and V only.
The boundary conditions must be specified in bndary.
The problem is subject to the following restrictions:
  1. (i)Pi,j, Qi,j and Si must not depend on any time derivatives;
  2. (ii)t0<tout, so that integration is in the forward direction;
  3. (iii) The evaluation of the function Gi is done approximately at the mid-points of the mesh x[i-1], for i=1,2,,npts, by calling pdedef for each mid-point in turn. Any discontinuities in the function must therefore be at one or more of the fixed mesh points specified by xfix;
  4. (iv)At least one of the functions Pi,j must be nonzero so that there is a time derivative present in the PDE problem.
The algebraic-differential equation system which is defined by the functions Ri must be specified in odedef. You must also specify the coupling points ξ in the array xi.
The first-order equations are approximated by a system of ODEs in time for the values of Ui at mesh points. In this method of lines approach the Keller box scheme is applied to each PDE in the space variable only, resulting in a system of ODEs in time for the values of Ui at each mesh point. In total there are npde×npts+nv 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 monitf which specifies in an analytic or numeric 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 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 along 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
Keller H B (1970) A new difference scheme for parabolic problems Numerical Solutions of Partial Differential Equations (ed J Bramble) 2 327–350 Academic Press
Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw. 20 63–99

5 Arguments

1: npde Integer Input
On entry: the number of PDEs to be solved.
Constraint: npde1.
2: ts double * Input/Output
On entry: the initial value of the independent variable t.
Constraint: ts<tout.
On exit: the value of t corresponding to the solution values in u. Normally ts=tout.
3: tout double Input
On entry: the final value of t to which the integration is to be carried out.
4: pdedef function, supplied by the user External Function
pdedef must evaluate the functions Gi which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by d03prc.
The specification of pdedef is:
void  pdedef (Integer npde, double t, double x, const double u[], const double ut[], const double ux[], Integer nv, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm)
1: npde Integer Input
On entry: the number of PDEs in the system.
2: t double Input
On entry: the current value of the independent variable t.
3: x double Input
On entry: the current value of the space variable x.
4: u[npde] const double Input
On entry: u[i-1] contains the value of the component Uix,t, for i=1,2,,npde.
5: ut[npde] const double Input
On entry: ut[i-1] contains the value of the component Uix,t t , for i=1,2,,npde.
6: ux[npde] const double Input
On entry: ux[i-1] contains the value of the component Uix,t x , for i=1,2,,npde.
7: nv Integer Input
On entry: the number of coupled ODEs in the system.
8: v[nv] const double Input
On entry: if nv>0, v[i-1] contains the value of the component Vit, for i=1,2,,nv.
9: vdot[nv] const double Input
On entry: if nv>0, vdot[i-1] contains the value of component V.it, for i=1,2,,nv.
10: res[npde] double Output
On exit: res[i-1] must contain the ith component of G, for i=1,2,,npde, where G is defined as
Gi=j=1npdePi,j Uj t +j=1nvQi,jV.j, (9)
i.e., only terms depending explicitly on time derivatives, or
Gi=j=1npdePi,j Uj t +j=1nvQi,jV.j+Si, (10)
i.e., all terms in equation (3).
The definition of G is determined by the input value of ires.
11: ires Integer * Input/Output
On entry: the form of Gi that must be returned in the array res.
ires=-1
Equation (9) must be used.
ires=1
Equation (10) must be used.
On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions, as described below:
ires=2
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to fail.code= NE_USER_STOP.
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 ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires=3, d03prc returns to the calling function with the error indicator set to fail.code= NE_FAILED_DERIV.
12: comm Nag_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 d03prc you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from d03prc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03prc. If your code inadvertently does return any NaNs or infinities, d03prc is likely to produce unexpected results.
5: bndary function, supplied by the user External Function
bndary must evaluate the functions GiL and GiR which describe the boundary conditions, as given in (5) and (6).
