```/* nag_pde_parab_1d_cd_ode_remesh (d03psc) 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>
#include <nagx01.h>

static int ex1(void);
static int ex2(void);

#ifdef __cplusplus
extern "C"
{
#endif
static void NAG_CALL uvin1(Integer, Integer, Integer, const double[],
const double[], double[], Integer, double[],
Nag_Comm *);

static void NAG_CALL uvin2(Integer, Integer, Integer, const double[],
const double[], double[], Integer, double[],
Nag_Comm *);

static void NAG_CALL pdef1(Integer, double, double, const double[],
const double[], Integer, const double[],
const double[], double[], double[], double[],
double[], Integer *, Nag_Comm *);

static void NAG_CALL pdef2(Integer, double, double, const double[],
const double[], Integer, const double[],
const double[], double[], double[], double[],
double[], Integer *, Nag_Comm *);

static void NAG_CALL bndry1(Integer, Integer, double, const double[],
const double[], Integer, const double[],
const double[], Integer, double[],
Integer *, Nag_Comm *);

static void NAG_CALL bndry2(Integer, Integer, double, const double[],
const double[], Integer, const double[],
const double[], Integer, double[],
Integer *, Nag_Comm *);

static void NAG_CALL monit1(double, Integer, Integer, const double[],
const double[], double[], Nag_Comm *);

static void NAG_CALL monit2(double, Integer, Integer, const double[],
const double[], double[], Nag_Comm *);

static void NAG_CALL nmflx1(Integer, double, double, Integer,
const double[], const double[],
const double[], double[], Integer *,
Nag_Comm *, Nag_D03_Save *);

static void NAG_CALL nmflx2(Integer, double, double, Integer,
const double[], const double[],
const double[], double[], Integer *,
Nag_Comm *, Nag_D03_Save *);
#ifdef __cplusplus
}
#endif

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

#define P(I, J)       p[npde*((J) -1)+(I) -1]
#define UE(I, J)      ue[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)
{
Integer exit_status_ex1 = 0;
Integer exit_status_ex2 = 0;

printf("nag_pde_parab_1d_cd_ode_remesh (d03psc) Example Program Results\n");
exit_status_ex1 = ex1();
exit_status_ex2 = ex2();

return (exit_status_ex1 == 0 && exit_status_ex2 == 0) ? 0 : 1;
}

int ex1(void)
{
const Integer iprint = 1;
const Integer npts = 61, intpts = 7, itype = 1;
const Integer npde = 1, ncode = 0, nxi = 0, nxfix = 0;
const Integer neqn = npde * npts + ncode, lisave = 25 + nxfix + neqn;
const Integer nwkres = npde * (3 * npts + 3 * npde + 32) + 7 * npts + 3;
const Integer lenode = 11 * neqn + 50, mlu = 3 * npde - 1;
const Integer lrsave = (3 * mlu + 1) * neqn + nwkres + lenode;
static double ruser1[5] = { -1.0, -1.0, -1.0, -1.0, -1.0 };
static Integer iuser1[1] = {1};
static double xout[7] = { .2, .3, .4, .5, .6, .7, .8 };
double        con, dxmesh, tout, trmesh, ts, xratio;
Integer       exit_status = 0;
Integer       i, ind, ipminf, it, itask, itol, itrace, m, nrmesh;
Nag_Boolean   remesh;
double        *algopt = 0, *atol = 0, *rsave = 0, *rtol = 0, *u = 0;
double        *uout = 0, *x = 0, *xfix = 0, *xi = 0;
Integer       *isave = 0;
NagError      fail;
Nag_Comm      comm;
Nag_D03_Save  saved;

INIT_FAIL(fail);

/* For communication with user-supplied functions: */
comm.user = ruser1;
comm.iuser = iuser1;

/* 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(npde * npts, double)) ||
!(uout = NAG_ALLOC(npde * intpts * itype, double)) ||
!(x = NAG_ALLOC(npts, double)) ||
!(xfix = NAG_ALLOC(1, double)) ||
!(xi = NAG_ALLOC(1, double)) ||
!(isave = NAG_ALLOC(lisave, Integer)))
{
printf("Allocation failure\n");
exit_status = 1;
goto END;
}

printf("\n\nExample 1\n\n");

itrace = 0;
itol = 1;
atol[0] = 1.0e-4;
rtol[0] = 1.0e-4;

printf("  npts = %4" NAG_IFMT "", npts);
printf(" atol = %12.3e", atol[0]);
printf(" rtol = %12.3e\n\n", rtol[0]);

/* Initialize mesh */

for (i = 0; i < npts; ++i)
x[i] = i / (npts - 1.0);
xfix[0] = 0.0;

/* Set remesh parameters */

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

xi[0] = 0.0;
ind = 0;

for (i = 0; i < 30; ++i)
algopt[i] = 0.0;

/* b.d.f. integration */

algopt[0] = 1.0;
algopt[12] = 0.005;

/* Loop over output value of t */

ts = 0.0;
for (it = 0; it < 3; ++it) {
tout = 0.1 * (it + 1);

/* nag_pde_parab_1d_cd_ode_remesh (d03psc).
