NAG Library Function Document

nag_pde_parab_1d_keller (d03pec)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     npde – IntegerInput

On entry: the number of PDEs in the system to be solved.

Constraint: $\mathbf{npde}\ge 1$.

2:     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:     tout – doubleInput

On entry: the final value of $t$ to which the integration is to be carried out.

4:     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 nag_pde_parab_1d_keller (d03pec).

The specification of pdedef is:

void  pdedef (Integer npde, double t, double x, const double u[], const double ut[], const double ux[], double res[], Integer *ires, Nag_Comm *comm)

1:     npde – IntegerInput

On entry: the number of PDEs in the system.

2:     t – doubleInput

On entry: the current value of the independent variable $t$.

3:     x – doubleInput

On entry: the current value of the space variable $x$.

4:     u[npde] – const doubleInput

On entry: $\mathbf{u}\left[\mathit{i}-1\right]$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

5:     ut[npde] – const doubleInput

On entry: $\mathbf{ut}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial t}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

6:     ux[npde] – const doubleInput

On entry: $\mathbf{ux}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial x}$, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

7:     res[npde] – doubleOutput

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

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}\text{,}$$

(8)

i.e., only terms depending explicitly on time derivatives, or

$$G_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathbf{npde}}{P}_{\mathit{i},\mathit{j}}\frac{\partial U_{\mathit{j}}}{\partial t}+Q_{\mathit{i}}\text{,}$$

(9)

i.e., all terms in equation (2).

The definition of $G$ is determined by the input value of ires.

8:     ires – Integer *Input/Output

On entry: the form of $G_i$ that must be returned in the array res.

$\mathbf{ires}=-1$

Equation (8) must be used.

$\mathbf{ires}=1$

Equation (9) must be used.

On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions, as described below:

$\mathbf{ires}=2$

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 $\mathbf{ires}=3$, then nag_pde_parab_1d_keller (d03pec) returns to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

9:     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 nag_pde_parab_1d_keller (d03pec) you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from nag_pde_parab_1d_keller (d03pec) (see Section 3.2.1.1 in the Essential Introduction).

5:     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).

The specification of bndary is:

void  bndary (Integer npde, double t, Integer ibnd, Integer nobc, const double u[], const double ut[], double res[], Integer *ires, Nag_Comm *comm)

1:     npde – IntegerInput

On entry: the number of PDEs in the system.

2:     t – doubleInput

On entry: the current value of the independent variable $t$.

3:     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:     nobc – IntegerInput

On entry: specifies the number of boundary conditions at the boundary specified by ibnd.

5:     u[npde] – const doubleInput

On entry: $\mathbf{u}\left[\mathit{i}-1\right]$ contains the value of the component $U_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

6:     ut[npde] – const doubleInput

On entry: $\mathbf{ut}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial U_{\mathit{i}}\left(x,t\right)}{\partial t}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,\mathbf{npde}$.

7:     res[nobc] – doubleOutput

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

$$G_{\mathit{i}}^L=\sum _{\mathit{j}=1}^{\mathbf{npde}}{E}_{\mathit{i},\mathit{j}}^L\frac{\partial U_{\mathit{j}}}{\partial t}\text{,}$$

(10)

i.e., only terms depending explicitly on time derivatives, or

$$G_{\mathit{i}}^L=\sum _{\mathit{j}=1}^{\mathbf{npde}}{E}_{\mathit{i},\mathit{j}}^L\frac{\partial U_{\mathit{j}}}{\partial t}+S_{\mathit{i}}^L\text{,}$$

(11)

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:     ires – Integer *Input/Output

On entry: the form $G_i^L$ (or $G_i^R$) that must be returned in the array res.

$\mathbf{ires}=-1$

Equation (10) must be used.

$\mathbf{ires}=1$

Equation (11) must be used.

On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions, as described below:

$\mathbf{ires}=2$

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 $\mathbf{ires}=3$, then nag_pde_parab_1d_keller (d03pec) returns to the calling function with the error indicator set to $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

9:     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 nag_pde_parab_1d_keller (d03pec) you may allocate memory and initialize these pointers with various quantities for use by bndary when called from nag_pde_parab_1d_keller (d03pec) (see Section 3.2.1.1 in the Essential Introduction).

6:     u[$\mathbf{npde}\times \mathbf{npts}$] – doubleInput/Output

On entry: the initial values of $U\left(x,t\right)$ 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: $\mathbf{u}\left[\mathbf{npde}\times \left(\mathit{j}-1\right)+\mathit{i}-1\right]$ will contain the computed solution at $t=\mathbf{ts}$.

7:     npts – IntegerInput

On entry: the number of mesh points in the interval $\left[a,b\right]$.

Constraint: $\mathbf{npts}\ge 3$.

8:     x[npts] – const doubleInput

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, $b$.

Constraint: $\mathbf{x}\left[0\right]<\mathbf{x}\left[1\right]<\cdots <\mathbf{x}\left[\mathbf{npts}-1\right]$.