The specification of bndary is:
void  bndary (Integer npde, double t, Integer ibnd, Integer nobc, const double u[], const double ut[], Integer nv, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm)
1: npde Integer Input
On entry: the number of PDEs in the system.
2: t double Input
On entry: the current value of the independent variable t.
3: ibnd Integer Input
On entry: specifies which boundary conditions are to be evaluated.
ibnd=0
bndary must compute the left-hand boundary condition at x=a.
ibnd0
bndary must compute of the right-hand boundary condition at x=b.
4: nobc Integer Input
On entry: specifies the number na of boundary conditions at the boundary specified by ibnd.
5: u[npde] const double Input
On entry: u[i-1] contains the value of the component Uix,t at the boundary specified by ibnd, for i=1,2,,npde.
6: ut[npde] const double Input
On entry: ut[i-1] contains the value of the component Uix,t t , for i=1,2,,npde.
7: nv Integer Input
On entry: the number of coupled ODEs in the system.
8: v[nv] const double Input
On entry: if nv>0, v[i-1] contains the value of the component Vit, for i=1,2,,nv.
9: vdot[nv] const double Input
On entry: if nv>0, vdot[i-1] contains the value of component V.it, for i=1,2,,nv.
Note: vdot[i-1], for i=1,2,,nv, may only appear linearly as in (11) and (12).
10: res[dim] double Output
On exit: res[i-1] must contain the ith component of GL or GR, depending on the value of ibnd, for i=1,2,,nobc, where GL is defined as
GiL=j=1npdeEi,jL Uj t +j=1nvHi,jLV.j, (11)
i.e., only terms depending explicitly on time derivatives, or
GiL=j=1npdeEi,jL Uj t +j=1nvHi,jLV.j+KiL, (12)
i.e., all terms in equation (7), and similarly for GiR.
The definitions of GL and GR are determined by the input value of ires.
11: ires Integer * Input/Output
On entry: the form of GiL (or GiR) that must be returned in the array res.
ires=-1
Equation (11) must be used.
ires=1
Equation (12) must be used.
On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:
ires=2
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to fail.code= NE_USER_STOP.
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 ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires=3, d03prc returns to the calling function with the error indicator set to fail.code= NE_FAILED_DERIV.
12: comm Nag_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 d03prc you may allocate memory and initialize these pointers with various quantities for use by bndary when called from d03prc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03prc. If your code inadvertently does return any NaNs or infinities, d03prc is likely to produce unexpected results.
6: uvinit function, supplied by the user External Function
uvinit must supply the initial t=t0 values of Ux,t and Vt for all values of x in the interval a,b.
The specification of uvinit is:
void  uvinit (Integer npde, Integer npts, Integer nxi, const double x[], const double xi[], double u[], Integer nv, double v[], Nag_Comm *comm)
1: npde Integer Input
On entry: the number of PDEs in the system.
2: npts Integer Input
On entry: the number of mesh points in the interval a,b.
3: nxi Integer Input
On entry: the number of ODE/PDE coupling points.
4: x[npts] const double Input
On entry: the current mesh. x[i-1] contains the value of xi, for i=1,2,,npts.
5: xi[nxi] const double Input
On entry: if nxi>0, xi[i-1] contains the ODE/PDE coupling point, ξi, for i=1,2,,nxi.
6: u[npde×npts] double Output
Note: the i,jth element of the matrix U is stored in u[j-1×npde+i-1].
On exit: u[j-1×npde+i-1] contains the value of the component Uixj,t0, for i=1,2,,npde and j=1,2,,npts.
7: nv Integer Input
On entry: the number of coupled ODEs in the system.
8: v[nv] double Output
On exit: if nv>0, v[i-1] must contain the value of component Vit0, for i=1,2,,nv.
9: comm Nag_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 d03prc you may allocate memory and initialize these pointers with various quantities for use by uvinit when called from d03prc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: uvinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03prc. If your code inadvertently does return any NaNs or infinities, d03prc is likely to produce unexpected results.
