nag_pde_parab_1d_cd (d03pfc) (PDF version)
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d03 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_pde_parab_1d_cd (d03pfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_pde_parab_1d_cd (d03pfc) integrates a system of linear or nonlinear convection-diffusion equations in one space dimension, with optional source terms. The system must be posed in conservative form. Convection terms are discretized using a sophisticated upwind scheme involving a user-supplied numerical flux function based on the solution of a Riemann problem at each mesh point. The method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs), and the resulting system is solved using a backward differentiation formula (BDF) method.

2  Specification

#include <nag.h>
#include <nagd03.h>
void  nag_pde_parab_1d_cd (Integer npde, double *ts, double tout,
void (*pdedef)(Integer npde, double t, double x, const double u[], const double ux[], double p[], double c[], double d[], double s[], Integer *ires, Nag_Comm *comm),
void (*numflx)(Integer npde, double t, double x, const double uleft[], const double uright[], double flux[], Integer *ires, Nag_Comm *comm, Nag_D03_Save *saved),
void (*bndary)(Integer npde, Integer npts, double t, const double x[], const double u[], Integer ibnd, double g[], Integer *ires, Nag_Comm *comm),
double u[], Integer npts, const double x[], const double acc[], double tsmax, 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_cd (d03pfc) integrates the system of convection-diffusion equations in conservative form:
j=1npdePi,j Uj t + Fi x =Ci Di x +Si, (1)
or the hyperbolic convection-only system:
Ui t + Fi x =0, (2)
for i=1,2,,npde,  axb,  tt0, where the vector U is the set of solution values
U x,t = U 1 x,t ,, U npde x,t T .
The functions Pi,j, Fi, Ci and Si depend on x, t and U; and Di depends on x, t, U and Ux, where Ux is the spatial derivative of U. Note that Pi,j, Fi, Ci and Si must not depend on any space derivatives; and none of the functions may depend on time derivatives. In terms of conservation laws, Fi, CiDi x  and Si are the convective flux, diffusion and source terms respectively.
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 user-defined mesh x1,x2,,xnpts. The initial values of the functions Ux,t must be given at t=t0.
The PDEs are approximated by a system of ODEs in time for the values of Ui at mesh points using a spatial discretization method similar to the central-difference scheme used in nag_pde_parab_1d_fd (d03pcc), nag_pde_parab_1d_fd_ode (d03phc) and nag_pde_parab_1d_fd_ode_remesh (d03ppc), but with the flux Fi replaced by a numerical flux, which is a representation of the flux taking into account the direction of the flow of information at that point (i.e., the direction of the characteristics). Simple central differencing of the numerical flux then becomes a sophisticated upwind scheme in which the correct direction of upwinding is automatically achieved.
The numerical flux vector, F^i say, must be calculated by you in terms of the left and right values of the solution vector U (denoted by UL and UR respectively), at each mid-point of the mesh xj-1/2=xj-1+xj/2, for j=2,3,,npts. The left and right values are calculated by nag_pde_parab_1d_cd (d03pfc) from two adjacent mesh points using a standard upwind technique combined with a Van Leer slope-limiter (see LeVeque (1990)). The physically correct value for F^i is derived from the solution of the Riemann problem given by
Ui t + Fi y =0, (3)
where y=x-xj-1/2, i.e., y=0 corresponds to x=xj-1/2, with discontinuous initial values U=UL for y<0 and U=UR for y>0, using an approximate Riemann solver. This applies for either of the systems (1) or (2); the numerical flux is independent of the functions Pi,j, Ci, Di and Si. A description of several approximate Riemann solvers can be found in LeVeque (1990) and Berzins et al. (1989). Roe's scheme (see Roe (1981)) is perhaps the easiest to understand and use, and a brief summary follows. Consider the system of PDEs Ut+Fx=0 or equivalently Ut+AUx=0. Provided the system is linear in U, i.e., the Jacobian matrix A does not depend on U, the numerical flux F^ is given by
