d03 Chapter Contents
d03 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_pde_parab_1d_keller_ode (d03pkc)

## 1  Purpose

nag_pde_parab_1d_keller_ode (d03pkc) integrates a system of linear or nonlinear, first-order, time-dependent partial differential equations (PDEs) in one space variable, with scope for coupled ordinary differential equations (ODEs). 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 ODEs. The resulting system is solved using a Backward Differentiation Formula (BDF) method or a Theta method (switching between Newton's method and functional iteration).

## 2  Specification

 #include #include
void  nag_pde_parab_1d_keller_ode (Integer npde, double *ts, double tout,
 void (*pdedef)(Integer npde, double t, double x, const double u[], const double ut[], const double ux[], Integer ncode, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm),
 void (*bndary)(Integer npde, double t, Integer ibnd, Integer nobc, const double u[], const double ut[], Integer ncode, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm),
double u[], Integer npts, const double x[], Integer nleft, Integer ncode,
 void (*odedef)(Integer npde, double t, Integer ncode, const double v[], const double vdot[], Integer nxi, const double xi[], const double ucp[], const double ucpx[], const double ucpt[], double r[], Integer *ires, Nag_Comm *comm),
Integer nxi, const double xi[], Integer neqn, const double rtol[], const double atol[], Integer itol, Nag_NormType norm, Nag_LinAlgOption laopt, const double algopt[], 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)

## 3  Description

nag_pde_parab_1d_keller_ode (d03pkc) integrates the system of first-order PDEs and coupled ODEs
 $Gi x,t,U,Ux,Ut,V,V. = 0 , i=1,2,…,npde , a≤x≤b,t≥t0 ,$ (1)
 $Ri t,V,V.,ξ,U*,Ux*,Ut* = 0 , i=1,2,…,ncode .$ (2)
In the PDE part of the problem given by (1), the functions ${G}_{i}$ must have the general form
 $Gi = ∑ j=1 npde Pi,j ∂Uj ∂t + ∑ j=1 ncode Qi,j V.j + Si = 0 , i=1,2,…,npde ,$ (3)
where ${P}_{i,j}$, ${Q}_{i,j}$ and ${S}_{i}$ depend on $x,t,U,{U}_{x}$ and $V$.
The vector $U$ is the set of PDE solution values
 $U x,t = U 1 x,t ,…, U npde x,t T ,$
and the vector ${U}_{x}$ is the partial derivative with respect to $x$. The vector $V$ is the set of ODE solution values
 $Vt=V1t,…,VncodetT,$
and $\stackrel{.}{V}$ denotes its derivative with respect to time.
In the ODE part given by (2), $\xi$ represents a vector of ${n}_{\xi }$ spatial coupling points at which the ODEs are coupled to the PDEs. These points may or may not be equal to some of the PDE spatial mesh points. ${U}^{*}$, ${U}_{x}^{*}$ and ${U}_{t}^{*}$ are the functions $U$, ${U}_{x}$ and ${U}_{t}$ evaluated at these coupling points. Each ${R}_{i}$ may only depend linearly on time derivatives. Hence equation (2) may be written more precisely as
 $R=A-BV.-CUt*,$ (4)
where $R={\left[{R}_{1},\dots ,{R}_{{\mathbf{ncode}}}\right]}^{\mathrm{T}}$, $A$ is a vector of length ncode, $B$ is an ncode by ncode matrix, $C$ is an ncode by $\left({n}_{\xi }×{\mathbf{npde}}\right)$ matrix. The entries in $A$, $B$ and $C$ may depend on $t$, $\xi$, ${U}^{*}$, ${U}_{x}^{*}$ and $V$. In practice you only need to supply a vector of information to define the ODEs and not the matrices $B$ and $C$. (See Section 5 for the specification of odedef.)
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 PDE system which is defined by the functions ${G}_{i}$ must be specified in pdedef.
The initial values of the functions $U\left(x,t\right)$ and $V\left(t\right)$ 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.
The boundary conditions have the form:
 $G i L x,t,U, U t ,V, V . = 0 at ​ x = a , i=1,2,…,na ,$ (5)
at the left-hand boundary, and
 $G i R x,t,U, U t ,V, V . = 0 at ​ x = b, i=1,2,…,nb,$ (6)
at the right-hand boundary.
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. 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:
 $∑j=1npdeEi,jL ∂Uj ∂t +∑j=1ncodeHi,jLV.j+KiL=0, i=1,2,…,na,$ (7)
at the left-hand boundary, and
 $∑j=1npdeEi,jR ∂Uj ∂t +∑j=1ncodeHi,jRV.j+KiR=0, i=1,2,…,nb,$ (8)
at the right-hand boundary, where ${E}_{i,j}^{L}$, ${E}_{i,j}^{R}$, ${H}_{i,j}^{L}$, ${H}_{i,j}^{R}$, ${K}_{i}^{L}$ and ${K}_{i}^{R}$ depend on $x,t,U$ and $V$ only.
The boundary conditions must be specified in bndary.
The problem is subject to the following restrictions:
 (i) ${P}_{i,j}$, ${Q}_{i,j}$ and ${S}_{i}$ must not depend on any time derivatives; (ii) ${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction; (iii) The evaluation of the function ${G}_{i}$ is done approximately at the mid-points of the mesh ${\mathbf{x}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$, 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 PDE problem.
The algebraic-differential equation system which is defined by the functions ${R}_{i}$ must be specified in odedef. You must also specify the coupling points $\xi$ in the array xi.
The parabolic equations are approximated by a system of ODEs in time for the values of ${U}_{i}$ at mesh points. 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 ${\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{ncode}}$ ODEs in time direction. This system is then integrated forwards in time using a Backward Differentiation Formula (BDF) or a Theta method.