9:     nleft – IntegerInput

On entry: the number $n_a$ of boundary conditions at the left-hand mesh point $\mathbf{x}\left[0\right]$.

Constraint: $0\le \mathbf{nleft}\le \mathbf{npde}$.

10:   acc – doubleInput

On entry: a positive quantity for controlling the local error estimate in the time integration. If $E\left(i,j\right)$ is the estimated error for $U_i$ at the $j$th mesh point, the error test is:

$$\left|E\left(i,j\right)\right|=\mathbf{acc}\times \left(1.0+\left|\mathbf{u}\left[\mathbf{npde}\times \left(j-1\right)+i-1\right]\right|\right)\text{.}$$

Constraint: $\mathbf{acc}>0.0$.

11:   rsave[lrsave] – doubleCommunication Array

If $\mathbf{ind}=0$, rsave need not be set on entry.

If $\mathbf{ind}=1$, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.

12:   lrsave – IntegerInput

On entry: the dimension of the array rsave.

Constraint: $$\begin{gathered} \mathbf{lrsave}\ge \left(4\times \mathbf{npde}+\mathbf{nleft}+14\right)\times \mathbf{npde}\times \mathbf{npts}+\left(3\times \mathbf{npde}+21\right)\times \mathbf{npde}+ \\ 7\times \mathbf{npts}+54 \end{gathered}$$.

13:   isave[lisave] – IntegerCommunication Array

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.

14:   lisave – IntegerInput

On entry: the dimension of the array isave.

Constraint: $\mathbf{lisave}\ge \mathbf{npde}\times \mathbf{npts}+24$.

15:   itask – IntegerInput

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:   itrace – IntegerInput

On entry: the level of trace information required from nag_pde_parab_1d_keller (d03pec) and the underlying ODE solver as follows:

$\mathbf{itrace}\le -1$

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:   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:   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 arguments tout and fail should be reset between calls to nag_pde_parab_1d_keller (d03pec).

Constraint: $\mathbf{ind}=0$ or $1$.

On exit: $\mathbf{ind}=1$.

19:   comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

20:   saved – Nag_D03_Save *Communication Structure

saved must remain unchanged following a previous call to a Chapter d03 function and prior to any subsequent call to a Chapter d03 function.

21:   fail – NagError *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 a small change in acc is unlikely to result in a changed solution. $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ 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 $\mathbf{ires}=3$ in pdedef or bndary.

NE_FAILED_START

acc was too small to start integration: $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.

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: $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.

Underlying ODE solver cannot make further progress from the point ts with the supplied value of acc. $\mathbf{ts}=⟨\mathit{\text{value}}⟩$, $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.

NE_INT

ires set to an invalid value in call to pdedef or bndary.

On entry, $\mathbf{ind}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ind}=0$ or $1$.

On entry, $\mathbf{itask}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{itask}=1$, $2$ or $3$.

On entry, $\mathbf{nleft}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nleft}\ge 0$.

On entry, $\mathbf{npde}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{npde}\ge 1$.

On entry, $\mathbf{npts}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{npts}\ge 3$.

NE_INT_2

On entry, lisave is too small: $\mathbf{lisave}=⟨\mathit{\text{value}}⟩$. Minimum possible dimension: $⟨\mathit{\text{value}}⟩$.

On entry, lrsave is too small: $\mathbf{lrsave}=⟨\mathit{\text{value}}⟩$. Minimum possible dimension: $⟨\mathit{\text{value}}⟩$.

On entry, $\mathbf{nleft}=⟨\mathit{\text{value}}⟩$, $\mathbf{npde}=⟨\mathit{\text{value}}⟩$.
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.

Serious error in internal call to an auxiliary. Increase itrace for further details.

NE_NOT_CLOSE_FILE

Cannot close file $⟨\mathit{\text{value}}⟩$.

NE_NOT_STRICTLY_INCREASING

On entry, mesh points x appear to be badly ordered: $\mathit{I}=⟨\mathit{\text{value}}⟩$, $\mathbf{x}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$, $\mathit{J}=⟨\mathit{\text{value}}⟩$ and $\mathbf{x}\left[\mathit{J}-1\right]=⟨\mathit{\text{value}}⟩$.

NE_NOT_WRITE_FILE

Cannot open file $⟨\mathit{\text{value}}⟩$ for writing.

NE_REAL

On entry, $\mathbf{acc}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{acc}>0.0$.

NE_REAL_2

On entry, $\mathbf{tout}=⟨\mathit{\text{value}}⟩$ and $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{tout}>\mathbf{ts}$.

On entry, $\mathbf{tout}-\mathbf{ts}$ is too small: $\mathbf{tout}=⟨\mathit{\text{value}}⟩$ and $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.

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: $\mathbf{ts}=⟨\mathit{\text{value}}⟩$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (d03pece.c)

10.2  Program Data

10.3  Program Results

Program Results (d03pece.r)