NAG Library Function Document

1Purpose

nag_pde_parab_1d_coll (d03pdc) integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable. The spatial discretization is performed using a Chebyshev ${C}^{0}$ collocation method, 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.

2Specification

 #include #include
void  nag_pde_parab_1d_coll (Integer npde, Integer m, double *ts, double tout,
 void (*pdedef)(Integer npde, double t, const double x[], Integer nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires, Nag_Comm *comm),
 void (*bndary)(Integer npde, double t, const double u[], const double ux[], Integer ibnd, double beta[], double gamma[], Integer *ires, Nag_Comm *comm),
double u[], Integer nbkpts, const double xbkpts[], Integer npoly, Integer npts, double x[],
 void (*uinit)(Integer npde, Integer npts, const double x[], double u[], Nag_Comm *comm),
double acc, 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)

3Description

nag_pde_parab_1d_coll (d03pdc) integrates the system of parabolic equations:
 $∑j=1npdePi,j ∂Uj ∂t +Qi=x-m ∂∂x xmRi, i=1,2,…,npde, a≤x≤b,t≥t0,$ (1)
where ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ depend on $x$, $t$, $U$, ${U}_{x}$ and the vector $U$ is the set of solution values
 $U x,t = U 1 x,t ,…, U npde x,t T ,$ (2)
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 $b={x}_{{\mathbf{nbkpts}}}$ are the leftmost and rightmost of a user-defined set of break-points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{nbkpts}}}$. 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 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 $U\left(x,t\right)$ must be given at $t={t}_{0}$, and must be specified in uinit.
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
 $βix,tRix,t,U,Ux=γix,t,U,Ux, i=1,2,…,npde,$ (3)
where $x=a$ or $x=b$.
The boundary conditions must be specified in bndary. Thus, 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 both the break-points and internally selected points for each element in turn, that is ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ are evaluated twice at each break-point. Any discontinuities in these functions must therefore be at one or more of the break-points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{nbkpts}}}$; (iv) at least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the problem; (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 the mesh points. This ODE system is obtained by approximating the PDE solution between each pair of break-points by a Chebyshev polynomial of degree npoly. The interval between each pair of break-points is treated by nag_pde_parab_1d_coll (d03pdc) as an element, and on this element, a polynomial and its space and time derivatives are made to satisfy the system of PDEs at ${\mathbf{npoly}}-1$ spatial points, which are chosen internally by the code and the break-points. In the case of just one element, the break-points are the boundaries. The user-defined break-points and the internally selected points together define the mesh. The smallest value that npoly can take is one, in which case, the solution is approximated by piecewise linear polynomials between consecutive break-points and the method is similar to an ordinary finite element method.
In total there are $\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$ mesh points in the spatial direction, and ${\mathbf{npde}}×\left(\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1\right)$ ODEs in the time direction; one ODE at each break-point for each PDE component and (${\mathbf{npoly}}-1$) ODEs for each PDE component between each pair of break-points. The system is then integrated forwards in time using a backward differentiation formula method.
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 and Dew P M (1991) Algorithm 690: Chebyshev polynomial software for elliptic-parabolic systems of PDEs ACM Trans. Math. Software 17 178–206
Zaturska N B, Drazin P G and Banks W H H (1988) On the flow of a viscous fluid driven along a channel by a suction at porous walls Fluid Dynamics Research 4

