d03pcc integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable. The spatial discretization is performed using finite differences, 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 method.
and the vector ${U}_{x}$ is its partial derivative with respect to $x$. Note that ${P}_{i,j}$,
${Q}_{i}$ and ${R}_{i}$ 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 coordinate system in space is defined by the value of $m$;
$m=0$ for Cartesian coordinates,
$m=1$ for cylindrical polar coordinates and $m=2$ for spherical polar coordinates. The mesh should be chosen in accordance with the expected behaviour of the solution.
The system is defined by the functions ${P}_{i,j}$,
${Q}_{i}$ and ${R}_{i}$ which must be specified in pdedef.
The initial values of the functions $U(x,t)$ must be given at $t={t}_{0}$. The functions
${R}_{i}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, which may be thought of as fluxes, are also used in the definition of the boundary conditions for each equation. The boundary conditions must have the form
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)${P}_{i,j}$,
${Q}_{i}$ and the flux ${R}_{i}$ must not depend on any time derivatives;
(iii)the evaluation of the functions ${P}_{i,j}$,
${Q}_{i}$ and ${R}_{i}$ is done at the mid-points of the mesh intervals by calling the pdedef for each mid-point in turn. Any discontinuities in these functions 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; and
(v)if $m>0$ and ${x}_{1}=0.0$, which is the left boundary point, then it must be ensured that the PDE solution is bounded at this point. This can be done by either specifying the solution at $x=0.0$ or by specifying a zero flux there, that is ${\beta}_{i}=1.0$ and ${\gamma}_{i}=0.0$. See also Section 9.
The parabolic equations are approximated by a system of ODEs in time for the values of ${U}_{i}$ at mesh points. For simple problems in Cartesian coordinates, this system is obtained by replacing the space derivatives by the usual central, three-point finite difference formula. However, for polar and spherical problems, or problems with nonlinear coefficients, the space derivatives are replaced by a modified three-point formula which maintains second-order accuracy. In total there are ${\mathbf{npde}}\times {\mathbf{npts}}$ ODEs in the time direction. This system is then integrated forwards in time using a backward differentiation formula 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
Dew P M and Walsh J (1981) A set of library routines for solving parabolic equations in one space variable ACM Trans. Math. Software7 295–314
Skeel R D and Berzins M (1990) A method for the spatial discretization of parabolic equations in one space variable SIAM J. Sci. Statist. Comput.11(1) 1–32
5Arguments
1: $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs in the system to be solved.
Constraint:
${\mathbf{npde}}\ge 1$.
2: $\mathbf{m}$ – IntegerInput
On entry: the coordinate system used:
${\mathbf{m}}=0$
Indicates Cartesian coordinates.
${\mathbf{m}}=1$
Indicates cylindrical polar coordinates.
${\mathbf{m}}=2$
Indicates spherical polar coordinates.
Constraint:
${\mathbf{m}}=0$, $1$ or $2$.
3: $\mathbf{ts}$ – double *Input/Output
On entry: the initial value of the independent variable $t$.
On exit: the value of $t$ corresponding to the solution values in u. Normally ${\mathbf{ts}}={\mathbf{tout}}$.
Constraint:
${\mathbf{ts}}<{\mathbf{tout}}$.
4: $\mathbf{tout}$ – doubleInput
On entry: the final value of $t$ to which the integration is to be carried out.
5: $\mathbf{pdedef}$ – function, supplied by the userExternal Function
pdedef must compute the functions ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by d03pcc.
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: ${\mathbf{ux}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial x}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
Note: the $(i,j)$th element of the matrix $P$ is stored in ${\mathbf{p}}\left[(j-1)\times {\mathbf{npde}}+i-1\right]$.
On exit: ${\mathbf{p}}\left[\left(\mathit{j}-1\right)\times {\mathbf{npde}}+\mathit{i}-1\right]$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}(x,t,U,{U}_{x})$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
On exit: ${\mathbf{q}}\left[\mathit{i}-1\right]$ must be set to the value of ${Q}_{\mathit{i}}(x,t,U,{U}_{x})$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
On exit: ${\mathbf{r}}\left[\mathit{i}-1\right]$ must be set to the value of ${R}_{\mathit{i}}(x,t,U,{U}_{x})$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
9: $\mathbf{ires}$ – Integer *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:
${\mathbf{ires}}=2$
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 ${\mathbf{ires}}=3$, d03pcc returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_FAILED_DERIV.
10: $\mathbf{comm}$ – 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 d03pcc you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from d03pcc (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 d03pcc. If your code inadvertently does return any NaNs or infinities, d03pcc is likely to produce unexpected results.
