NAG Library Manual, Mark 28.4
```/* nag_pde_dim1_parab_remesh_fd (d03ppc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.4, 2022.
*/

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

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

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

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

static void NAG_CALL monitf(double, Integer, Integer, const double[],
const double[], const double[], double[],
Nag_Comm *);
#ifdef __cplusplus
}
#endif

static void exact(double, double *, Integer, double *, Nag_Comm *);

#define P(I, J) p[npde * ((J)-1) + (I)-1]
#define R(I, J) r[npde * ((J)-1) + (I)-1]
#define U(I, J) u[npde * ((J)-1) + (I)-1]
#define UOUT(I, J, K) uout[npde * (intpts * ((K)-1) + (J)-1) + (I)-1]

int main(void) {
const Integer npde = 1, npts = 61, ncode = 0, m = 0, nxi = 0, nxfix = 0;
const Integer itype = 1, neqn = npde * npts + ncode, intpts = 5;
const Integer lisave = 25 + nxfix;
const Integer nwkres = npde * (npts + 3 * npde + 21) + 7 * npts + nxfix + 3;
const Integer lenode = 11 * neqn + 50,
lrsave = neqn * neqn + neqn + nwkres + lenode;
static double ruser[4] = {-1.0, -1.0, -1.0, -1.0};
double con, dxmesh, e, tout, trmesh, ts, xratio;
Integer exit_status, i, ind, ipminf, it, itask, itol, itrace, nrmesh;
Nag_Boolean remesh, theta;
double *algopt = 0, *atol = 0, *rsave = 0, *rtol = 0, *u = 0, *ue = 0;
double *uout = 0, *x = 0, *xfix = 0, *xi = 0, *xout = 0;
Integer *isave = 0;
NagError fail;
Nag_Comm comm;
Nag_D03_Save saved;

INIT_FAIL(fail);

exit_status = 0;

/* Allocate memory */

if (!(algopt = NAG_ALLOC(30, double)) || !(atol = NAG_ALLOC(1, double)) ||
!(rsave = NAG_ALLOC(lrsave, double)) || !(rtol = NAG_ALLOC(1, double)) ||
!(u = NAG_ALLOC(neqn, double)) || !(ue = NAG_ALLOC(intpts, double)) ||
!(uout = NAG_ALLOC(npde * intpts * itype, double)) ||
!(x = NAG_ALLOC(npts, double)) || !(xfix = NAG_ALLOC(1, double)) ||
!(xi = NAG_ALLOC(1, double)) || !(xout = NAG_ALLOC(intpts, double)) ||
!(isave = NAG_ALLOC(lisave, Integer))) {
printf("Allocation failure\n");
exit_status = 1;
goto END;
}

printf("nag_pde_dim1_parab_remesh_fd (d03ppc) Example Program"
" Results\n\n");

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

e = 0.005;
comm.p = (Pointer)&e;
itrace = 0;
itol = 1;
atol[0] = 5e-5;
rtol[0] = atol[0];

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

/* Initialize mesh */

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

/* Set remesh parameters */

remesh = Nag_TRUE;
nrmesh = 3;
dxmesh = 0.5;
trmesh = 0.0;
con = 2.0 / (npts - 1.0);
xratio = 1.5;
ipminf = 0;

printf("  Remeshing every %3" NAG_IFMT " time steps\n\n", nrmesh);
printf("  e =%8.3f\n\n\n", e);

xi[0] = 0.0;
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;
}

/* Loop over output value of t */

ts = 0.0;
for (it = 0; it < 5; ++it) {
tout = 0.2 * (it + 1);

/* nag_pde_dim1_parab_remesh_fd (d03ppc).