## 4  References

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
Berzins M and Furzeland R M (1992) An adaptive theta method for the solution of stiff and nonstiff differential equations Appl. Numer. Math. 9 1–19
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

## 5  Arguments

1:     npdeIntegerInput
On entry: the number of PDEs to be solved.
Constraint: ${\mathbf{npde}}\ge 1$.
2:     tsdouble *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 in u. Normally ${\mathbf{ts}}={\mathbf{tout}}$.
3:     toutdoubleInput
On entry: the final value of $t$ to which the integration is to be carried out.
4:     pdedeffunction, supplied by the userExternal Function
pdedef must evaluate 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_ode (d03pkc).
The specification of pdedef is:
 void pdedef (Integer npde, double t, double x, const double u[], const double ut[], const double ux[], Integer ncode, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm)
1:     npdeIntegerInput
On entry: the number of PDEs in the system.
2:     tdoubleInput
On entry: the current value of the independent variable $t$.
3:     xdoubleInput
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:     ncodeIntegerInput
On entry: the number of coupled ODEs in the system.
8:     v[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{v}}\left[\mathit{i}-1\right]$ contains the value of component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
9:     vdot[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{vdot}}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
10:   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
 $Gi=∑j=1npdePi,j ∂Uj ∂t +∑j=1ncodeQi,jV.j,$ (9)
i.e., only terms depending explicitly on time derivatives, or
 $Gi=∑j=1npdePi,j ∂Uj ∂t +∑j=1ncodeQi,jV.j+Si,$ (10)
i.e., all terms in equation (3).
The definition of $G$ is determined by the input value of ires.
11:   iresInteger *Input/Output
On entry: the form of ${G}_{i}$ that must be returned in the array res.
${\mathbf{ires}}=-1$
Equation (9) must be used.
${\mathbf{ires}}=1$
Equation (10) 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 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_ode (d03pkc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
12:   commNag_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_keller_ode (d03pkc) you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from nag_pde_parab_1d_keller_ode (d03pkc) (see Section 3.2.1 in the Essential Introduction).
5:     bndaryfunction, supplied by the userExternal Function
bndary must evaluate the functions ${G}_{i}^{L}$ and ${G}_{i}^{R}$ which describe the boundary conditions, as given in (5) and (6).
The specification of bndary is:
 void bndary (Integer npde, double t, Integer ibnd, Integer nobc, const double u[], const double ut[], Integer ncode, const double v[], const double vdot[], double res[], Integer *ires, Nag_Comm *comm)
1:     npdeIntegerInput
On entry: the number of PDEs in the system.
2:     tdoubleInput
On entry: the current value of the independent variable $t$.
3:     ibndIntegerInput
On entry: specifies which boundary conditions are to be evaluated.
${\mathbf{ibnd}}=0$
bndary must compute the left-hand boundary condition at $x=a$.
${\mathbf{ibnd}}\ne 0$
bndary must compute the right-hand boundary condition at $x=b$.
4:     nobcIntegerInput
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:     ncodeIntegerInput
On entry: the number of coupled ODEs in the system.
8:     v[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{v}}\left[\mathit{i}-1\right]$ contains the value of component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
9:     vdot[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{vdot}}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
Note: ${\mathbf{vdot}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$, may only appear linearly as in (7) and (8).
10:   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
 $GiL=∑j=1npdeEi,jL ∂Uj ∂t +∑j=1ncodeHi,jLV.j,$ (11)
i.e., only terms depending explicitly on time derivatives, or
 $GiL=∑j=1npdeEi,jL ∂Uj ∂t +∑j=1ncodeHi,jLV.j+KiL,$ (12)
i.e., all terms in equation (7), and similarly for ${G}_{\mathit{i}}^{R}$.
The definitions of ${G}^{L}$ and ${G}^{R}$ are determined by the input value of ires.
11:   iresInteger *Input/Output
On entry: the form of ${G}_{i}^{L}$ (or ${G}_{i}^{R}$) that must be returned in the array res.