* General system of convection-diffusion PDEs with source
* terms in conservative form, coupled DAEs, method of
* lines, upwind scheme using numerical flux function based
* on Riemann solver, remeshing, one space variable
*/
nag_pde_parab_1d_cd_ode_remesh(npde, &ts, tout, pdef1, nmflx1, bndry1,
uvin1, u, npts, x, ncode, NULLFN, nxi, xi,
neqn, rtol, atol, itol, Nag_OneNorm,
Nag_LinAlgBand, algopt, remesh, nxfix,
xfix, nrmesh, dxmesh, trmesh, ipminf,
xratio, con, monit1, rsave, lrsave, isave,
&comm, &saved, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_parab_1d_cd_ode_remesh (d03psc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

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

for (i = 1; i <= intpts; ++i) {
printf("%9.4f", xout[i - 1]);
printf(i % 7 == 0 || i == 7 ? "\n" : "");
}

/* Interpolate at output points */

m = 0;
/* nag_pde_interp_1d_fd (d03pzc). PDEs, spatial interpolation with
* nag_pde_parab_1d_cd_ode_remesh (d03psc).
*/
nag_pde_interp_1d_fd(npde, m, u, npts, x, xout, intpts, itype, uout,
&fail);

if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_interp_1d_fd (d03pzc).\n%s\n", fail.message);
exit_status = 1;
goto END;
}

printf(" Approx u ");

for (i = 1; i <= intpts; ++i) {
printf("%9.4f", UOUT(1, i, 1));
printf(i % 7 == 0 || i == 7 ? "\n" : "");
}
printf("\n");
}

if (iprint>0) {
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(uout);
NAG_FREE(x);
NAG_FREE(xfix);
NAG_FREE(xi);
NAG_FREE(isave);

return exit_status;
}

static void NAG_CALL uvin1(Integer npde, Integer npts, Integer nxi,
const double x[], const double xi[], double u[],
Integer ncode, double v[], Nag_Comm *comm)
{
Integer i;

if (comm->user[0] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback uvin1, first invocation.)\n");
}
comm->user[0] = 0.0;
}
for (i = 1; i <= npts; ++i) {
if (x[i - 1] > 0.2 && x[i - 1] <= 0.4) {
U(1, i) = sin(nag_pi * (5.0 * x[i - 1] - 1.0));
}
else {
U(1, i) = 0.0;
}
}
return;
}

static void NAG_CALL pdef1(Integer npde, double t, double x, const double u[],
const double ux[], Integer ncode, const double v[],
const double vdot[], double p[], double c[],
double d[], double s[], Integer *ires,
Nag_Comm *comm)
{
if (comm->user[1] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback pdef1, first invocation.)\n");
}
comm->user[1] = 0.0;
}
P(1, 1) = 1.0;
c[0] = 0.002;
d[0] = ux[0];
s[0] = 0.0;

return;
}

static void NAG_CALL bndry1(Integer npde, Integer npts, double t,
const double x[], const double u[], Integer ncode,
const double v[], const double vdot[],
Integer ibnd, double g[], Integer *ires,
Nag_Comm *comm)
{
/* Zero solution at both boundaries */

if (comm->user[2] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback bndry1, first invocation.)\n");
}
comm->user[2] = 0.0;
}
if (ibnd == 0) {
g[0] = U(1, 1);
}
else {
g[0] = U(1, npts);
}
return;
}

static void NAG_CALL monit1(double t, Integer npts, Integer npde,
const double x[], const double u[], double fmon[],
Nag_Comm *comm)
{
double h1, h2, h3;
Integer i;