* General system of parabolic PDEs, coupled DAEs, method of
* lines, finite differences, remeshing, one space variable
*/
nag_pde_dim1_parab_remesh_fd(
npde, m, &ts, tout, pdedef, bndary, uvinit, u, npts, x, ncode, NULLFN,
nxi, xi, neqn, rtol, atol, itol, Nag_TwoNorm, Nag_LinAlgFull, algopt,
remesh, nxfix, xfix, nrmesh, dxmesh, trmesh, ipminf, xratio, con,
monitf, rsave, lrsave, isave, lisave, itask, itrace, 0, &ind, &comm,
&saved, &fail);

if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_dim1_parab_remesh_fd (d03ppc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* Set output points */

switch (it) {
case 0:
for (i = 0; i < 5; ++i)
xout[i] = 0.3 + 0.1 * i;
break;
case 1:
for (i = 0; i < 5; ++i)
xout[i] = 0.4 + 0.1 * i;
break;
case 2:
for (i = 0; i < 5; ++i)
xout[i] = 0.6 + 0.05 * i;
break;
case 3:
for (i = 0; i < 5; ++i)
xout[i] = 0.7 + 0.05 * i;
break;
case 4:
for (i = 0; i < 5; ++i)
xout[i] = 0.8 + 0.05 * i;
break;
}

printf(" t = %6.3f\n", ts);
printf(" x           ");

for (i = 0; i < 5; ++i) {
printf("%9.4f", xout[i]);
printf((i + 1) % 5 == 0 || i == 4 ? "\n" : " ");
}

/* Interpolate at output points */

/* nag_pde_dim1_parab_fd_interp (d03pzc). PDEs, spatial interpolation with
* nag_pde_dim1_parab_remesh_fd (d03ppc),
*/
nag_pde_dim1_parab_fd_interp(npde, m, u, npts, x, xout, intpts, itype, uout,
&fail);

if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_dim1_parab_fd_interp (d03pzc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* Check against exact solution */

exact(ts, xout, intpts, ue, &comm);

printf(" Approx sol. ");

for (i = 1; i <= intpts; ++i) {
printf("%9.4f", UOUT(1, i, 1));
printf(i % 5 == 0 || i == 5 ? "\n" : " ");
}

printf(" Exact  sol. ");

for (i = 1; i <= intpts; ++i) {
printf("%9.4f", ue[i - 1]);
printf(i % 5 == 0 || i == 5 ? "\n" : " ");
}
printf("\n");
}

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(rsave);
NAG_FREE(rtol);
NAG_FREE(u);
NAG_FREE(ue);
NAG_FREE(uout);
NAG_FREE(x);
NAG_FREE(xfix);
NAG_FREE(xi);
NAG_FREE(xout);
NAG_FREE(isave);

return exit_status;
}

static void NAG_CALL uvinit(Integer npde, Integer npts, Integer nxi,
const double x[], const double xi[], double u[],
Integer ncode, double v[], Nag_Comm *comm) {
double *e = (double *)comm->p;
double a, b, c, t;
Integer i;

if (comm->user[0] == -1.0) {
printf("(User-supplied callback uvinit, first invocation.)\n");
comm->user[0] = 0.0;
}
t = 0.0;
for (i = 1; i <= npts; ++i) {
a = (x[i - 1] - 0.25 - 0.75 * t) / (*e * 4.0);
b = (0.9 * x[i - 1] - 0.325 - 0.495 * t) / (*e * 2.0);
if (a > 0.0 && a > b) {
a = exp(-a);
c = (0.8 * x[i - 1] - 0.4 - 0.24 * t) / (*e * 4.0);
c = exp(c);
U(1, i) = (0.1 * c + 0.5 + a) / (c + 1.0 + a);
} else if (b > 0.0 && b >= a) {
b = exp(-b);
c = (-0.8 * x[i - 1] + 0.4 + 0.24 * t) / (*e * 4.0);
c = exp(c);
U(1, i) = (0.5 * c + 0.1 + b) / (c + 1.0 + b);
} else {
a = exp(a);
b = exp(b);
U(1, i) = (0.5 * a + 1.0 + 0.1 * b) / (a + 1.0 + b);
}
}
return;
}

static void NAG_CALL pdedef(Integer npde, double t, double x, const double u[],