${\mathbf{ires}}=-1$
Equation (11) must be used.
${\mathbf{ires}}=1$
Equation (12) 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 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_ode (d03pkc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
12:   commNag_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_keller_ode (d03pkc) you may allocate memory and initialize these pointers with various quantities for use by bndary when called from nag_pde_parab_1d_keller_ode (d03pkc) (see Section 3.2.1 in the Essential Introduction).
6:     u[neqn]doubleInput/Output
On entry: the initial values of the dependent variables defined as follows:
• ${\mathbf{u}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contain ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and
• ${\mathbf{u}}\left[{\mathbf{npts}}×{\mathbf{npde}}+\mathit{i}-1\right]$ contain ${V}_{\mathit{i}}\left({t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
On exit: the computed solution ${U}_{\mathit{i}}\left({x}_{\mathit{j}},t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and ${V}_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,{\mathbf{ncode}}$, evaluated at $t={\mathbf{ts}}$.
7:     nptsIntegerInput
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 space 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:     nleftIntegerInput
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:   ncodeIntegerInput
On entry: the number of coupled ODE components.
Constraint: ${\mathbf{ncode}}\ge 0$.
11:   odedeffunction, supplied by the userExternal Function
odedef must evaluate the functions $R$, which define the system of ODEs, as given in (4).
If ${\mathbf{ncode}}=0$, odedef will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to nag_pde_parab_1d_keller_ode (d03pkc).
The specification of odedef is:
 void odedef (Integer npde, double t, Integer ncode, const double v[], const double vdot[], Integer nxi, const double xi[], const double ucp[], const double ucpx[], const double ucpt[], double r[], Integer *ires, Nag_Comm *comm)
1:     npdeIntegerInput
On entry: the number of PDEs in the system.
2:     tdoubleInput
On entry: the current value of the independent variable $t$.
3:     ncodeIntegerInput
On entry: the number of coupled ODEs in the system.
4:     v[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{v}}\left[\mathit{i}-1\right]$ contains the value of component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
5:     vdot[ncode]const doubleInput
On entry: if ${\mathbf{ncode}}>0$, ${\mathbf{vdot}}\left[\mathit{i}-1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$.
6:     nxiIntegerInput
On entry: the number of ODE/PDE coupling points.
7:     xi[nxi]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left[\mathit{i}-1\right]$ contains the ODE/PDE coupling points, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$.
8:     ucp[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucp}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of ${U}_{\mathit{i}}\left(x,t\right)$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
9:     ucpx[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpx}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
10:   ucpt[${\mathbf{npde}}×{\mathbf{nxi}}$]const doubleInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpt}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ contains the value of $\frac{\partial {U}_{\mathit{i}}}{\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
11:   r[ncode]doubleOutput
On exit: if ${\mathbf{ncode}}>0$, ${\mathbf{r}}\left[\mathit{i}-1\right]$ must contain the $\mathit{i}$th component of $R$, for $\mathit{i}=1,2,\dots ,{\mathbf{ncode}}$, where $R$ is defined as
 $R=-BV.-CUt*,$ (13)
i.e., only terms depending explicitly on time derivatives, or
 $R=A-BV.-CUt*,$ (14)
i.e., all terms in equation (4). The definition of $R$ is determined by the input value of ires.
12:   iresInteger *Input/Output
On entry: the form of $R$ that must be returned in the array r.
${\mathbf{ires}}=-1$
Equation (13) must be used.
${\mathbf{ires}}=1$
Equation (14) must be used.
On exit: should usually remain unchanged. However, you may reset 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 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_ode (d03pkc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
