d03pef integrates a system of linear or nonlinear, first-order, time-dependent partial differential equations (PDEs) in one space variable. The spatial discretization is performed using the Keller box scheme and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a Backward Differentiation Formula (BDF) method.
and the vector ${U}_{x}$ is its partial derivative with respect to $x$. Note that ${P}_{i,j}$ and ${Q}_{i}$ must not depend on $\frac{\partial U}{\partial t}$.
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 user-defined mesh ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$. The mesh should be chosen in accordance with the expected behaviour of the solution.
The PDE system which is defined by the functions ${G}_{i}$ must be specified in pdedef.
The initial values of the functions $U(x,t)$ must be given at $t={t}_{0}$. For a first-order system of PDEs, only one boundary condition is required for each PDE component ${U}_{i}$. The npde boundary conditions are separated into ${n}_{a}$ at the left-hand boundary $x=a$, and ${n}_{b}$ at the right-hand boundary $x=b$, such that ${n}_{a}+{n}_{b}={\mathbf{npde}}$. The position of the boundary condition for each component should be chosen with care; the general rule is that if the characteristic direction of ${U}_{i}$ at the left-hand boundary (say) points into the interior of the solution domain, then the boundary condition for ${U}_{i}$ 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 routines.
Note that the functions ${G}_{i}^{L}$ and ${G}_{i}^{R}$ must not depend on ${U}_{x}$, since spatial derivatives are not determined explicitly in the Keller box scheme (see Keller (1970)). 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 ${G}_{i}^{L}$ and ${G}_{i}^{R}$ must be linear with respect to time derivatives, so that the boundary conditions have the general form
at the right-hand boundary, where ${E}_{i,j}^{L}$, ${E}_{i,j}^{R}$, ${S}_{i}^{L}$, and ${S}_{i}^{R}$ depend on $x$, $t$ and $U$ only.
The boundary conditions must be specified in bndary.
The problem is subject to the following restrictions:
(i)${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction;
(ii)${P}_{i,j}$ and ${Q}_{i}$ must not depend on any time derivatives;
(iii)The evaluation of the function ${G}_{i}$ is done at the mid-points of the mesh intervals by calling the pdedef for each mid-point in turn. Any discontinuities in the function must, therefore, be at one or more of the mesh points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$;
(iv)At least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the problem.
In this method of lines approach the Keller box scheme (see Keller (1970)) is applied to each PDE in the space variable only, resulting in a system of ODEs in time for the values of ${U}_{i}$ at each mesh point. In total there are ${\mathbf{npde}}\times {\mathbf{npts}}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF method.
4References
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
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
5Arguments
1: $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs in the system to be solved.
Constraint:
${\mathbf{npde}}\ge 1$.
2: $\mathbf{ts}$ – Real (Kind=nag_wp)Input/Output
On entry: the initial value of the independent variable $t$.
Constraint:
${\mathbf{ts}}<{\mathbf{tout}}$.
On exit: the value of $t$ corresponding to the solution values in u. Normally ${\mathbf{ts}}={\mathbf{tout}}$.
3: $\mathbf{tout}$ – Real (Kind=nag_wp)Input
On entry: the final value of $t$ to which the integration is to be carried out.
4: $\mathbf{pdedef}$ – Subroutine, supplied by the user.External Procedure
pdedef must compute the functions ${G}_{i}$ which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by d03pef.
On entry: the current value of the independent variable $t$.
3: $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the current value of the space variable $x$.
4: $\mathbf{u}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}\left(\mathit{i}\right)$ contains the value of the component ${U}_{\mathit{i}}(x,t)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
5: $\mathbf{ut}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{ut}}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial t}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
6: $\mathbf{ux}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{ux}}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial x}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
7: $\mathbf{res}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{res}}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $G$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, where $G$ is defined as
On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions, as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
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 ${\mathbf{ires}}=3$, d03pef returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
pdedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pef is called. Arguments denoted as Input must not be changed by this procedure.