7: u[neqn] double Input/Output
On entry: if ind=1 the value of u must be unchanged from the previous call.
On exit: the computed solution Uixj,t, for i=1,2,,npde and j=1,2,,npts, and Vkt, for k=1,2,,nv, evaluated at t=ts, as follows:
  • u[npde×j-1+i-1] contain Uixj,t, for i=1,2,,npde and j=1,2,,npts, and
  • u[npts×npde+i-1] contain Vit, for i=1,2,,nv.
8: npts Integer Input
On entry: the number of mesh points in the interval [a,b].
Constraint: npts3.
9: x[npts] double Input/Output
On entry: the initial mesh points in the space direction. x[0] must specify the left-hand boundary, a, and x[npts-1] must specify the right-hand boundary, b.
Constraint: x[0]<x[1]<<x[npts-1].
On exit: the final values of the mesh points.
10: nleft Integer Input
On entry: the number na of boundary conditions at the left-hand mesh point x[0].
Constraint: 0nleftnpde.
11: nv Integer Input
On entry: the number of coupled ODE components.
Constraint: nv0.
12: odedef function, supplied by the user External Function
odedef must evaluate the functions R, which define the system of ODEs, as given in (4).
If nv=0, odedef will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to d03prc.
The specification of odedef is:
void  odedef (Integer npde, double t, Integer nv, const double v[], const double vdot[], Integer nxi, const double xi[], const double ucp[], const double ucpx[], const double ucpt[], double r[], Integer *ires, Nag_Comm *comm)
1: npde Integer Input
On entry: the number of PDEs in the system.
2: t double Input
On entry: the current value of the independent variable t.
3: nv Integer Input
On entry: the number of coupled ODEs in the system.
4: v[nv] const double Input
On entry: if nv>0, v[i-1] contains the value of the component Vit, for i=1,2,,nv.
5: vdot[nv] const double Input
On entry: if nv>0, vdot[i-1] contains the value of component V.it, for i=1,2,,nv.
6: nxi Integer Input
On entry: the number of ODE/PDE coupling points.
7: xi[nxi] const double Input
On entry: if nxi>0, xi[i-1] contains the ODE/PDE coupling point, ξi, for i=1,2,,nxi.
8: ucp[npde×nxi] const double Input
Note: the i,jth element of the matrix is stored in ucp[j-1×npde+i-1].
On entry: if nxi>0, ucp[j-1×npde+i-1] contains the value of Uix,t at the coupling point x=ξj, for i=1,2,,npde and j=1,2,,nxi.
9: ucpx[npde×nxi] const double Input
Note: the i,jth element of the matrix is stored in ucpx[j-1×npde+i-1].
On entry: if nxi>0, ucpx[j-1×npde+i-1] contains the value of Uix,t x at the coupling point x=ξj, for i=1,2,,npde and j=1,2,,nxi.
10: ucpt[npde×nxi] const double Input
Note: the i,jth element of the matrix is stored in ucpt[j-1×npde+i-1].
On entry: if nxi>0, ucpt[j-1×npde+i-1] contains the value of Ui t at the coupling point x=ξj, for i=1,2,,npde and j=1,2,,nxi.
11: r[nv] double Output
On exit: if nv>0, r[i-1] must contain the ith component of R, for i=1,2,,nv, where R is defined as
R=-BV.-CUt*, (13)
i.e., only terms depending explicitly on time derivatives, or
R=A-BV.-CUt*, (14)
i.e., all terms in equation (4). The definition of R is determined by the input value of ires.
12: ires Integer * Input/Output
On entry: the form of R that must be returned in the array r.
ires=-1
Equation (13) must be used.
ires=1
Equation (14) 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:
ires=2
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to fail.code= NE_USER_STOP.
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 ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires=3, d03prc returns to the calling function with the error indicator set to fail.code= NE_FAILED_DERIV.
13: comm Nag_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 d03prc you may allocate memory and initialize these pointers with various quantities for use by odedef when called from d03prc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03prc. If your code inadvertently does return any NaNs or infinities, d03prc is likely to produce unexpected results.