13:   commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to odedef.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_pde_parab_1d_keller_ode (d03pkc) you may allocate memory and initialize these pointers with various quantities for use by odedef when called from nag_pde_parab_1d_keller_ode (d03pkc) (see Section 3.2.1 in the Essential Introduction).
12:   nxiIntegerInput
On entry: the number of ODE/PDE coupling points.
Constraints:
• if ${\mathbf{ncode}}=0$, ${\mathbf{nxi}}=0$;
• if ${\mathbf{ncode}}>0$, ${\mathbf{nxi}}\ge 0$.
13:   xi[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array xi must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nxi}}\right)$.
On entry: ${\mathbf{xi}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$, must be set to the ODE/PDE coupling points, ${\xi }_{\mathit{i}}$.
Constraint: ${\mathbf{x}}\left[0\right]\le {\mathbf{xi}}\left[0\right]<{\mathbf{xi}}\left[1\right]<\cdots <{\mathbf{xi}}\left[{\mathbf{nxi}}-1\right]\le {\mathbf{x}}\left[{\mathbf{npts}}-1\right]$.
14:   neqnIntegerInput
On entry: the number of ODEs in the time direction.
Constraint: ${\mathbf{neqn}}={\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{ncode}}$.
15:   rtol[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array rtol must be at least
• $1$ when ${\mathbf{itol}}=1$ or $2$;
• ${\mathbf{neqn}}$ when ${\mathbf{itol}}=3$ or $4$.
On entry: the relative local error tolerance.
Constraint: ${\mathbf{rtol}}\left[i-1\right]\ge 0.0$ for all relevant $i$.
16:   atol[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array atol must be at least
• $1$ when ${\mathbf{itol}}=1$ or $3$;
• ${\mathbf{neqn}}$ when ${\mathbf{itol}}=2$ or $4$.
On entry: the absolute local error tolerance.
Constraint: ${\mathbf{atol}}\left[i-1\right]\ge 0.0$ for all relevant $i$.
Note: corresponding elements of rtol and atol cannot both be $0.0$.
17:   itolIntegerInput
On entry: a value to indicate the form of the local error test. itol indicates to nag_pde_parab_1d_keller_ode (d03pkc) whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $‖{e}_{i}/{w}_{i}‖<1.0$, where ${w}_{i}$ is defined as follows:
 itol rtol atol ${w}_{i}$ 1 scalar scalar ${\mathbf{rtol}}\left[0\right]×\left|{\mathbf{u}}\left[i-1\right]\right|+{\mathbf{atol}}\left[0\right]$ 2 scalar vector ${\mathbf{rtol}}\left[0\right]×\left|{\mathbf{u}}\left[i-1\right]\right|+{\mathbf{atol}}\left[i-1\right]$ 3 vector scalar ${\mathbf{rtol}}\left[i-1\right]×\left|{\mathbf{u}}\left[i-1\right]\right|+{\mathbf{atol}}\left[0\right]$ 4 vector vector ${\mathbf{rtol}}\left[i-1\right]×\left|{\mathbf{u}}\left[i-1\right]\right|+{\mathbf{atol}}\left[i-1\right]$
In the above, ${e}_{\mathit{i}}$ denotes the estimated local error for the $\mathit{i}$th component of the coupled PDE/ODE system in time, ${\mathbf{u}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{neqn}}$.
The choice of norm used is defined by the argument norm.
Constraint: $1\le {\mathbf{itol}}\le 4$.
18:   normNag_NormTypeInput
On entry: the type of norm to be used.
${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$
Maximum norm.
${\mathbf{norm}}=\mathrm{Nag_TwoNorm}$
Averaged ${L}_{2}$ norm.
If ${{\mathbf{u}}}_{\mathrm{norm}}$ denotes the norm of the vector u of length neqn, then for the averaged ${L}_{2}$ norm
 $unorm=1neqn∑i=1neqnu[i-1]/wi2,$
while for the maximum norm
 $u norm = maxi u[i-1] / wi .$
See the description of itol for the formulation of the weight vector $w$.
Constraint: ${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$ or $\mathrm{Nag_TwoNorm}$.
19:   laoptNag_LinAlgOptionInput
On entry: the type of matrix algebra required.
${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$
Full matrix methods to be used.
${\mathbf{laopt}}=\mathrm{Nag_LinAlgBand}$
Banded matrix methods to be used.
${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$
Sparse matrix methods to be used.
Constraint: ${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$, $\mathrm{Nag_LinAlgBand}$ or $\mathrm{Nag_LinAlgSparse}$.
Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., ${\mathbf{ncode}}=0$).
20:   algopt[$30$]const doubleInput
On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, then ${\mathbf{algopt}}\left[0\right]$ should be set to $0.0$. Default values will also be used for any other elements of algopt set to zero. The permissible values, default values, and meanings are as follows:
${\mathbf{algopt}}\left[0\right]$
