```/* nag_pde_parab_1d_keller_ode (d03pkc) Example Program.
*
* Copyright 2017 Numerical Algorithms Group.
*
* Mark 26.2, 2017.
*/

#include <stdio.h>
#include <math.h>
#include <nag.h>
#include <nag_stdlib.h>
#include <nagd03.h>

#ifdef __cplusplus
extern "C"
{
#endif
static void NAG_CALL pdedef(Integer, double, double, const double[],
const double[], const double[], Integer,
const double[], const double[], double[],
Integer *, Nag_Comm *);

static void NAG_CALL 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 *);

static void NAG_CALL odedef(Integer, double, Integer, const double[],
const double[], Integer, const double[],
const double[], const double[], const double[],
double[], Integer *, Nag_Comm *);

static void NAG_CALL uvinit(Integer npde, Integer npts, double *x,
double *u, Integer ncode, Integer neqn,
double ts);

static void NAG_CALL exact(double, Integer, Integer, double *, double *);
#ifdef __cplusplus
}
#endif

#define UCP(I, J) ucp[npde*((J) -1)+(I) -1]

int main(void)
{
const Integer npde = 2, npts = 21, ncode = 1, nxi = 1, nleft = 1;
const Integer neqn = npde * npts + ncode, lisave = 24;
const Integer nwkres =
npde * (npts + 6 * nxi + 3 * npde + 15) + ncode + nxi + 7 * npts + 2;
const Integer lenode = 11 * neqn + 50, lrsave =
neqn * neqn + neqn + nwkres + lenode;
static double ruser[3] = { -1.0, -1.0, -1.0 };
double tout, ts;
Integer exit_status = 0, i, ind, it, itask, itol, itrace;
Nag_Boolean theta;
double *algopt = 0, *atol = 0, *exy = 0, *rsave = 0, *rtol = 0;
double *u = 0, *x = 0, *xi = 0;
Integer *isave = 0;
NagError fail;
Nag_Comm comm;
Nag_D03_Save saved;

INIT_FAIL(fail);

printf("nag_pde_parab_1d_keller_ode (d03pkc) Example Program Results\n\n");

/* For communication with user-supplied functions: */
comm.user = ruser;

/* Allocate memory */

if (!(algopt = NAG_ALLOC(30, double)) ||
!(atol = NAG_ALLOC(1, double)) ||
!(exy = NAG_ALLOC(neqn, double)) ||
!(rsave = NAG_ALLOC(lrsave, double)) ||
!(rtol = NAG_ALLOC(1, double)) ||
!(u = NAG_ALLOC(neqn, double)) ||
!(x = NAG_ALLOC(npts, double)) ||
!(xi = NAG_ALLOC(nxi, double)) || !(isave = NAG_ALLOC(lisave, Integer)))
{
printf("Allocation failure\n");
exit_status = 1;
goto END;
}

itrace = 0;
itol = 1;
atol[0] = 1e-4;
rtol[0] = atol[0];

printf("  Accuracy requirement =%12.3e", atol[0]);
printf(" Number of points = %3" NAG_IFMT "\n\n", npts);

/* Set spatial-mesh points */

for (i = 0; i < npts; ++i)
x[i] = i / (npts - 1.0);

xi[0] = 1.0;
ind = 0;

/* Set THETA to TRUE if the Theta integrator is required */

theta = Nag_FALSE;
for (i = 0; i < 30; ++i)
algopt[i] = 0.0;
if (theta) {
algopt[0] = 2.0;
}
else {
algopt[0] = 0.0;
}
algopt[0] = 1.0;
algopt[12] = 0.005;

/* Loop over output value of t */

ts = 1e-4;
printf("  x        %9.3f%9.3f%9.3f%9.3f%9.3f\n\n",
x[0], x[4], x[8], x[12], x[20]);

uvinit(npde, npts, x, u, ncode, neqn, ts);

for (it = 0; it < 5; ++it) {
tout = 0.1 * pow(2.0, (it + 1.0));
/* nag_pde_parab_1d_keller_ode (d03pkc).