13: nxi Integer Input
On entry: the number of ODE/PDE coupling points.
Constraints:
  • if nv=0, nxi=0;
  • if nv>0, nxi0.
14: xi[nxi] const double Input
On entry: xi[i-1], for i=1,2,,nxi, must be set to the ODE/PDE coupling points, ξi.
Constraint: x[0]xi[0]<xi[1]<<xi[nxi-1]x[npts-1].
15: neqn Integer Input
On entry: the number of ODEs in the time direction.
Constraint: neqn=npde×npts+nv.
16: rtol[dim] const double Input
Note: the dimension, dim, of the array rtol must be at least
  • 1 when itol=1 or 2;
  • neqn when itol=3 or 4.
On entry: the relative local error tolerance.
Constraint: rtol[i-1]0.0 for all relevant i.
17: atol[dim] const double Input
Note: the dimension, dim, of the array atol must be at least
  • 1 when itol=1 or 3;
  • neqn when itol=2 or 4.
On entry: the absolute local error tolerance.
Constraint: atol[i-1]0.0 for all relevant i.
Note: corresponding elements of rtol and atol cannot both be 0.0.
18: itol Integer Input
A value to indicate the form of the local error test. itol indicates to d03prc whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is ei/wi<1.0, where wi is defined as follows:
On entry:
itol rtol atol wi
1 scalar scalar rtol[0]×u[i-1]+atol[0]
2 scalar vector rtol[0]×u[i-1]+atol[i-1]
3 vector scalar rtol[i-1]×u[i-1]+atol[0]
4 vector vector rtol[i-1]×u[i-1]+atol[i-1]
In the above, ei denotes the estimated local error for the ith component of the coupled PDE/ODE system in time, u[i-1], for i=1,2,,neqn.
The choice of norm used is defined by the argument norm.
Constraint: itol=1, 2, 3 or 4.
19: norm Nag_NormType Input
On entry: the type of norm to be used.
norm=Nag_MaxNorm
Maximum norm.
norm=Nag_TwoNorm
Averaged L2 norm.
If Unorm denotes the norm of the vector u of length neqn, then for the averaged L2 norm
Unorm= 1neqn i=1neqn Ui/wi 2 ,  
while for the maximum norm
Unorm=maxiu[i-1]/wi.  
See the description of itol for the formulation of the weight vector w.
Constraint: norm=Nag_MaxNorm or Nag_TwoNorm.
20: laopt Nag_LinAlgOption Input
On entry: the type of matrix algebra required.
laopt=Nag_LinAlgFull
Full matrix methods to be used.
laopt=Nag_LinAlgBand
Banded matrix methods to be used.
laopt=Nag_LinAlgSparse
Sparse matrix methods to be used.
Constraint: laopt=Nag_LinAlgFull, Nag_LinAlgBand or Nag_LinAlgSparse.
Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., nv=0).
21: algopt[30] const double Input
On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, algopt[0] 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:
algopt[0]
Selects the ODE integration method to be used. If algopt[0]=1.0, a BDF method is used and if algopt[0]=2.0, a Theta method is used. The default value is algopt[0]=1.0.
If algopt[0]=2.0, then algopt[i-1], for i=2,3,4, are not used.
algopt[1]
Specifies the maximum order of the BDF integration formula to be used. algopt[1] may be 1.0, 2.0, 3.0, 4.0 or 5.0. The default value is algopt[1]=5.0.
algopt[2]
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If algopt[2]=1.0 a modified Newton iteration is used and if algopt[2]=2.0 a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, there is an automatic switch to the modified Newton iteration. The default value is algopt[2]=1.0.
algopt[3]
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 Pi,j=0.0, for j=1,2,,npde, for some i or when there is no V.it dependence in the coupled ODE system. If algopt[3]=1.0, the Petzold test is used. If algopt[3]=2.0, the Petzold test is not used. The default value is algopt[3]=1.0.