Note:pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pef. If your code inadvertently does return any NaNs or infinities, d03pef is likely to produce unexpected results.
5: $\mathbf{bndary}$ – Subroutine, supplied by the user.External Procedure
bndary must compute the functions ${G}_{i}^{L}$ and ${G}_{i}^{R}$ which define the boundary conditions as in equations (4) and (5).
On entry: the current value of the independent variable $t$.
3: $\mathbf{ibnd}$ – IntegerInput
On entry: determines the position of the boundary conditions.
${\mathbf{ibnd}}=0$
bndary must compute the left-hand boundary condition at $x=a$.
${\mathbf{ibnd}}\ne 0$
Indicates that bndary must compute the right-hand boundary condition at $x=b$.
4: $\mathbf{nobc}$ – IntegerInput
On entry: specifies the number of boundary conditions at the boundary specified by ibnd.
5: $\mathbf{u}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}\left(\mathit{i}\right)$ contains the value of the component ${U}_{\mathit{i}}(x,t)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
6: $\mathbf{ut}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{ut}}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial t}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
7: $\mathbf{res}\left({\mathbf{nobc}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{res}}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of ${G}^{L}$ or ${G}^{R}$, depending on the value of ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{nobc}}$, where ${G}^{L}$ is defined as
i.e., all terms in equation (6), and similarly for ${G}_{\mathit{i}}^{R}$. The definitions of ${G}^{L}$ and ${G}^{R}$ are determined by the input value of ires.
8: $\mathbf{ires}$ – IntegerInput/Output
On entry: the form ${G}_{i}^{L}$ (or ${G}_{i}^{R}$) that must be returned in the array res.
On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions, as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
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 ${\mathbf{ires}}=3$, d03pef returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
bndary must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pef is called. Arguments denoted as Input must not be changed by this procedure.
Note:bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pef. If your code inadvertently does return any NaNs or infinities, d03pef is likely to produce unexpected results.
6: $\mathbf{u}({\mathbf{npde}},{\mathbf{npts}})$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the initial values of $U(x,t)$ at $t={\mathbf{ts}}$ and the mesh points
${\mathbf{x}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
On exit: ${\mathbf{u}}(\mathit{i},\mathit{j})$ will contain the computed solution at $t={\mathbf{ts}}$.
7: $\mathbf{npts}$ – IntegerInput
On entry: the number of mesh points in the interval $[a,b]$.
Constraint:
${\mathbf{npts}}\ge 3$.
8: $\mathbf{x}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the mesh points in the spatial direction. ${\mathbf{x}}\left(1\right)$ must specify the left-hand boundary, $a$, and ${\mathbf{x}}\left({\mathbf{npts}}\right)$ must specify the right-hand boundary, $b$.
On entry: a positive quantity for controlling the local error estimate in the time integration. If $E(i,j)$ is the estimated error for ${U}_{i}$ at the $j$th mesh point, the error test is:
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 routine because it contains required information about the iteration. In particular:
${\mathbf{isave}}\left(1\right)$
Contains the number of steps taken in time.
${\mathbf{isave}}\left(2\right)$
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.
${\mathbf{isave}}\left(3\right)$
Contains the number of Jacobian evaluations performed by the time integrator.
${\mathbf{isave}}\left(4\right)$
Contains the order of the last backward differentiation formula method used.
${\mathbf{isave}}\left(5\right)$
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.
14: $\mathbf{lisave}$ – IntegerInput
On entry: the dimension of the array isave as declared in the (sub)program from which d03pef is called.
On entry: specifies the task to be performed by the ODE integrator.
${\mathbf{itask}}=1$
Normal computation of output values ${\mathbf{u}}$ at $t={\mathbf{tout}}$.
${\mathbf{itask}}=2$
Take one step and return.