F^=12 FL+FR-12k=1npdeαkλkek, (4)
where FL (FR) is the flux F calculated at the left (right) value of U, denoted by UL (UR); the λk are the eigenvalues of A; the ek are the right eigenvectors of A; and the αk are defined by
UR-UL=k=1npdeαkek. (5)
An example is given in Section 9.
If the system is nonlinear, Roe's scheme requires that a linearized Jacobian is found (see Roe (1981)).
The functions Pi,j, Ci, Di and Si (but not Fi) must be specified in a pdedef. The numerical flux F^i must be supplied in a separate numflx. For problems in the form (2)) the NAG defined null void function pointer, NULLFN, can be supplied in the call to nag_pde_parab_1d_cd (d03pfc).
The boundary condition specification has sufficient flexibility to allow for different types of problems. For second-order problems, i.e., Di depending on Ux, a boundary condition is required for each PDE at both boundaries for the problem to be well-posed. If there are no second-order terms present, then the continuous PDE problem generally requires exactly one boundary condition for each PDE, that is npde boundary conditions in total. However, in common with most discretization schemes for first-order problems, a numerical boundary condition is required at the other boundary for each PDE. In order to be consistent with the characteristic directions of the PDE system, the numerical boundary conditions must be derived from the solution inside the domain in some manner (see below). You must supply both types of boundary conditions, i.e., a total of npde conditions at each boundary point.
The position of each boundary condition should be chosen with care. In simple terms, if information is flowing into the domain then a physical boundary condition is required at that boundary, and a numerical boundary condition is required at the other boundary. In many cases the boundary conditions are simple, e.g., for the linear advection equation. In general you should calculate the characteristics of the PDE system and specify a physical boundary condition for each of the characteristic variables associated with incoming characteristics, and a numerical boundary condition for each outgoing characteristic.
A common way of providing numerical boundary conditions is to extrapolate the characteristic variables from the inside of the domain. Note that only linear extrapolation is allowed in this function (for greater flexibility the function nag_pde_parab_1d_cd_ode (d03plc) should be used). For problems in which the solution is known to be uniform (in space) towards a boundary during the period of integration then extrapolation is unnecessary; the numerical boundary condition can be supplied as the known solution at the boundary. Examples can be found in Section 9.
The boundary conditions must be specified in bndary in the form
GiLx,t,U=0  at ​x=a,  i=1,2,,npde, (6)
at the left-hand boundary, and
GiRx,t,U=0  at ​x=b,  i=1,2,,npde, (7)
at the right-hand boundary.
Note that spatial derivatives at the boundary are not passed explicitly to bndary, but they can be calculated using values of U at and adjacent to the boundaries if required. However, it should be noted that instabilities may occur if such one-sided differencing opposes the characteristic direction at the boundary.
The problem is subject to the following restrictions:
(i) Pi,j, Fi, Ci and Si must not depend on any space derivatives;
(ii) Pi,j, Fi, Ci, Di and Si must not depend on any time derivatives;
(iii) t0<tout, so that integration is in the forward direction;
(iv) The evaluation of the terms Pi,j, Ci, Di and Si is done by calling the pdedef at a point approximately midway between each pair of mesh points in turn. Any discontinuities in these functions must therefore be at one or more of the mesh points x1,x2,,xnpts;
(v) At least one of the functions Pi,j must be nonzero so that there is a time derivative present in the PDE problem.
In total there are npde×npts ODEs in the time direction. This system is then integrated forwards in time using a BDF method.
For further details of the algorithm, see Pennington and Berzins (1994) and the references therein.