if (comm->user[3] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback monit1, first invocation.)\n");
}
comm->user[3] = 0.0;
}
for (i = 2; i <= npts - 1; ++i) {
h1 = x[i - 1] - x[i - 2];
h2 = x[i] - x[i - 1];
h3 = 0.5 * (x[i] - x[i - 2]);

/* Second derivatives */

fmon[i - 1] = fabs(((U(1, i + 1) - U(1, i)) / h2 -
(U(1, i) - U(1, i - 1)) / h1) / h3);
}
fmon[0] = fmon[1];
fmon[npts - 1] = fmon[npts - 2];

return;
}

static void NAG_CALL nmflx1(Integer npde, double t, double x, Integer ncode,
const double v[], const double uleft[],
const double uright[], double flux[],
Integer *ires, Nag_Comm *comm,
Nag_D03_Save *saved)
{
if (comm->user[4] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback nmflx1, first invocation.)\n");
}
comm->user[4] = 0.0;
}
flux[0] = uleft[0];

return;
}

int ex2(void)
{
const Integer iprint = 0;
const Integer npts = 61, intpts = 7, itype = 1;
const Integer npde = 1, ncode = 0, nxi = 0, nxfix = 0;
const Integer neqn = npde * npts + ncode, lisave = 25 + nxfix + neqn;
const Integer nwkres = npde * (3 * npts + 3 * npde + 32) + 7 * npts + 3;
const Integer lenode = 11 * neqn + 50, mlu = 3 * npde - 1;
const Integer lrsave = (3 * mlu + 1) * neqn + nwkres + lenode;
static double ruser2[5] = { -1.0, -1.0, -1.0, -1.0, -1.0 };
static Integer iuser2[1] = {0};
double        con, dxmesh, tout, trmesh, ts, xratio, dx, du, pu, xx;
Integer       exit_status = 0;
Integer       i, ind, ipminf, it, itask, itol, itrace, nrmesh;
Nag_Boolean   remesh;
double        *algopt = 0, *atol = 0, *rsave = 0, *rtol = 0, *u = 0, *ue = 0;
double        *uout = 0, *x = 0, *xfix = 0, *xi = 0;
Integer       *isave = 0;
NagError      fail;
Nag_Comm      comm;
Nag_D03_Save  saved;

INIT_FAIL(fail);

/* For communication with user-supplied functions: */
comm.user = ruser2;
comm.iuser = iuser2;

/* 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(npts, double)) ||
!(ue = NAG_ALLOC(npde * intpts, double)) ||
!(uout = NAG_ALLOC(npde * intpts * itype, double)) ||
!(x = NAG_ALLOC(npts, double)) ||
!(xfix = NAG_ALLOC(1, double)) ||
!(xi = NAG_ALLOC(1, double)) ||
!(isave = NAG_ALLOC(lisave, Integer)))
{
printf("Allocation failure\n");
exit_status = 1;
goto END;
}

printf("\n\nExample 2\n\n");

itrace = 0;
itol = 1;
atol[0] = 5e-4;
rtol[0] = 5e-2;
printf(" npts = %4" NAG_IFMT "", npts);
printf(" atol = %12.3e", atol[0]);
printf(" rtol = %12.3e\n\n", rtol[0]);

/* Initialize mesh */

for (i = 0; i < npts; ++i)
x[i] = i / (npts - 1.0);
xfix[0] = 0.0;

/* Set remesh parameters */

remesh = Nag_TRUE;
nrmesh = 5;
dxmesh = 0.0;
trmesh = 0.0;
con = 1.0 / (npts - 1.0);
xratio = 1.5;
ipminf = 0;
xi[0] = 0.0;
ind = 0;

for (i = 0; i < 30; ++i)
algopt[i] = 0.0;

/* Theta integration */

algopt[0] = 2.0;
algopt[5] = 2.0;
algopt[6] = 2.0;

/* Max. time step */

algopt[12] = 0.0025;

printf(" Track centre point, x,  of moving front\n\n");
printf("    t        x        Exact x\n\n");
ts = 0.0;
tout = 0.0;
for (it = 0; it < 10; ++it) {
tout = tout + 0.02;

/* nag_pde_parab_1d_cd_ode_remesh (d03psc), see above. */
