nag_pde_parab_1d_keller (d03pec) (PDF version)
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d03 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_pde_parab_1d_keller (d03pec)

## 1  Purpose

nag_pde_parab_1d_keller (d03pec) integrates a system of linear or nonlinear, first-order, time-dependent partial differential equations (PDEs) in one space variable. The spatial discretization is performed using the Keller box scheme and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a Backward Differentiation Formula (BDF) method.

## 2  Specification

 #include #include
void  nag_pde_parab_1d_keller (Integer npde, double *ts, double tout,
 void (*pdedef)(Integer npde, double t, double x, const double u[], const double ut[], const double ux[], double res[], Integer *ires, Nag_Comm *comm),
 void (*bndary)(Integer npde, double t, Integer ibnd, Integer nobc, const double u[], const double ut[], double res[], Integer *ires, Nag_Comm *comm),
double u[], Integer npts, const double x[], Integer nleft, 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)

## 3  Description

nag_pde_parab_1d_keller (d03pec) integrates the system of first-order PDEs
 $Gix,t,U,Ux,Ut=0, i=1,2,…,npde.$ (1)
In particular the functions ${G}_{i}$ must have the general form
 $Gi=∑j=1npdePi,j ∂Uj ∂t +Qi, i=1,2,…,npde, a≤x≤b,t≥t0,$ (2)
where ${P}_{i,j}$ and ${Q}_{i}$ depend on $x$, $t$, $U$, ${U}_{x}$ and the vector $U$ is the set of solution values
 $U x,t = U 1 x,t ,…, U npde x,t T ,$ (3)
and the vector ${U}_{x}$ is its partial derivative with respect to $x$. Note that ${P}_{i,j}$ and ${Q}_{i}$ must not depend on $\frac{\partial U}{\partial t}$.
The integration in time is from ${t}_{0}$ to ${t}_{\mathrm{out}}$, over the space interval $a\le x\le b$, where $a={x}_{1}$ and $b={x}_{{\mathbf{npts}}}$ are the leftmost and rightmost points of a user-defined mesh ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$. The mesh should be chosen in accordance with the expected behaviour of the solution.
The PDE system which is defined by the functions ${G}_{i}$ must be specified in pdedef.
The initial values of the functions $U\left(x,t\right)$ 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:
 $GiLx,t,U,Ut=0 at ​x=a, i=1,2,…,na$ (4)
at the left-hand boundary, and
 $GiRx,t,U,Ut=0 at ​x=b, i=1,2,…,nb$ (5)
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 (see Keller (1970)). If the problem involves derivative (Neumann) boundary conditions then it is generally possible to restate such boundary conditions in terms of permissible variables. Also note that ${G}_{i}^{L}$ and ${G}_{i}^{R}$ must be linear with respect to time derivatives, so that the boundary conditions have the general form
 $∑j=1npdeEi,jL ∂Uj ∂t +SiL=0, i=1,2,…,na$ (6)
at the left-hand boundary, and
 $∑j=1npdeEi,jR ∂Uj ∂t +SiR=0, i=1,2,…,nb$ (7)
at the right-hand boundary, where ${E}_{i,j}^{L}$, ${E}_{i,j}^{R}$, ${S}_{i}^{L}$, and ${S}_{i}^{R}$ depend on $x$, $t$ and $U$ only.
The boundary conditions must be specified in bndary.
The problem is subject to the following restrictions:
 (i) ${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction; (ii) ${P}_{i,j}$ and ${Q}_{i}$ must not depend on any time derivatives; (iii) The evaluation of the function ${G}_{i}$ is done at the mid-points of the mesh intervals by calling the pdedef for each mid-point in turn. Any discontinuities in the function must therefore be at one or more of the mesh points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$; (iv) At least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the problem.
In this method of lines approach the Keller box scheme (see Keller (1970)) is applied to each PDE in the space variable only, resulting in a system of ODEs in time for the values of ${U}_{i}$ at each mesh point. In total there are ${\mathbf{npde}}×{\mathbf{npts}}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF 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
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 in the system 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 values 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 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:     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:     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
 $Gi=∑j=1npdePi,j ∂Uj ∂t ,$ (8)
i.e., only terms depending explicitly on time derivatives, or
 $Gi=∑j=1npdePi,j ∂Uj ∂t +Qi,$ (9)
i.e., all terms in equation (2).
The definition of $G$ is determined by the input value of ires.
8:     iresInteger *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 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 NE_FAILED_DERIV.
9:     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 (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 in the Essential Introduction).
5:     bndaryfunction, 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:     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: 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:     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:     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
 $GiL=∑j=1npde Ei,jL ∂Uj ∂t ,$ (10)
i.e., only terms depending explicitly on time derivatives, or
 $GiL=∑j=1npde Ei,jL ∂Uj ∂t +SiL,$ (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:     iresInteger *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 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 NE_FAILED_DERIV.
9:     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 (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 in the Essential Introduction).
6:     u[${\mathbf{npde}}×{\mathbf{npts}}$]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: ${\mathbf{u}}\left[{\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$ will contain the computed solution 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 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:     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:   accdoubleInput
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:
 $Ei,j=acc×1.0+u[npde×j-1+i-1].$
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:   lrsaveIntegerInput
On entry: the dimension of the array rsave.
Constraint: ${\mathbf{lrsave}}\ge \left(4×{\mathbf{npde}}+{\mathbf{nleft}}+14\right)×{\mathbf{npde}}×{\mathbf{npts}}+\left(3×{\mathbf{npde}}+21\right)×{\mathbf{npde}}+\phantom{\rule{0ex}{0ex}}7×{\mathbf{npts}}+54$.
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:   lisaveIntegerInput
On entry: the dimension of the array isave.
Constraint: ${\mathbf{lisave}}\ge {\mathbf{npde}}×{\mathbf{npts}}+24$.
15:   itaskIntegerInput
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:   itraceIntegerInput
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:   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.
18:   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 (d03pec).
Constraint: ${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
19:   commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
20:   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.
21:   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 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: $i=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left[i-1\right]=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{x}}\left[j-1\right]=〈\mathit{\text{value}}〉$.
NE_NOT_WRITE_FILE
Cannot open file $〈\mathit{\text{value}}〉$ for writing.
NE_REAL
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

nag_pde_parab_1d_keller (d03pec) controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the accuracy argument, acc.

## 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 problem in nag_pde_parab_1d_keller_ode (d03pkc)). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (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 (d03pfc) for example), or the addition of a second-order artificial dissipation term.
The time taken depends on the complexity of the system and on the accuracy requested.

## 9  Example

This example is the simple first-order system
 $∂U1 ∂t + ∂U1 ∂x + ∂U2 ∂x =0, ∂U2 ∂t +4 ∂U1 ∂x + ∂U2 ∂x =0,$
for $t\in \left[0,1\right]$ and $x\in \left[0,1\right]$.
The initial conditions are
 $U1x,0=expx, U2x,0=sinx,$
and the Dirichlet boundary conditions for ${U}_{1}$ at $x=0$ and ${U}_{2}$ at $x=1$ are given by the exact solution:
 $U1x,t=12 expx+t+expx-3t+14 sinx-3t-sinx+t , U2x,t=expx-3t-expx+t+12 sinx+t+sinx-3t .$

### 9.1  Program Text

Program Text (d03pece.c)

None.

### 9.3  Program Results

Program Results (d03pece.r)

nag_pde_parab_1d_keller (d03pec) (PDF version)
d03 Chapter Contents
d03 Chapter Introduction
NAG C Library Manual