4  References

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
Hirsch C (1990) Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows John Wiley
LeVeque R J (1990) Numerical Methods for Conservation Laws Birkhäuser Verlag
Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw. 20 63–99
Roe P L (1981) Approximate Riemann solvers, parameter vectors, and difference schemes J. Comput. Phys. 43 357–372

5  Arguments

1:     npdeIntegerInput
On entry: the number of PDEs to be solved.
Constraint: npde1.
2:     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 ts=tout.
Constraint: ts<tout.
3:     toutdoubleInput
On entry: the final value of t to which the integration is to be carried out.
4:     pdedeffunction, supplied by the userExternal Function
pdedef must evaluate the functions Pi,j, Ci, Di and Si which partially define the system of PDEs. Pi,j, Ci and Si may depend on x, t and U; Di may depend on x, t, U and Ux. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_parab_1d_cd (d03pfc). For problems in the form (2)) the NAG defined null void function pointer, NULLFN, can be supplied in the call to nag_pde_parab_1d_cd (d03pfc).
The specification of pdedef is:
void  pdedef (Integer npde, double t, double x, const double u[], const double ux[], double p[], double c[], double d[], double s[], 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: u[i-1] contains the value of the component Uix,t, for i=1,2,,npde.
5:     ux[npde]const doubleInput
On entry: ux[i-1] contains the value of the component Ui x,t x , for i=1,2,,npde.
6:     p[npde×npde]doubleOutput
On exit: p[npde×j-1+i-1] must be set to the value of Pi,jx,t,U, for i=1,2,,npde and j=1,2,,npde.
7:     c[npde]doubleOutput
On exit: c[i-1] must be set to the value of Cix,t,U, for i=1,2,,npde.
8:     d[npde]doubleOutput
On exit: d[i-1] must be set to the value of Dix,t,U,Ux, for i=1,2,,npde.
9:     s[npde]doubleOutput
On exit: s[i-1] must be set to the value of Six,t,U, for i=1,2,,npde.
10:   iresInteger *Input/Output
On entry: set to -1​ 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:
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, then nag_pde_parab_1d_cd (d03pfc) returns to the calling function with the error indicator set to fail.code= NE_FAILED_DERIV.
11:   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_cd (d03pfc) you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from nag_pde_parab_1d_cd (d03pfc) (see Section 3.2.1 in the Essential Introduction).
5:     numflxfunction, supplied by the userExternal Function
numflx must supply the numerical flux for each PDE given the left and right values of the solution vector u. numflx is called approximately midway between each pair of mesh points in turn by nag_pde_parab_1d_cd (d03pfc).
The specification of numflx is:
void  numflx (Integer npde, double t, double x, const double uleft[], const double uright[], double flux[], Integer *ires, Nag_Comm *comm, Nag_D03_Save *saved)
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:     uleft[npde]const doubleInput
On entry: uleft[i-1] contains the left value of the component Uix, for i=1,2,,npde.
5:     uright[npde]const doubleInput
On entry: uright[i-1] contains the right value of the component Uix, for i=1,2,,npde.
6:     flux[npde]doubleOutput
On exit: flux[i-1] must be set to the numerical flux F^i, for i=1,2,,npde.
7:     iresInteger *Input/Output
On entry: set to -1​ 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:
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, then nag_pde_parab_1d_cd (d03pfc) returns to the calling function with the error indicator set to fail.code= NE_FAILED_DERIV.
8:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to numflx.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling nag_pde_parab_1d_cd (d03pfc) you may allocate memory and initialize these pointers with various quantities for use by numflx when called from nag_pde_parab_1d_cd (d03pfc) (see Section 3.2.1 in the Essential Introduction).
9:     savedNag_D03_Save *Communication Structure
If numflx calls one of the approximate Riemann solvers nag_pde_parab_1d_euler_roe (d03puc), nag_pde_parab_1d_euler_osher (d03pvc), nag_pde_parab_1d_euler_hll (d03pwc) or nag_pde_parab_1d_euler_exact (d03pxc) then saved is used to pass data concerning the computation to the solver. You should not change the components of saved.
6:     bndaryfunction, supplied by the userExternal Function
bndary must evaluate the functions GiL and GiR which describe the physical and numerical boundary conditions, as given by (6) and (7).
The specification of bndary is:
void  bndary (Integer npde, Integer npts, double t, const double x[], const double u[], Integer ibnd, double g[], Integer *ires, 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 a,b.
3:     tdoubleInput
On entry: the current value of the independent variable t.
4:     x[npts]const doubleInput
On entry: the mesh points in the spatial direction. x[0] corresponds to the left-hand boundary, a, and x[npts-1] corresponds to the right-hand boundary, b.
5:     u[3×npde]const doubleInput
On entry: contains the value of solution components in the boundary region.
If ibnd=0, u[3×j-1+i-1] contains the value of the component Uix ,t at x=x[j-1], for i=1,2,,npde and j=1,2,3.
If ibnd0, u[3×j-1+i-1] contains the value of the component Uix,t at x=x[npts-j], for i=1,2,,npde and j=1,2,3.
6:     ibndIntegerInput
On entry: specifies which boundary conditions are to be evaluated.