If algopt[0]=1.0, algopt[i-1], for i=5,6,7, are not used.
algopt[4]
Specifies the value of Theta to be used in the Theta integration method. 0.51algopt[4]0.99. The default value is algopt[4]=0.55.
algopt[5]
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If algopt[5]=1.0, a modified Newton iteration is used and if algopt[5]=2.0, a functional iteration method is used. The default value is algopt[5]=1.0.
algopt[6]
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 algopt[6]=1.0, switching is allowed and if algopt[6]=2.0, switching is not allowed. The default value is algopt[6]=1.0.
algopt[10]
Specifies a point in the time direction, tcrit, beyond which integration must not be attempted. The use of tcrit is described under the argument itask. If algopt[0]0.0, a value of 0.0, for algopt[10], say, should be specified even if itask subsequently specifies that tcrit will not be used.
algopt[11]
Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, algopt[11] should be set to 0.0.
algopt[12]
Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, algopt[12] should be set to 0.0.
algopt[13]
Specifies the initial step size to be attempted by the integrator. If algopt[13]=0.0, the initial step size is calculated internally.
algopt[14]
Specifies the maximum number of steps to be attempted by the integrator in any one call. If algopt[14]=0.0, no limit is imposed.
algopt[22]
Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of U, Ut, V and V.. If algopt[22]=1.0, a modified Newton iteration is used and if algopt[22]=2.0, functional iteration is used. The default value is algopt[22]=1.0.
algopt[28] and algopt[29] are used only for the sparse matrix algebra option, i.e., laopt=Nag_LinAlgSparse.
algopt[28]
Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range 0.0<algopt[28]<1.0, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If algopt[28] lies outside this range then the default value is used. If the functions regard the Jacobian matrix as numerically singular then increasing algopt[28] towards 1.0 may help, but at the cost of increased fill-in. The default value is algopt[28]=0.1.
algopt[29]
Used as a relative pivot threshold during subsequent Jacobian decompositions (see algopt[28]) below which an internal error is invoked. algopt[29] must be greater than zero, otherwise the default value is used. If algopt[29] 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 algopt[28]). The default value is algopt[29]=0.0001.
22: remesh Nag_Boolean Input
On entry: indicates whether or not spatial remeshing should be performed.
remesh=Nag_TRUE
Indicates that spatial remeshing should be performed as specified.
remesh=Nag_FALSE
Indicates that spatial remeshing should be suppressed.
Note: remesh should not be changed between consecutive calls to d03prc. Remeshing can be switched off or on at specified times by using appropriate values for the arguments nrmesh and trmesh at each call.
23: nxfix Integer Input
On entry: the number of fixed mesh points.
Constraint: 0nxfixnpts-2.
Note: the end points x[0] and x[npts-1] are fixed automatically and hence should not be specified as fixed points.
24: xfix[nxfix] const double Input
On entry: xfix[i-1], for i=1,2,,nxfix, must contain the value of the x coordinate at the ith fixed mesh point.
Constraint: xfix[i-1]<xfix[i], for i=1,2,,nxfix-1, and each fixed mesh point must coincide with a user-supplied initial mesh point, that is xfix[i-1]=x[j-1] for some j, 2jnpts-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: nrmesh Integer Input
On entry: indicates the form of meshing to be performed.
nrmesh<0
Indicates that a new mesh is adopted according to the argument dxmesh. The mesh is tested every nrmesh timesteps.
nrmesh=0
Indicates that remeshing should take place just once at the end of the first time step reached when t>trmesh.
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 d03prc to give greater flexibility over the times of remeshing.
26: dxmesh double Input
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 nrmesh time steps, but is adopted only if
xinew>xiold+dxmesh×xi+1old-xiold,  
or
xinew<xiold-dxmesh×xiold-xi- 1old.  
dxmesh thus imposes a lower limit on the difference between one mesh and the next.
Constraint: dxmesh0.0.