Selects the ODE integration method to be used. If ${\mathbf{algopt}}\left[0\right]=1.0$, a BDF method is used and if ${\mathbf{algopt}}\left[0\right]=2.0$, a Theta method is used. The default value is ${\mathbf{algopt}}\left[0\right]=1.0$.
If ${\mathbf{algopt}}\left[0\right]=2.0$, then ${\mathbf{algopt}}\left[\mathit{i}-1\right]$, for $\mathit{i}=2,3,4$, are not used.
${\mathbf{algopt}}\left[1\right]$
Specifies the maximum order of the BDF integration formula to be used. ${\mathbf{algopt}}\left[1\right]$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is ${\mathbf{algopt}}\left[1\right]=5.0$.
${\mathbf{algopt}}\left[2\right]$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If ${\mathbf{algopt}}\left[2\right]=1.0$ a modified Newton iteration is used and if ${\mathbf{algopt}}\left[2\right]=2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, then there is an automatic switch to the modified Newton iteration. The default value is ${\mathbf{algopt}}\left[2\right]=1.0$.
${\mathbf{algopt}}\left[3\right]$
Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as ${P}_{i,\mathit{j}}=0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, for some $i$ or when there is no ${\stackrel{.}{V}}_{i}\left(t\right)$ dependence in the coupled ODE system. If ${\mathbf{algopt}}\left[3\right]=1.0$, then the Petzold test is used. If ${\mathbf{algopt}}\left[3\right]=2.0$, then the Petzold test is not used. The default value is ${\mathbf{algopt}}\left[3\right]=1.0$.
If ${\mathbf{algopt}}\left[0\right]=1.0$, then ${\mathbf{algopt}}\left[\mathit{i}-1\right]$, for $\mathit{i}=5,6,7$, are not used.
${\mathbf{algopt}}\left[4\right]$
Specifies the value of Theta to be used in the Theta integration method. $0.51\le {\mathbf{algopt}}\left[4\right]\le 0.99$. The default value is ${\mathbf{algopt}}\left[4\right]=0.55$.
${\mathbf{algopt}}\left[5\right]$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If ${\mathbf{algopt}}\left[5\right]=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left[5\right]=2.0$, a functional iteration method is used. The default value is ${\mathbf{algopt}}\left[5\right]=1.0$.
${\mathbf{algopt}}\left[6\right]$
Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If ${\mathbf{algopt}}\left[6\right]=1.0$, then switching is allowed and if ${\mathbf{algopt}}\left[6\right]=2.0$, then switching is not allowed. The default value is ${\mathbf{algopt}}\left[6\right]=1.0$.
${\mathbf{algopt}}\left[10\right]$
Specifies a point in the time direction, ${t}_{\mathrm{crit}}$, beyond which integration must not be attempted. The use of ${t}_{\mathrm{crit}}$ is described under the argument itask. If ${\mathbf{algopt}}\left[0\right]\ne 0.0$, a value of $0.0$, for ${\mathbf{algopt}}\left[10\right]$, say, should be specified even if itask subsequently specifies that ${t}_{\mathrm{crit}}$ will not be used.
${\mathbf{algopt}}\left[11\right]$
Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left[11\right]$ should be set to $0.0$.
${\mathbf{algopt}}\left[12\right]$
Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left[12\right]$ should be set to $0.0$.
${\mathbf{algopt}}\left[13\right]$
Specifies the initial step size to be attempted by the integrator. If ${\mathbf{algopt}}\left[13\right]=0.0$, then the initial step size is calculated internally.
${\mathbf{algopt}}\left[14\right]$
Specifies the maximum number of steps to be attempted by the integrator in any one call. If ${\mathbf{algopt}}\left[14\right]=0.0$, then no limit is imposed.
${\mathbf{algopt}}\left[22\right]$
Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of $U$, ${U}_{t}$, $V$ and $\stackrel{.}{V}$. If ${\mathbf{algopt}}\left[22\right]=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left[22\right]=2.0$, functional iteration is used. The default value is ${\mathbf{algopt}}\left[22\right]=1.0$.
${\mathbf{algopt}}\left[28\right]$ and ${\mathbf{algopt}}\left[29\right]$ are used only for the sparse matrix algebra option, i.e., ${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$.
${\mathbf{algopt}}\left[28\right]$
Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<{\mathbf{algopt}}\left[28\right]<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If ${\mathbf{algopt}}\left[28\right]$ lies outside this range then the default value is used. If the functions regard the Jacobian matrix as numerically singular then increasing ${\mathbf{algopt}}\left[28\right]$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is ${\mathbf{algopt}}\left[28\right]=0.1$.