6: $\mathbf{bndary}$ – function, supplied by the userExternal Function
bndary must compute the functions ${\beta}_{i}$ and ${\gamma}_{i}$ which define the boundary conditions as in equation (3).
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: ${\mathbf{ux}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial x}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
5: $\mathbf{ibnd}$ – IntegerInput
On entry: determines the position of the boundary conditions.
${\mathbf{ibnd}}=0$
bndary must set up the coefficients of the left-hand boundary, $x=a$.
${\mathbf{ibnd}}\ne 0$
Indicates that bndary must set up the coefficients of the right-hand boundary, $x=b$.
On exit: ${\mathbf{beta}}\left[\mathit{i}-1\right]$ must be set to the value of ${\beta}_{\mathit{i}}(x,t)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
On exit: ${\mathbf{gamma}}\left[\mathit{i}-1\right]$ must be set to the value of ${\gamma}_{\mathit{i}}(x,t,U,{U}_{x})$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
8: $\mathbf{ires}$ – Integer *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:
${\mathbf{ires}}=2$
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 ${\mathbf{ires}}=3$, d03pcc 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 d03pcc you may allocate memory and initialize these pointers with various quantities for use by bndary when called from d03pcc (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 d03pcc. If your code inadvertently does return any NaNs or infinities, d03pcc 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 $t={\mathbf{ts}}$.
8: $\mathbf{npts}$ – 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 u at $t={\mathbf{tout}}$.
${\mathbf{itask}}=2$
One step and return.
${\mathbf{itask}}=3$
Stop at first internal integration point at or beyond $t={\mathbf{tout}}$.
Constraint:
${\mathbf{itask}}=1$, $2$ or $3$.
16: $\mathbf{itrace}$ – IntegerInput
On entry: the level of trace information required from d03pcc and the underlying ODE solver. itrace may take the value $-1$, $0$, $1$, $2$ or $3$.
${\mathbf{itrace}}=-1$
No output is generated.
${\mathbf{itrace}}=0$
Only warning messages from the PDE solver are printed.
${\mathbf{itrace}}>0$
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 ${\mathbf{itrace}}<-1$, $-1$ is assumed and similarly if ${\mathbf{itrace}}>3$, $3$ is assumed.
The advisory messages are given in greater detail as itrace increases.
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 d03pcc.
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.
${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
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:
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
Underlying ODE solver cannot make further progress from the point ts with the supplied value of acc.
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{acc}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_INCOMPAT_PARAM
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left[0\right]=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\le 0$ or ${\mathbf{x}}\left[0\right]\ge 0.0$
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, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}=0$, $1$ or $2$.
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: ${\mathbf{lrsave}}\ge \u27e8\mathit{\text{value}}\u27e9$.
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_TIME_DERIV_DEP
Flux function appears to depend on time derivatives.
NE_USER_STOP
In evaluating residual of ODE system, ${\mathbf{ires}}=2$ has been set in pdedef or bndary. Integration is successful as far as ts:
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
d03pcc 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
d03pcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pcc 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
d03pcc is designed to solve parabolic systems (possibly including some elliptic equations) with second-order derivatives in space. The argument specification allows you to include equations with only first-order derivatives in the space direction but there is no guarantee that the method of integration will be satisfactory for such systems. The position and nature of the boundary conditions in particular are critical in defining a stable problem. It may be advisable in such cases to reduce the whole system to first-order and to use the Keller box scheme function d03pec.
The time taken depends on the complexity of the parabolic system and on the accuracy requested.
10Example
We use the example given in Dew and Walsh (1981) which consists of an elliptic-parabolic pair of PDEs. The problem was originally derived from a single third-order in space PDE. The elliptic equation is
$$\frac{\partial}{\partial r}\left(r{U}_{1}\right)=0\text{\hspace{1em} and \hspace{1em}}{U}_{2}=0\text{\hspace{1em} at}r=1\text{.}$$
The first of these boundary conditions implies that the flux term in the second PDE,
$(\frac{\partial {U}_{2}}{\partial r}-{U}_{2}{U}_{1})$, is zero at $r=0$.
The initial conditions at $t=0$ are given by
$${U}_{1}=2\alpha r\text{\hspace{1em} and \hspace{1em}}{U}_{2}=1.0\text{, \hspace{1em}}r\in [0,1]\text{.}$$
The value $\alpha =1$ was used in the problem definition. A mesh of $20$ points was used with a circular mesh spacing to cluster the points towards the right-hand side of the spatial interval,
$r=1$.