27: trmesh double Input
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 d03prc to force remeshing at several specified times.
28: ipminf Integer Input
On entry: the level of trace information regarding the adaptive remeshing.
ipminf=0
No trace information.
ipminf=1
Brief summary of mesh characteristics.
ipminf=2
More detailed information, including old and new mesh points, mesh sizes and monitor function values.
Constraint: ipminf=0, 1 or 2.
29: xratio double Input
On entry: input bound on 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<xi+1-xi<xratio×xi-xi-1.  
Suggested value: xratio=1.5.
Constraint: xratio>1.0.
30: con double Input
On entry: an input bound on the sub-integral of the monitor function Fmonx over each space step. The remeshing functions will attempt to ensure that
x1xi+1Fmonxdxconx1xnptsFmonxdx,  
(see Furzeland (1984)). con gives you more control over the mesh distribution e.g., decreasing con allows more clustering. A typical value is 2/npts-1, 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: con=2.0/npts-1.
Constraint: 0.1/npts-1con10.0/npts-1.
31: monitf function, supplied by the user External Function
monitf must supply and evaluate a remesh monitor function to indicate the solution behaviour of interest.
If nv=0, monitf will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to d03prc.
The specification of monitf is:
void  monitf (double t, Integer npts, Integer npde, const double x[], const double u[], double fmon[], Nag_Comm *comm)
1: t double Input
On entry: the current value of the independent variable t.
2: npts Integer Input
On entry: the number of mesh points in the interval a,b.
3: npde Integer Input
On entry: the number of PDEs in the system.
4: x[npts] const double Input
On entry: the current mesh. x[i-1] contains the value of xi, for i=1,2,,npts.
5: u[npde×npts] const double Input
Note: the i,jth element of the matrix U is stored in u[j-1×npde+i-1].
On entry: u[j-1×npde+i-1] contains the value of Uix,t at x=x[j-1] and time t, for i=1,2,,npde and j=1,2,,npts.
6: fmon[npts] double Output
On exit: fmon[i-1] must contain the value of the monitor function Fmonx at mesh point x=x[i-1].
Constraint: fmon[i-1]0.0.
7: comm Nag_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 d03prc you may allocate memory and initialize these pointers with various quantities for use by monitf when called from d03prc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: monitf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03prc. If your code inadvertently does return any NaNs or infinities, d03prc is likely to produce unexpected results.
32: rsave[lrsave] double Communication Array
If ind=0, rsave need not be set on entry.
If ind=1, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
33: lrsave Integer Input
On entry: the dimension of the array rsave. Its size depends on the type of matrix algebra selected.
If laopt=Nag_LinAlgFull, lrsaveneqn×neqn+neqn+nwkres+lenode.
If laopt=Nag_LinAlgBand, lrsave3ml+mu+2×neqn+nwkres+lenode.
If laopt=Nag_LinAlgSparse, lrsave4neqn+11neqn/2+1+nwkres+lenode.
Where ml and mu are the lower and upper half bandwidths given by ml=npde+nleft-1 such that
for problems involving PDEs only,
mu=2npde-nleft-1;
for coupled PDE/ODE problems,
ml=mu=neqn-1.
Where nwkres is defined by
if nv>0​ and ​nxi>0,
nwkres=npde3npde+6nxi+npts+15+nxi+nv+7npts+nxfix+1;
if nv>0​ and ​nxi=0,
nwkres=npde3npde+npts+21+nv+7npts+nxfix+2;
if nv=0,
nwkres=npde3npde+npts+21+7npts+nxfix+3.
Where lenode is defined by
if the BDF method is used,
lenode=6+intalgopt[1]×neqn+50;
if the Theta method is used,
lenode=9neqn+50.
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 itrace>0 and the function returns with fail.code= NE_INT_2.
34: isave[lisave] Integer Communication Array
If ind=0, isave need not be set.