${\mathbf{algopt}}\left[29\right]$
Used as a relative pivot threshold during subsequent Jacobian decompositions (see ${\mathbf{algopt}}\left[28\right]$) below which an internal error is invoked. ${\mathbf{algopt}}\left[29\right]$ must be greater than zero, otherwise the default value is used. If ${\mathbf{algopt}}\left[29\right]$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian is found to be numerically singular (see ${\mathbf{algopt}}\left[28\right]$). The default value is ${\mathbf{algopt}}\left[29\right]=0.0001$.
21:   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.
22:   lrsaveIntegerInput
On entry: the dimension of the array rsave. Its size depends on the type of matrix algebra selected.
If ${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$, ${\mathbf{lrsave}}\ge {\mathbf{neqn}}×{\mathbf{neqn}}+{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\mathrm{Nag_LinAlgBand}$, ${\mathbf{lrsave}}\ge \left(2×\mathit{ml}+\mathit{mu}+2\right)×{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$, ${\mathbf{lrsave}}\ge 4×{\mathbf{neqn}}+11×{\mathbf{neqn}}/2+1+\mathit{nwkres}+\mathit{lenode}$.
Where
 $\mathit{ml}$ and $\mathit{mu}$ are the lower and upper half bandwidths given by $\mathit{ml}={\mathbf{npde}}+{\mathbf{nleft}}-1$ such that $\mathit{mu}=2×{\mathbf{npde}}-{\mathbf{nleft}}-1$, for problems involving PDEs only; or $\mathit{ml}=\mathit{mu}={\mathbf{neqn}}-1$, for coupled PDE/ODE problems. $\mathit{nwkres}=\left\{\begin{array}{ll}{\mathbf{npde}}×\left(3×{\mathbf{npde}}+6×{\mathbf{nxi}}+{\mathbf{npts}}+15\right)+{\mathbf{nxi}}+{\mathbf{ncode}}+7×{\mathbf{npts}}+2\text{,}& \text{when ​}{\mathbf{ncode}}>0\text{​ and ​}{\mathbf{nxi}}>0\text{; or}\\ {\mathbf{npde}}×\left(3×{\mathbf{npde}}+{\mathbf{npts}}+21\right)+{\mathbf{ncode}}+7×{\mathbf{npts}}+3\text{,}& \text{when ​}{\mathbf{ncode}}>0\text{​ and ​}{\mathbf{nxi}}=0\text{; or}\\ {\mathbf{npde}}×\left(3×{\mathbf{npde}}+{\mathbf{npts}}+21\right)+7×{\mathbf{npts}}+4\text{,}& \text{when ​}{\mathbf{ncode}}=0\text{.}\end{array}\right\$ $\mathit{lenode}=\left\{\begin{array}{ll}\left(6+\mathrm{int}\left({\mathbf{algopt}}\left[1\right]\right)\right)×{\mathbf{neqn}}+50\text{,}& \text{when the BDF method is used; or}\\ 9×{\mathbf{neqn}}+50\text{,}& \text{when the Theta method is used.}\end{array}\right\$
Note: when using the sparse option, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if ${\mathbf{itrace}}>0$ and the function returns with NE_INT_2.
23:   isave[lisave]IntegerCommunication Array
If ${\mathbf{ind}}=0$, isave need not be set.
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 the following components of the array isave concern the efficiency of the integration:
${\mathbf{isave}}\left[0\right]$
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 evaluating 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 ODE method last used in the time integration.
${\mathbf{isave}}\left[4\right]$
Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.
24:   lisaveIntegerInput
On entry: the dimension of the array isave. Its size depends on the type of matrix algebra selected:
• if ${\mathbf{laopt}}=\mathrm{Nag_LinAlgFull}$, ${\mathbf{lisave}}\ge 24$;
• if ${\mathbf{laopt}}=\mathrm{Nag_LinAlgBand}$, ${\mathbf{lisave}}\ge {\mathbf{neqn}}+24$;
• if ${\mathbf{laopt}}=\mathrm{Nag_LinAlgSparse}$, ${\mathbf{lisave}}\ge 25×{\mathbf{neqn}}+24$.
Note: when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed if ${\mathbf{itrace}}>0$ and the function returns with NE_INT_2.
On entry: the task to be performed by the ODE integrator.
${\mathbf{itask}}=1$
Normal computation of output values u at $t={\mathbf{tout}}$ (by overshooting and interpolating).
${\mathbf{itask}}=2$
Take one step in the time direction and return.
${\mathbf{itask}}=3$
Stop at first internal integration point at or beyond $t={\mathbf{tout}}$.
${\mathbf{itask}}=4$
Normal computation of output values u at $t={\mathbf{tout}}$ but without overshooting $t={t}_{\mathrm{crit}}$, where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
${\mathbf{itask}}=5$
Take one step in the time direction and return, without passing ${t}_{\mathrm{crit}}$, where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
Constraint: ${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