* General system of first-order PDEs, coupled DAEs, method
* of lines, Keller box discretization, one space variable
*/
nag_pde_parab_1d_keller_ode(npde, &ts, tout, pdedef, bndary, u, npts, x,
nleft, ncode, odedef, nxi, xi, neqn, rtol,
atol, itol, Nag_TwoNorm, Nag_LinAlgFull,
algopt, rsave, lrsave, isave, lisave, itask,
itrace, 0, &ind, &comm, &saved, &fail);

if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_parab_1d_keller_ode (d03pkc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* Check against the exact solution */

exact(tout, neqn, npts, x, exy);

printf(" t = %6.3f\n", ts);
printf(" App.  sol.  %7.3f%9.3f%9.3f%9.3f%9.3f",
u[0], u[8], u[16], u[24], u[40]);
printf("  ODE sol. =%8.3f\n", u[42]);
printf(" Exact sol.  %7.3f%9.3f%9.3f%9.3f%9.3f",
exy[0], exy[8], exy[16], exy[24], exy[40]);
printf("  ODE sol. =%8.3f\n\n", ts);
}
printf(" Number of integration steps in time = %6" NAG_IFMT "\n", isave[0]);
printf(" Number of function evaluations = %6" NAG_IFMT "\n", isave[1]);
printf(" Number of Jacobian evaluations =%6" NAG_IFMT "\n", isave[2]);
printf(" Number of iterations = %6" NAG_IFMT "\n\n", isave[4]);
END:
NAG_FREE(algopt);
NAG_FREE(atol);
NAG_FREE(exy);
NAG_FREE(rsave);
NAG_FREE(rtol);
NAG_FREE(u);
NAG_FREE(x);
NAG_FREE(xi);
NAG_FREE(isave);

return exit_status;
}

static void NAG_CALL uvinit(Integer npde, Integer npts, double *x,
double *u, Integer ncode, Integer neqn, double ts)
{
Integer i, k;

/* Routine for PDE initial values */

k = 0;
for (i = 0; i < npts; ++i) {
u[k] = exp(ts * (1.0 - x[i])) - 1.0;
u[k + 1] = -ts * exp(ts * (1.0 - x[i]));
k += 2;
}
u[neqn - 1] = ts;

return;
}

static void NAG_CALL 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 f[], Integer *ires,
Nag_Comm *comm)
{
if (comm->user[0] == -1.0) {
printf("(User-supplied callback odedef, first invocation.)\n");
comm->user[0] = 0.0;
}
if (*ires == -1) {
f[0] = vdot[0];
}
else {
f[0] = vdot[0] - v[0] * UCP(1, 1) - UCP(2, 1) - 1.0 - t;
}
return;
}

static void NAG_CALL 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)
{
if (comm->user[1] == -1.0) {
printf("(User-supplied callback pdedef, first invocation.)\n");
comm->user[1] = 0.0;
}
if (*ires == -1) {
res[0] = v[0] * v[0] * ut[0] - x * u[1] * v[0] * vdot[0];
res[1] = 0.0;
}
else {
res[0] = v[0] * v[0] * ut[0] - x * u[1] * v[0] * vdot[0] - ux[1];
res[1] = u[1] - ux[0];
}
return;
}

static void NAG_CALL 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)
{
if (comm->user[2] == -1.0) {
printf("(User-supplied callback bndary, first invocation.)\n");
comm->user[2] = 0.0;
}
if (ibnd == 0) {
if (*ires == -1) {
res[0] = 0.0;
}
else {
res[0] = u[1] + v[0] * exp(t);
}
}
else {
if (*ires == -1) {
res[0] = v[0] * vdot[0];
}
else {
res[0] = u[1] + v[0] * vdot[0];
}
}
return;
}

static void NAG_CALL exact(double time, Integer neqn, Integer npts, double *x,
double *u)
{
/* Exact solution (for comparison purposes) */

Integer i, k;

k = 0;
for (i = 0; i < npts; ++i) {
u[k] = exp(time * (1.0 - x[i])) - 1.0;
k += 2;
}
return;
}
```