If ind=1, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular the following components of the array isave concern the efficiency of the integration:
isave[0]
Contains the number of steps taken in time.
isave[1]
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves evaluating the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
isave[2]
Contains the number of Jacobian evaluations performed by the time integrator.
isave[3]
Contains the order of the ODE method last used in the time integration.
isave[4]
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: lisave Integer Input
On entry: the dimension of the array isave. Its size depends on the type of matrix algebra selected:
  • if laopt=Nag_LinAlgFull, lisave25+nxfix;
  • if laopt=Nag_LinAlgBand, lisaveneqn+25+nxfix;
  • if laopt=Nag_LinAlgSparse, lisave25×neqn+25+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 itrace>0 and the function returns with fail.code= NE_INT_2.
36: itask Integer Input
On entry: the task to be performed by the ODE integrator.
itask=1
Normal computation of output values u at t=tout (by overshooting and interpolating).
itask=2
Take one step in the time direction and return.
itask=3
Stop at first internal integration point at or beyond t=tout.
itask=4
Normal computation of output values u at t=tout but without overshooting t=tcrit where tcrit is described under the argument algopt.
itask=5
Take one step in the time direction and return, without passing tcrit, where tcrit is described under the argument algopt.
Constraint: itask=1, 2, 3, 4 or 5.
37: itrace Integer Input
On entry: the level of trace information required from d03prc and the underlying ODE solver as follows:
itrace-1
No output is generated.
itrace=0
Only warning messages from the PDE solver are printed.
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.
itrace=2
Output from the underlying ODE solver is similar to that produced when itrace=1, except that the advisory messages are given in greater detail.
itrace3
The output from the underlying ODE solver is similar to that produced when itrace=2, except that the advisory messages are given in greater detail.
38: outfile const 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: ind Integer * Input/Output
On entry: indicates whether this is a continuation call or a new integration.
ind=0
Starts or restarts the integration in time.
ind=1
Continues the integration after an earlier exit from the function. In this case, only the argument tout and the remeshing arguments nrmesh, dxmesh, trmesh, xratio and con may be reset between calls to d03prc.
Constraint: ind=0 or 1.
On exit: ind=1.
40: comm Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
41: saved Nag_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: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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.
The required task has been completed, but it is estimated that a small change in atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when itask2 or 5.)
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument 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 ires=3 in pdedef or bndary.
NE_FAILED_START
atol and rtol were too small to start integration.
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: ts=value.
In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as t=ts. The problem may have a singularity, or the error requirement may be inappropriate. Incorrect positioning of boundary conditions may also result in this error.
Underlying ODE solver cannot make further progress from the point ts with the supplied values of atol and rtol. ts=value.
NE_INCOMPAT_PARAM
On entry, con=value, npts=value.
Constraint: con10.0/npts-1.
On entry, con=value, npts=value.
Constraint: con0.1/npts-1.
On entry, the point xfix[i-1] does not coincide with any x[j-1]: i=value and xfix[i-1]=value.
NE_INT
ires set to an invalid value in call to pdedef, bndary, or odedef.
On entry, ind=value.
Constraint: ind=0 or 1.
On entry, ipminf=value.
Constraint: ipminf=0, 1 or 2.
On entry, itask=value.
Constraint: itask=1, 2, 3, 4 or 5.
On entry, itol=value.
Constraint: itol=1, 2, 3 or 4.
On entry, nleft=value.
Constraint: nleft0.
On entry, npde=value.
Constraint: npde1.
On entry, npts=value.
Constraint: npts3.
On entry, nv=value.
Constraint: nv0.
On entry, nxfix=value.
Constraint: nxfix0.
On entry, on initial entry ind=1.
Constraint: on initial entry ind=0.
NE_INT_2
On entry, i=value and j=value.
Constraint: corresponding elements atol[i-1] and rtol[j-1] cannot both be 0.0.
On entry, lisave=value.
Constraint: lisavevalue.
On entry, lrsave=value.
Constraint: lrsavevalue.
On entry, nleft=value, npde=value.
Constraint: nleftnpde.