ibnd=0
bndary must evaluate the left-hand boundary condition at x=a.
ibnd0
bndary must evaluate the right-hand boundary condition at x=b.
7:     g[npde]doubleOutput
On exit: g[i-1] must contain the ith component of either gL or gR in (6) and (7), depending on the value of ibnd, for i=1,2,,npde.
8:     iresInteger *Input/Output
On entry: set to -1​ 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:
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, then nag_pde_parab_1d_cd (d03pfc) returns to the calling function with the error indicator set to fail.code= NE_FAILED_DERIV.
9:     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_cd (d03pfc) you may allocate memory and initialize these pointers with various quantities for use by bndary when called from nag_pde_parab_1d_cd (d03pfc) (see Section 3.2.1 in the Essential Introduction).
7:     u[npde×npts]doubleInput/Output
On entry: u[npde×j-1+i-1] must contain the initial value of Uix,t at x=x[j-1] and t=ts, for i=1,2,,npde and j=1,2,,npts.
On exit: u[npde×j-1+i-1] will contain the computed solution Uix,t at x=x[j-1] and t=ts, for i=1,2,,npde and j=1,2,,npts.
8:     nptsIntegerInput
On entry: the number of mesh points in the interval a,b.
Constraint: npts3.
9:     x[npts]const doubleInput
On entry: the 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].
10:   acc[2]const doubleInput
On entry: the components of acc contain the relative and absolute error tolerances used in the local error test in the time integration.
If Ei,j is the estimated error for Ui at the jth mesh point, the error test is
Ei,j=acc[0]×u[npde×j-1+i-1]+acc[1].
Constraint: acc[0] and acc[1]0.0 (but not both zero).
11:   tsmaxdoubleInput
On entry: the maximum absolute step size to be allowed in the time integration. If tsmax=0.0 then no maximum is imposed.
Constraint: tsmax0.0.
12:   rsave[lrsave]doubleCommunication 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.
13:   lrsaveIntegerInput
On entry: the dimension of the array rsave.
Constraint: lrsave11+9×npde×npde×npts+32+3×npde×npde+7×npts+54.
14:   isave[lisave]IntegerCommunication Array
If ind=0, isave need not be set on entry.
If ind=1, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:
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 computing 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 last backward differentiation formula method used.
isave[4]
Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the LU decomposition of the Jacobian matrix.
15:   lisaveIntegerInput
On entry: the dimension of the array isave.
Constraint: lisavenpde×npts+24.
16:   itaskIntegerInput
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.
Constraint: itask=1, 2 or 3.
17:   itraceIntegerInput
On entry: the level of trace information required from nag_pde_parab_1d_cd (d03pfc) and the underlying ODE solver. itrace may take the value -1, 0, 1, 2 or 3.
itrace=-1
No output is generated.
itrace=0
Only warning messages from the PDE solver are printed.
itrace>0
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.
If itrace<-1, then -1 is assumed and similarly if itrace>3, then 3 is assumed.
The advisory messages are given in greater detail as itrace increases.
18:   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.
19:   indInteger *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 arguments tout and fail should be reset between calls to nag_pde_parab_1d_cd (d03pfc).
Constraint: ind=0 or 1.
On exit: ind=1.
20:   commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
21:   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.
22:   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 acc are unlikely to result in a changed solution. acc[0]=value, acc[1]=value.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
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, numflx, or bndary.
NE_FAILED_START
Values in acc are too small to start integration: acc[0]=value, acc[1]=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: ts=value.
Underlying ODE solver cannot make further progress from the point ts with the supplied values of acc. ts=value, acc[0]=value, acc[1]=value.
NE_INCOMPAT_PARAM
On entry, acc[0] and acc[1] are both zero.
NE_INT
ires set to an invalid value in call to pdedef, numflx, or bndary.
On entry, ind=value.
Constraint: ind=0 or 1.
On entry, itask=value.
Constraint: itask=1, 2 or 3.
On entry, npde=value.
Constraint: npde1.
On entry, npts=value.
Constraint: npts3.
NE_INT_2
On entry, lisave is too small: lisave=value. Minimum possible dimension: value.
On entry, lrsave is too small: lrsave=value. Minimum possible dimension: value.
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_NOT_CLOSE_FILE
Cannot close file value.
NE_NOT_STRICTLY_INCREASING
On entry, mesh points x appear to be badly ordered: i=value, x[i-1]=value, j=value and x[j-1]=value.
NE_NOT_WRITE_FILE
Cannot open file value for writing.
NE_REAL
On entry, acc[0]<0.0: acc[0]=value.
On entry, acc[1]<0.0: acc[1]=value.
On entry, tsmax=value.
Constraint: tsmax0.0.
NE_REAL_2
On entry, tout=value and ts=value.
Constraint: tout>ts.
On entry, tout-ts is too small: tout=value and ts=value.
NE_SING_JAC
Singular Jacobian of ODE system. Check problem formulation.
NE_TIME_DERIV_DEP
The functions P, D, or C appear to depend on time derivatives.
NE_USER_STOP
In evaluating residual of ODE system, ires=2 has been set in pdedef, numflx, or bndary. Integration is successful as far as ts: ts=value.