26:   itraceIntegerInput
On entry: the level of trace information required from nag_pde_parab_1d_keller_ode (d03pkc) 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.
27:   outfileconst 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.
28:   indInteger *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_ode (d03pkc).
Constraint: ${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
29:   commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
30:   savedNag_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.
31:   failNagError *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 small changes in atol or rtol are unlikely to result in a changed solution.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
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
atol and rtol were too small to start integration.
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 values of atol and rtol. ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.
NE_INT
ires set to an invalid value in call to pdedef, bndary, or odedef.
On entry, ${\mathbf{ind}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On entry, ${\mathbf{itask}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
On entry, ${\mathbf{itol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{itol}}=1$, $2$, $3$ or $4$.
On entry, ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncode}}\ge 0$.
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, corresponding elements ${\mathbf{atol}}\left[i-1\right]$ and ${\mathbf{rtol}}\left[j-1\right]$ are both zero: $i=〈\mathit{\text{value}}〉$ and $j=〈\mathit{\text{value}}〉$.
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, ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nxi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nxi}}=0$ when ${\mathbf{ncode}}=0$.
On entry, ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nxi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nxi}}\ge 0$ when ${\mathbf{ncode}}>0$.
On entry, ${\mathbf{nleft}}=〈\mathit{\text{value}}〉$ and ${\mathbf{npde}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{nleft}}\le {\mathbf{npde}}$.
On entry, ${\mathbf{nleft}}=〈\mathit{\text{value}}〉$, ${\mathbf{npde}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nleft}}\le {\mathbf{npde}}$.
When using the sparse option lisave or lrsave is too small: ${\mathbf{lisave}}=〈\mathit{\text{value}}〉$, ${\mathbf{lrsave}}=〈\mathit{\text{value}}〉$.
NE_INT_4
On entry, ${\mathbf{npde}}=〈\mathit{\text{value}}〉$, ${\mathbf{npts}}=〈\mathit{\text{value}}〉$, ${\mathbf{ncode}}=〈\mathit{\text{value}}〉$ and ${\mathbf{neqn}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{neqn}}={\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{ncode}}$.
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_ITER_FAIL
In solving ODE system, the maximum number of steps ${\mathbf{algopt}}\left[14\right]$ has been exceeded. ${\mathbf{algopt}}\left[14\right]=〈\mathit{\text{value}}〉$.
NE_NOT_CLOSE_FILE
Cannot close file $〈\mathit{\text{value}}〉$.
NE_NOT_STRICTLY_INCREASING
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{xi}}\left[i\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{xi}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xi}}\left[i\right]>{\mathbf{xi}}\left[i-1\right]$.
On entry, mesh points x badly ordered: $i=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left[i-1\right]=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[j-1\right]=〈\mathit{\text{value}}〉$.
NE_NOT_WRITE_FILE
Cannot open file $〈\mathit{\text{value}}〉$ for writing.
NE_REAL_2
On entry, at least one point in xi lies outside $\left[{\mathbf{x}}\left[0\right],{\mathbf{x}}\left[{\mathbf{npts}}-1\right]\right]$: ${\mathbf{x}}\left[0\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[{\mathbf{npts}}-1\right]=〈\mathit{\text{value}}〉$.
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_REAL_ARRAY
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{atol}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{atol}}\left[i-1\right]\ge 0.0$.
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{rtol}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rtol}}\left[i-1\right]\ge 0.0$.
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, bndary, or odedef. Integration is successful as far as ts: ${\mathbf{ts}}=〈\mathit{\text{value}}〉$.
NE_ZERO_WTS
Zero error weights encountered during time integration.