On entry, nv=value and nxi=value.
Constraint: nxi=0 when nv=0.
On entry, nv=value and nxi=value.
Constraint: nxi0 when nv>0.
On entry, nxfix=value, npts=value.
Constraint: nxfixnpts-2.
When using the sparse option lisave or lrsave is too small: lisave=value, lrsave=value.
NE_INT_4
On entry, neqn=value, npde=value, npts=value and nv=value.
Constraint: neqn=npde×npts+nv.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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 algopt[14] has been exceeded. algopt[14]=value.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_CLOSE_FILE
Cannot close file value.
NE_NOT_STRICTLY_INCREASING
On entry, i=value, xfix[i]=value and xfix[i-1]=value.
Constraint: xfix[i]>xfix[i-1].
On entry, i=value, x[i-1]=value, j=value and x[j-1]=value.
Constraint: x[0]<x[1]<<x[npts-1].
On entry, i=value, xi[i]=value and xi[i-1]=value.
Constraint: xi[i]>xi[i-1].
NE_NOT_WRITE_FILE
Cannot open file value for writing.
NE_REAL
On entry, dxmesh=value.
Constraint: dxmesh0.0.
On entry, xratio=value.
Constraint: xratio>1.0.
NE_REAL_2
On entry, at least one point in xi lies outside x[0],x[npts-1]: x[0]=value and x[npts-1]=value.
On entry, tout=value and ts=value.
Constraint: tout>ts.
On entry, tout-ts is too small: tout=value and ts=value.
NE_REAL_ARRAY
On entry, i=value and atol[i-1]=value.
Constraint: atol[i-1]0.0.
On entry, i=value and rtol[i-1]=value.
Constraint: rtol[i-1]0.0.
NE_REMESH_CHANGED
remesh has been changed between calls to d03prc.
NE_SING_JAC
Singular Jacobian of ODE system. Check problem formulation.
NE_USER_STOP
In evaluating residual of ODE system, ires=2 has been set in pdedef, bndary, or odedef. Integration is successful as far as ts: ts=value.
NE_ZERO_WTS
Zero error weights encountered during time integration.
Some error weights wi became zero during the time integration (see the description of itol). Pure relative error control atol[i-1]=0.0 was requested on a variable (the ith) which has become zero. The integration was successful as far as t=ts.

7 Accuracy

d03prc 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.

8 Parallelism and Performance

d03prc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03prc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The Keller box scheme can be used to solve higher-order problems which have been reduced to first-order by the introduction of new variables (see the example in Section 10). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (d03ppc for example), but at the expense of increased CPU time due to the larger number of function evaluations required.
It should be noted that the Keller box scheme, in common with other central-difference schemes, may be unsuitable for some hyperbolic first-order problems such as the apparently simple linear advection equation Ut+aUx=0, where a is a constant, resulting in spurious oscillations due to the lack of dissipation. This type of problem requires a discretization scheme with upwind weighting
(d03psc for example), or the addition of a second-order artificial dissipation term.
The time taken depends on the complexity of the 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.

10 Example

This example is the first-order system
U1 t + U1 x + U2 x = 0, U2 t +4 U1 x + U2 x = 0,  
for x0,1 and t0.
The initial conditions are
U1x,0 = ex, U2x,0 = x2+sin2πx2,  
and the Dirichlet boundary conditions for U1 at x=0 and U2 at x=1 are given by the exact solution:
U1x,t = 12 ex+t+ex-3t+14 sin2π x-3t 2-sin2π x+t 2+2t2-2xt, U2x,t = ex-3t-ex+t+12 sin2π x-3t 2+sin2π x+t 2+x2+5t2-2xt.  

10.1 Program Text

Program Text (d03prce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (d03prce.r)
GnuplotProduced by GNUPLOT 5.0 patchlevel 0 Example Program Solution of First-order System using Moving Mesh U(1,x,t) U(1,x,t) gnuplot_plot_1 gnuplot_plot_2