${\mathbf{itask}}=3$
Stop at the first internal integration point at or beyond $t={\mathbf{tout}}$.
Constraint:
${\mathbf{itask}}=1$, $2$ or $3$.
16: $\mathbf{itrace}$ – IntegerInput
On entry: the level of trace information required from d03pef and the underlying ODE solver as follows:
${\mathbf{itrace}}\le \mathrm{-1}$
No output is generated.
${\mathbf{itrace}}=0$
Only warning messages from the PDE solver are printed on the current error message unit (see x04aaf).
${\mathbf{itrace}}=1$
Output from the underlying ODE solver is printed on the current advisory message unit (see x04abf). 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.
You are advised to set ${\mathbf{itrace}}=0$, unless you are experienced with Sub-chapter D02M–N.
17: $\mathbf{ind}$ – IntegerInput/Output
On entry: indicates whether this is a continuation call or a new integration.
${\mathbf{ind}}=0$
Starts or restarts the integration in time.
${\mathbf{ind}}=1$
Continues the integration after an earlier exit from the routine. In this case, only the arguments tout and ifail should be reset between calls to d03pef.
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
18: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{acc}}>0.0$.
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{x}}\left(\mathit{i}\right)=\u27e8\mathit{\text{value}}\u27e9$, $\mathit{j}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left(\mathit{j}\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{npts}}\right)$.
On entry, ${\mathbf{ind}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On entry, ${\mathbf{itask}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{itask}}=1$, $2$ or $3$.
On entry, ${\mathbf{lisave}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lisave}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{lrsave}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lrsave}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{nleft}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{npde}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nleft}}\le {\mathbf{npde}}$.
On entry, ${\mathbf{nleft}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nleft}}\ge 0$.
On entry, ${\mathbf{npde}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{npde}}\ge 1$.
On entry, ${\mathbf{npts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{npts}}\ge 3$.
On entry, on initial entry ${\mathbf{ind}}=1$.
Constraint: on initial entry ${\mathbf{ind}}=0$.
On entry, ${\mathbf{tout}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tout}}>{\mathbf{ts}}$.
On entry, ${\mathbf{tout}}-{\mathbf{ts}}$ is too small:
${\mathbf{tout}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=2$
Underlying ODE solver cannot make further progress from the point ts with the supplied value of acc.
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=3$
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts:
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
In the underlying ODE solver, there were repeated errors or corrector convergence test failures on an attempted step, before completing the requested task. The problem may have a singularity or acc is too small for the integration to continue. Incorrect positioning of boundary conditions may also result in this error. Integration was successful as far as $t={\mathbf{ts}}$.
${\mathbf{ifail}}=4$
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.
${\mathbf{ifail}}=5$
Singular Jacobian of ODE system. Check problem formulation.
${\mathbf{ifail}}=6$
In evaluating residual of ODE system, ${\mathbf{ires}}=2$ has been set in pdedef or bndary. Integration is successful as far as ts:
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=7$
acc was too small to start integration:
${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
Serious error in internal call to an auxiliary. Increase itrace for further details.
${\mathbf{ifail}}=10$
Integration completed, but a small change in acc is unlikely to result in a changed solution.
${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
The required task has been completed, but it is estimated that a small change in acc is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when ${\mathbf{itask}}\ne 2$.)
${\mathbf{ifail}}=11$
Error during Jacobian formulation for ODE system. Increase itrace for further details.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
d03pef 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 argument, acc.
8Parallelism and Performance
d03pef is not thread safe and should not be called from a multithreaded user program. Please see Section 1 in FL Interface Multithreading for more information on thread safety.
d03pef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pef 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further 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 problem in d03pkf). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (d03pcf/d03pcaord03phf/d03pha 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 ${U}_{t}+a{U}_{x}=0$, 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 (d03pff for example), or the addition of a second-order artificial dissipation term.
The time taken depends on the complexity of the system and on the accuracy requested.