## 7  Accuracy

nag_pde_parab_1d_keller_ode (d03pkc) 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 arguments, atol and rtol.

## 8  Further 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 in Section 9). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (see nag_pde_parab_1d_fd (d03pcc) or nag_pde_parab_1d_fd_ode (d03phc) 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 (nag_pde_parab_1d_cd_ode (d03plc) 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. For a given system and a fixed accuracy it is approximately proportional to neqn.

## 9  Example

This example provides a simple coupled system of two PDEs and one ODE.
 $V1 2 ∂U1 ∂t -xV1V.1U2- ∂U2 ∂x =0, U2- ∂U1 ∂x =0, V.1-V1U1-U2-1-t=0,$
for $t\in \left[{10}^{-4},0.1×{2}^{i}\right]$, for $i=1,2,\dots ,5,x\in \left[0,1\right]$. The left boundary condition at $x=0$ is
 $U2=-V1exp⁡t,$
and the right boundary condition at $x=1$ is
 $U2=-V1V.1.$
The initial conditions at $t={10}^{-4}$ are defined by the exact solution:
 $V1 = t , U1 x,t = exp t 1-x - 1.0 and U2 x,t = - t exp t 1-x , ​ x ∈ 0,1 ,$
and the coupling point is at ${\xi }_{1}=1.0$.
This problem is exactly the same as the nag_pde_parab_1d_fd_ode (d03phc) example problem, but reduced to first-order by the introduction of a second PDE variable (as mentioned in Section 8).

### 9.1  Program Text

Program Text (d03pkce.c)

None.

### 9.3  Program Results

Program Results (d03pkce.r)