const double ux[], Integer ncode, const double v[],
const double vdot[], double p[], double q[],
double r[], Integer *ires, Nag_Comm *comm) {
double *e = (double *)comm->p;

if (comm->user[1] == -1.0) {
printf("(User-supplied callback pdedef, first invocation.)\n");
comm->user[1] = 0.0;
}
P(1, 1) = 1.0;
r[0] = *e * ux[0];
q[0] = u[0] * ux[0];

return;
}

static void NAG_CALL bndary(Integer npde, double t, const double u[],
const double ux[], Integer ncode, const double v[],
const double vdot[], Integer ibnd, double beta[],
double gamma[], Integer *ires, Nag_Comm *comm) {
double a, b, c, ue, x;
double *e = (double *)comm->p;

if (comm->user[2] == -1.0) {
printf("(User-supplied callback bndary, first invocation.)\n");
comm->user[2] = 0.0;
}
beta[0] = 0.0;
if (ibnd == 0) {
x = 0.0;
a = (x - 0.25 - 0.75 * t) / (*e * 4.0);
b = (0.9 * x - 0.325 - 0.495 * t) / (*e * 2.0);
if (a > 0. && a > b) {
a = exp(-a);
c = (0.8 * x - 0.4 - 0.24 * t) / (*e * 4.0);
c = exp(c);
ue = (0.1 * c + 0.5 + a) / (c + 1.0 + a);
} else if (b > 0.0 && b >= a) {
b = exp(-b);
c = (-0.8 * x + 0.4 + 0.24 * t) / (*e * 4.0);
c = exp(c);
ue = (0.5 * c + 0.1 + b) / (c + 1.0 + b);
} else {
a = exp(a);
b = exp(b);
ue = (0.5 * a + 1.0 + 0.1 * b) / (a + 1.0 + b);
}
} else {
x = 1.0;
a = (x - 0.25 - 0.75 * t) / (*e * 4.0);
b = (0.9 * x - 0.325 - 0.495 * t) / (*e * 2.0);
if (a > 0.0 && a > b) {
a = exp(-a);
c = (0.8 * x - 0.4 - 0.24 * t) / (*e * 4.0);
c = exp(c);
ue = (0.1 * c + 0.5 + a) / (c + 1.0 + a);
} else if (b > 0.0 && b >= a) {
b = exp(-b);
c = (-0.8 * x + 0.4 + 0.24 * t) / (*e * 4.0);
c = exp(c);
ue = (0.5 * c + 0.1 + b) / (c + 1.0 + b);
} else {
a = exp(a);
b = exp(b);
ue = (0.5 * a + 1.0 + 0.1 * b) / (a + 1.0 + b);
}
}
gamma[0] = u[0] - ue;

return;
}

static void exact(double t, double *x, Integer npts, double *u,
Nag_Comm *comm) {
/* Exact solution (for comparison purposes) */

double a, b, c;
double *e = (double *)comm->p;
Integer i;

for (i = 0; i < npts; ++i) {
a = (x[i] - 0.25 - 0.75 * t) / (*e * 4.0);
b = (0.9 * x[i] - 0.325 - 0.495 * t) / (*e * 2.0);
if (a > 0. && a > b) {
a = exp(-a);
c = (0.8 * x[i] - 0.4 - 0.24 * t) / (*e * 4.0);
c = exp(c);
u[i] = (0.1 * c + 0.5 + a) / (c + 1.0 + a);
} else if (b > 0. && b >= a) {
b = exp(-b);
c = (-0.8 * x[i] + 0.4 + 0.24 * t) / (*e * 4.0);
c = exp(c);
u[i] = (0.5 * c + 0.1 + b) / (c + 1.0 + b);
} else {
a = exp(a);
b = exp(b);
u[i] = (0.5 * a + 1.0 + 0.1 * b) / (a + 1.0 + b);
}
}
return;
}

static void NAG_CALL monitf(double t, Integer npts, Integer npde,
const double x[], const double u[],
const double r[], double fmon[], Nag_Comm *comm) {
double drdx, h;
Integer i, k, l;

if (comm->user[3] == -1.0) {
printf("(User-supplied callback monitf, first invocation.)\n");
comm->user[3] = 0.0;
}
for (i = 1; i <= npts - 1; ++i) {
k = i - 1;
if (i == 1)
k = 1;
l = i + 1;
h = 0.5 * (x[l - 1] - x[k - 1]);

/* Second derivative */

drdx = (R(1, i + 1) - R(1, i)) / h;
fmon[i - 1] = drdx;
if (fmon[i - 1] < 0)
fmon[i - 1] = -drdx;
}
fmon[npts - 1] = fmon[npts - 2];

return;
}
```