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 values of the functions ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$ and ${U}_{x}$ and must be evaluated at a set of points.
The specification of pdedef is:
 void pdedef (Integer npde, double t, const double x[], Integer nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires, Nag_Comm *comm)
1:    $\mathbf{npde}$IntegerInput
On entry: the number of PDEs in the system.
2:    $\mathbf{t}$doubleInput
On entry: the current value of the independent variable $t$.
3:    $\mathbf{x}\left[{\mathbf{nptl}}\right]$const doubleInput
On entry: contains a set of mesh points at which ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ are to be evaluated. ${\mathbf{x}}\left[0\right]$ and ${\mathbf{x}}\left[{\mathbf{nptl}}-1\right]$ contain successive user-supplied break-points and the elements of the array will satisfy ${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{nptl}}-1\right]$.
4:    $\mathbf{nptl}$IntegerInput
On entry: the number of points at which evaluations are required (the value of ${\mathbf{npoly}}+1$).
5:    $\mathbf{u}\left[{\mathbf{npde}}×{\mathbf{nptl}}\right]$const doubleInput
Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On entry: ${\mathbf{u}}\left[\left(\mathit{j}-1\right)×{\mathbf{npde}}+\mathit{i}-1\right]$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$ where $x={\mathbf{x}}\left[\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
6:    $\mathbf{ux}\left[{\mathbf{npde}}×{\mathbf{nptl}}\right]$const doubleInput
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{ux}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On entry: ${\mathbf{ux}}\left[\left(\mathit{j}-1\right)×{\mathbf{npde}}+\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ where $x={\mathbf{x}}\left[\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
7:    $\mathbf{p}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array p is ${\mathbf{npde}}×{\mathbf{npde}}×{\mathbf{nptl}}$.
Where ${\mathbf{P}}\left(i,j,k\right)$ appears in this document, it refers to the array element ${\mathbf{p}}\left[\left(k-1\right)×{\mathbf{npde}}×{\mathbf{npde}}+\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On exit: ${\mathbf{p}}\left[{\mathbf{npde}}×{\mathbf{npde}}×\left(\mathit{k}-1\right)+{\mathbf{npde}}×\left(\mathit{j}-1\right)+\left(\mathit{i}-1\right)\right]$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left[\mathit{k}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{k}=1,2,\dots ,{\mathbf{nptl}}$.
8:    $\mathbf{q}\left[{\mathbf{npde}}×{\mathbf{nptl}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix $Q$ is stored in ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On exit: ${\mathbf{q}}\left[\left(\mathit{j}-1\right)×{\mathbf{npde}}+\mathit{i}-1\right]$ must be set to the value of ${Q}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left[\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
9:    $\mathbf{r}\left[{\mathbf{npde}}×{\mathbf{nptl}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix $R$ is stored in ${\mathbf{r}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On exit: ${\mathbf{r}}\left[\left(\mathit{j}-1\right)×{\mathbf{npde}}+\mathit{i}-1\right]$ must be set to the value of ${R}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left[\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
10:  $\mathbf{ires}$Integer *Input/Output
On entry: set to $-1\text{​ or ​}1$.
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 ${\mathbf{ires}}=3$, nag_pde_parab_1d_coll (d03pdc) returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.
11:  $\mathbf{comm}$Nag_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_coll (d03pdc) you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from nag_pde_parab_1d_coll (d03pdc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_pde_parab_1d_coll (d03pdc). If your code inadvertently does return any NaNs or infinities, nag_pde_parab_1d_coll (d03pdc) 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).
The specification of bndary is:
 void bndary (Integer npde, double t, const double u[], const double ux[], Integer ibnd, double beta[], double gamma[], Integer *ires, Nag_Comm *comm)
1:    $\mathbf{npde}$IntegerInput
On entry: the number of PDEs in the system.
2:    $\mathbf{t}$doubleInput
On entry: the current value of the independent variable $t$.
3:    $\mathbf{u}\left[{\mathbf{npde}}\right]$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 $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
4:    $\mathbf{ux}\left[{\mathbf{npde}}\right]$const doubleInput
On entry: ${\mathbf{ux}}\left[\mathit{i}-1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
5:    $\mathbf{ibnd}$IntegerInput
On entry: specifies which boundary conditions are to be evaluated.
${\mathbf{ibnd}}=0$
bndary must set up the coefficients of the left-hand boundary, $x=a$.
${\mathbf{ibnd}}\ne 0$
bndary must set up the coefficients of the right-hand boundary, $x=b$.
6:    $\mathbf{beta}\left[{\mathbf{npde}}\right]$doubleOutput
On exit: ${\mathbf{beta}}\left[\mathit{i}-1\right]$ must be set to the value of ${\beta }_{\mathit{i}}\left(x,t\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
7:    $\mathbf{gamma}\left[{\mathbf{npde}}\right]$doubleOutput
On exit: ${\mathbf{gamma}}\left[\mathit{i}-1\right]$ must be set to the value of ${\gamma }_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
8:    $\mathbf{ires}$Integer *Input/Output
On entry: set to $-1\text{​ or ​}1$.
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 ${\mathbf{ires}}=3$, nag_pde_parab_1d_coll (d03pdc) 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.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_coll (d03pdc) you may allocate memory and initialize these pointers with various quantities for use by bndary when called from nag_pde_parab_1d_coll (d03pdc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_pde_parab_1d_coll (d03pdc). If your code inadvertently does return any NaNs or infinities, nag_pde_parab_1d_coll (d03pdc) is likely to produce unexpected results.
7:    $\mathbf{u}\left[{\mathbf{npde}}×{\mathbf{npts}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On entry: if ${\mathbf{ind}}=1$ the value of u must be unchanged from the previous call.
On exit: ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$ will contain the computed solution at $t={\mathbf{ts}}$.
