d03pec 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 functions.
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: $\mathbf{ts}$ – double *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: $\mathbf{tout}$ – doubleInput
On entry: the final value of $t$ to which the integration is to be carried out.
4: $\mathbf{pdedef}$ – function, supplied by the userExternal Function
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 d03pec.
On entry: ${\mathbf{u}}\left[\mathit{i}-1\right]$ contains the value of the component ${U}_{\mathit{i}}(x,t)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
On entry: ${\mathbf{ut}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial t}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
On entry: ${\mathbf{ux}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial x}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
On exit: ${\mathbf{res}}\left[\mathit{i}-1\right]$ 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 function to take certain actions, as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_USER_STOP.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03pec returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_FAILED_DERIV.
9: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to pdedef.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling d03pec you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from d03pec (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 d03pec. If your code inadvertently does return any NaNs or infinities, d03pec is likely to produce unexpected results.
5: $\mathbf{bndary}$ – function, supplied by the userExternal Function
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: ${\mathbf{u}}\left[\mathit{i}-1\right]$ contains the value of the component ${U}_{\mathit{i}}(x,t)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
On entry: ${\mathbf{ut}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial t}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
On exit: ${\mathbf{res}}\left[\mathit{i}-1\right]$ 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: $\mathbf{ires}$ – Integer *Input/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 function to take certain actions, as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_USER_STOP.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03pec returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_FAILED_DERIV.
9: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to bndary.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling d03pec you may allocate memory and initialize these pointers with various quantities for use by bndary when called from d03pec (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 d03pec. If your code inadvertently does return any NaNs or infinities, d03pec is likely to produce unexpected results.
Note: the $(i,j)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[(j-1)\times {\mathbf{npde}}+i-1\right]$.
On entry: the initial values of $U(x,t)$ at $t={\mathbf{ts}}$ and the mesh points
${\mathbf{x}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
On exit: ${\mathbf{u}}\left[\left(\mathit{j}-1\right)\times {\mathbf{npde}}+\mathit{i}-1\right]$ will contain the computed solution at $t={\mathbf{ts}}$.
7: $\mathbf{npts}$ – IntegerInput
On entry: the number of mesh points in the interval $[a,b]$.
On entry: the mesh points in the spatial direction. ${\mathbf{x}}\left[0\right]$ must specify the left-hand boundary, $a$, and ${\mathbf{x}}\left[{\mathbf{npts}}-1\right]$ must specify the right-hand boundary, $b$.
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 function because it contains required information about the iteration. In particular:
${\mathbf{isave}}\left[0\right]$
Contains the number of steps taken in time.
${\mathbf{isave}}\left[1\right]$
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
${\mathbf{isave}}\left[2\right]$
Contains the number of Jacobian evaluations performed by the time integrator.
${\mathbf{isave}}\left[3\right]$
Contains the order of the last backward differentiation formula method used.
${\mathbf{isave}}\left[4\right]$
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.
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 d03pec 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.
${\mathbf{itrace}}=1$
Output from the underlying ODE solver is printed. This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
${\mathbf{itrace}}=2$
Output from the underlying ODE solver is similar to that produced when ${\mathbf{itrace}}=1$, except that the advisory messages are given in greater detail.
${\mathbf{itrace}}\ge 3$
Output from the underlying ODE solver is similar to that produced when ${\mathbf{itrace}}=2$, except that the advisory messages are given in greater detail.
You are advised to set ${\mathbf{itrace}}=0$.
17: $\mathbf{outfile}$ – 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.
18: $\mathbf{ind}$ – Integer *Input/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 function. In this case, only the argument tout should be reset between calls to d03pec.
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
19: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
saved must remain unchanged following a previous call to a Chapter D03 function and prior to any subsequent call to a Chapter D03 function.
21: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ACC_IN_DOUBT
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 ${\mathbf{itask}}\ne 2$.)
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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_FAILED_DERIV
In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting ${\mathbf{ires}}=3$ in pdedef or bndary.
NE_FAILED_START
acc was too small to start integration:
${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_FAILED_STEP
Error during Jacobian formulation for ODE system. Increase itrace for further details.
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts:
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
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 $t={\mathbf{ts}}$.
Underlying ODE solver cannot make further progress from the point ts with the supplied value of acc.
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
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, ${\mathbf{nleft}}=\u27e8\mathit{\text{value}}\u27e9$.
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 ${\mathbf{ind}}=0$.
NE_INT_2
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: ${\mathbf{nleft}}\le {\mathbf{npde}}$.
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_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 $\u27e8\mathit{\text{value}}\u27e9$.
NE_NOT_STRICTLY_INCREASING
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{x}}\left[\mathit{i}-1\right]=\u27e8\mathit{\text{value}}\u27e9$, $\mathit{j}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left[\mathit{j}-1\right]=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{npts}}-1\right]$.
NE_NOT_WRITE_FILE
Cannot open file $\u27e8\mathit{\text{value}}\u27e9$ for writing.
NE_REAL
On entry, ${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{acc}}>0.0$.
NE_REAL_2
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 ${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_SING_JAC
Singular Jacobian of ODE system. Check problem formulation.
NE_USER_STOP
In evaluating residual of ODE system, ${\mathbf{ires}}=2$ has been set in pdedef or bndary. Integration is successful as far as ts:
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
d03pec 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
d03pec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pec 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.
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 d03pkc). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (d03pccord03phc 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 (d03pfc 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.