7  Accuracy

nag_pde_parab_1d_cd (d03pfc) 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 components of the accuracy argument, acc.

8  Further Comments

nag_pde_parab_1d_cd (d03pfc) is designed to solve systems of PDEs in conservative form, with optional source terms which are independent of space derivatives, and optional second-order diffusion terms. The use of the function to solve systems which are not naturally in this form is discouraged, and you are advised to use one of the central-difference schemes for such problems.
You should be aware of the stability limitations for hyperbolic PDEs. For most problems with small error tolerances the ODE integrator does not attempt unstable time steps, but in some cases a maximum time step should be imposed using tsmax. It is worth experimenting with this argument, particularly if the integration appears to progress unrealistically fast (with large time steps). Setting the maximum time step to the minimum mesh size is a safe measure, although in some cases this may be too restrictive.
Problems with source terms should be treated with caution, as it is known that for large source terms stable and reasonable looking solutions can be obtained which are in fact incorrect, exhibiting non-physical speeds of propagation of discontinuities (typically one spatial mesh point per time step). It is essential to employ a very fine mesh for problems with source terms and discontinuities, and to check for non-physical propagation speeds by comparing results for different mesh sizes. Further details and an example can be found in Pennington and Berzins (1994).
The time taken depends on the complexity of the system and on the accuracy requested.

9  Example

For this function two examples are presented. There is a single example program for nag_pde_parab_1d_cd (d03pfc), with a main program and the code to solve the two example problems given in Example 1 (ex1) and Example 2 (ex2).
Example 1 (ex1)
This example is a simple first-order system which illustrates the calculation of the numerical flux using Roe's approximate Riemann solver, and the specification of numerical boundary conditions using extrapolated characteristic variables. The PDEs are
U1 t + U1 x + U2 x = 0, U2 t +4 U1 x + U2 x = 0,
for x 0,1  and t 0 . The PDEs have an exact solution given by
U1 x,t = 12 exp x + t + expx-3t + 14 sin 2 π x-3t 2 - sin 2 π x+t 2 + 2 t2 - 2 x t , U2 x,t = expx-3t - expx+t + 12 sin 2 π x-3t 2 + sin 2 π x - 3 t 2 + x2 + 5 t2 - 2 x t .
The initial conditions are given by the exact solution. The characteristic variables are 2U1+U2 and 2U1-U2 corresponding to the characteristics given by dx/dt=3 and dx/dt=-1 respectively. Hence a physical boundary condition is required for 2U1+U2 at the left-hand boundary, and for 2U1-U2 at the right-hand boundary (corresponding to the incoming characteristics); and a numerical boundary condition is required for 2U1-U2 at the left-hand boundary, and for 2U1+U2 at the right-hand boundary (outgoing characteristics). The physical boundary conditions are obtained from the exact solution, and the numerical boundary conditions are calculated by linear extrapolation of the appropriate characteristic variable. The numerical flux is calculated using Roe's approximate Riemann solver: Using the notation in Section 3, the flux vector F and the Jacobian matrix A are
F= U1+U2 4U1+U2   and   A= 1 1 4 1 ,
and the eigenvalues of A are 3 and -1 with right eigenvectors 12T and -12T respectively. Using equation (4) the αk are given by
U1R-U1L U2R-U2L =α1 1 2 +α2 -1 2 ,
that is
α1 = 14 2 U1R - 2 U1L + U2R - U2L   and   α2 = 14 -2 U1R + 2 U1L + U2R - U2L .
FL is given by
FL = U1L+U2L 4U1L+U2L ,
and similarly for FR. From equation (4), the numerical flux vector is
F^ = 12 U1L+U2L+0U1R+U2R 4U1L+U2L+4U1R+U2R - 12 α1 3 1 2 - 12 α2 -1 -1 2 ,
that is
F^ = 12 3U1L-0U1R+32U2L+12 U2R 6U1L+ 2U1R+ 3U2L-0U2R .
Example 2 (ex2)
This example is an advection-diffusion equation in which the flux term depends explicitly on x:
U t +x U x =ε 2U x2 ,
for x-1,1 and 0t10. The argument ε is taken to be 0.01. The two physical boundary conditions are U-1,t=3.0 and U1,t=5.0 and the initial condition is Ux,0=x+4. The integration is run to steady state at which the solution is known to be U=4 across the domain with a narrow boundary layer at both boundaries. In order to write the PDE in conservative form, a source term must be introduced, i.e.,
U t + xU x =ε 2U x2 +U.
As in Example 1, the numerical flux is calculated using the Roe approximate Riemann solver. The Riemann problem to solve locally is
U t + x U x =0.
The x in the flux term is assumed to be constant at a local level, and so using the notation in Section 3, F=xU and A=x. The eigenvalue is x and the eigenvector (a scalar in this case) is 1. The numerical flux is therefore
F^= xUL if ​x0, xUR if ​x<0.

9.1  Program Text

Program Text (d03pfce.c)

9.2  Program Data

None.

9.3  Program Results

Program Results (d03pfce.r)


nag_pde_parab_1d_cd (d03pfc) (PDF version)
d03 Chapter Contents
d03 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012