8:    $\mathbf{nbkpts}$IntegerInput
On entry: the number of break-points in the interval $\left[a,b\right]$.
Constraint: ${\mathbf{nbkpts}}\ge 2$.
9:    $\mathbf{xbkpts}\left[{\mathbf{nbkpts}}\right]$const doubleInput
On entry: the values of the break-points in the space direction. ${\mathbf{xbkpts}}\left[0\right]$ must specify the left-hand boundary, $a$, and ${\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}-1\right]$ must specify the right-hand boundary, $b$.
Constraint: ${\mathbf{xbkpts}}\left[0\right]<{\mathbf{xbkpts}}\left[1\right]<\cdots <{\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}-1\right]$.
10:  $\mathbf{npoly}$IntegerInput
On entry: the degree of the Chebyshev polynomial to be used in approximating the PDE solution between each pair of break-points.
Constraint: $1\le {\mathbf{npoly}}\le 49$.
11:  $\mathbf{npts}$IntegerInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
Constraint: ${\mathbf{npts}}=\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$.
12:  $\mathbf{x}\left[{\mathbf{npts}}\right]$doubleOutput
On exit: the mesh points chosen by nag_pde_parab_1d_coll (d03pdc) in the spatial direction. The values of x will satisfy ${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{npts}}-1\right]$.
13:  $\mathbf{uinit}$function, supplied by the userExternal Function
uinit must compute the initial values of the PDE components ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
The specification of uinit is:
 void uinit (Integer npde, Integer npts, const double x[], double u[], Nag_Comm *comm)
1:    $\mathbf{npde}$IntegerInput
On entry: the number of PDEs in the system.
2:    $\mathbf{npts}$IntegerInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
3:    $\mathbf{x}\left[{\mathbf{npts}}\right]$const doubleInput
On entry: ${\mathbf{x}}\left[\mathit{j}-1\right]$, contains the values of the $\mathit{j}$th mesh point, for $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
4:    $\mathbf{u}\left[{\mathbf{npde}}×{\mathbf{npts}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On exit: ${\mathbf{u}}\left[\left(\mathit{j}-1\right)×{\mathbf{npde}}+\mathit{i}-1\right]$ must be set to the initial value ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
5:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to uinit.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_coll (d03pdc) you may allocate memory and initialize these pointers with various quantities for use by uinit when called from nag_pde_parab_1d_coll (d03pdc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: uinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_pde_parab_1d_coll (d03pdc). If your code inadvertently does return any NaNs or infinities, nag_pde_parab_1d_coll (d03pdc) is likely to produce unexpected results.
14:  $\mathbf{acc}$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:
 $Ei,j=acc×1.0+u[j-1×npde+i-1].$
Constraint: ${\mathbf{acc}}>0.0$.
15:  $\mathbf{rsave}\left[{\mathbf{lrsave}}\right]$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.
16:  $\mathbf{lrsave}$IntegerInput
On entry: the dimension of the array rsave.
Constraint: ${\mathbf{lrsave}}\ge 11×{\mathbf{npde}}×{\mathbf{npts}}+50+\mathit{nwkres}+\mathit{lenode}$.
17:  $\mathbf{isave}\left[{\mathbf{lisave}}\right]$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.
18:  $\mathbf{lisave}$IntegerInput
On entry: the dimension of the array isave.
Constraint: ${\mathbf{lisave}}\ge {\mathbf{npde}}×{\mathbf{npts}}+24$.
19:  $\mathbf{itask}$IntegerInput
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 $3$.
20:  $\mathbf{itrace}$IntegerInput
On entry: the level of trace information required from nag_pde_parab_1d_coll (d03pdc) 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.
21:  $\mathbf{outfile}$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.
22:  $\mathbf{ind}$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 arguments tout and fail should be reset between calls to nag_pde_parab_1d_coll (d03pdc).
Constraint: ${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
23:  $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
24:  $\mathbf{saved}$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.
25:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ACC_IN_DOUBT
Integration completed, but a small change in acc is unlikely to result in a changed solution. ${\mathbf{acc}}=〈\mathit{\text{value}}〉$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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_INCOMPAT_PARAM
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{xbkpts}}\left[0\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\le 0$ or ${\mathbf{xbkpts}}\left[0\right]\ge 0.0$
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, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}=0$, $1$ or $2$.
On entry, ${\mathbf{nbkpts}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nbkpts}}\ge 2$.
On entry, ${\mathbf{npde}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npde}}\ge 1$.
On entry, ${\mathbf{npoly}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{npoly}}\le 49$.
On entry, ${\mathbf{npoly}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npoly}}\le 49$.
On entry, ${\mathbf{npoly}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npoly}}\ge 1$.
On entry, on initial entry ${\mathbf{ind}}=1$.
Constraint: on initial entry ${\mathbf{ind}}=0$.
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: $〈\mathit{\text{value}}〉$.
NE_INT_3
On entry, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$, ${\mathbf{nbkpts}}=〈\mathit{\text{value}}〉$ and ${\mathbf{npoly}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{npts}}=\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$.
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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_CLOSE_FILE
Cannot close file $〈\mathit{\text{value}}〉$.
NE_NOT_STRICTLY_INCREASING
On entry, break-points xbkpts are badly ordered: $\mathit{I}=〈\mathit{\text{value}}〉$, ${\mathbf{xbkpts}}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$, $\mathit{J}=〈\mathit{\text{value}}〉$ and ${\mathbf{xbkpts}}\left[\mathit{J}-1\right]=〈\mathit{\text{value}}〉$.
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_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: ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.

7Accuracy

nag_pde_parab_1d_coll (d03pdc) controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on the degree of the polynomial approximation npoly, and on both the number of break-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

nag_pde_parab_1d_coll (d03pdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_pde_parab_1d_coll (d03pdc) 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.

nag_pde_parab_1d_coll (d03pdc) is designed to solve parabolic systems (possibly including 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.
The time taken depends on the complexity of the parabolic system and on the accuracy requested.

10Example

The problem consists of a fourth-order PDE which can be written as a pair of second-order elliptic-parabolic PDEs for ${U}_{1}\left(x,t\right)$ and ${U}_{2}\left(x,t\right)$,
 $0= ∂2U1 ∂x2 -U2$ (4)
 $∂U2 ∂t = ∂2U2 ∂x2 +U2 ∂U1 ∂x -U1 ∂U2 ∂x$ (5)
where $-1\le x\le 1$ and $t\ge 0$. The boundary conditions are given by
 $∂U1 ∂x =0 and U1=1 at ​x=-1, and ∂U1 ∂x =0 and U1=-1 at ​x=1.$
The initial conditions at $t=0$ are given by
 $U1=-sin⁡πx2 and U2=π24sin⁡πx2.$
The absence of boundary conditions for ${U}_{2}\left(x,t\right)$ does not pose any difficulties provided that the derivative flux boundary conditions are assigned to the first PDE (4) which has the correct flux, $\frac{\partial {U}_{1}}{\partial x}$. The conditions on ${U}_{1}\left(x,t\right)$ at the boundaries are assigned to the second PDE by setting ${\beta }_{2}=0.0$ in equation (3) and placing the Dirichlet boundary conditions on ${U}_{1}\left(x,t\right)$ in the function ${\gamma }_{2}$.

10.1Program Text

Program Text (d03pdce.c)

None.

10.3Program Results

Program Results (d03pdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017