nag_pde_parab_1d_cd_ode_remesh(npde, &ts, tout, pdef2, nmflx2, bndry2,
uvin2, u, npts, x, ncode, NULLFN, nxi, xi,
neqn, rtol, atol, itol, Nag_OneNorm,
Nag_LinAlgBand, algopt, remesh, nxfix,
xfix, nrmesh, dxmesh, trmesh, ipminf,
xratio, con, monit2, rsave, lrsave, isave,
&comm, &saved, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_parab_1d_cd_ode_remesh (d03psc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}

/* Find point x at which u = 0.5 */
i = 0;
while (u[i]>0.5) {
i++;
}
du = u[i-1] - u[i];
dx = x[i] - x[i-1];
pu = (1.0 - u[i-1])/du;
xx = x[i-1] + pu*dx;

/* Check against exact solution */

printf(" %6.2f", ts);
printf("  %8.2f", xx);
printf("   %8.3f\n", ts + 21.0/200.0);
}

if (iprint>0) {
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(isave);

return exit_status;
}

static void NAG_CALL uvin2(Integer npde, Integer npts, Integer nxi,
const double x[], const double xi[], double u[],
Integer ncode, double v[], Nag_Comm *comm)
{
double t;

if (comm->user[0] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback uvin2, first invocation.)\n");
}
comm->user[0] = 0.0;
}
t = 0.0;
exact(t, u, x, npde, npts);

return;
}

static void NAG_CALL pdef2(Integer npde, double t, double x, const double u[],
const double ux[], Integer ncode, const double v[],
const double vdot[], double p[], double c[],
double d[], double s[], Integer *ires,
Nag_Comm *comm)
{
if (comm->user[1] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback pdef2, first invocation.)\n");
}
comm->user[1] = 0.0;
}
P(1, 1) = 1.0;
c[0] = 0.0;
d[0] = 0.0;
s[0] = -100.0 * u[0] * (u[0] - 1.0) * (u[0] - 0.5);

return;
}

static void NAG_CALL bndry2(Integer npde, Integer npts, double t,
const double x[], const double u[],
Integer ncode, const double v[],
const double vdot[], Integer ibnd, double g[],
Integer *ires, Nag_Comm *comm)
{
/* Solution known to be constant at both boundaries */

double ue[1];

if (comm->user[2] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback bndry2, first invocation.)\n");
}
comm->user[2] = 0.0;
}
if (ibnd == 0) {
exact(t, ue, &x[0], npde, 1);
g[0] = UE(1, 1) - U(1, 1);
}
else {
exact(t, ue, &x[npts - 1], npde, 1);
g[0] = UE(1, 1) - U(1, npts);
}

return;
}

static void NAG_CALL nmflx2(Integer npde, double t, double x, Integer ncode,
const double v[], const double uleft[],
const double uright[], double flux[],
Integer *ires, Nag_Comm *comm,
Nag_D03_Save *saved)
{
if (comm->user[3] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback nmflx2, first invocation.)\n");
}
comm->user[3] = 0.0;
}
flux[0] = uleft[0];

return;
}

static void NAG_CALL monit2(double t, Integer npts, Integer npde,
const double x[], const double u[],
double fmon[], Nag_Comm *comm)
{
static double xa = 0.0;
double h1, ux, uxmax, xl, xmax, xr;
Integer i;

if (comm->user[4] == -1.0) {
if (comm->iuser[0] > 0) {
printf("(User-supplied callback monit2, first invocation.)\n");
}
comm->user[4] = 0.0;
}
/* Locate shock */

uxmax = 0.0;
xmax = 0.0;
for (i = 2; i <= npts - 1; ++i) {
h1 = x[i - 1] - x[i - 2];
ux = fabs((U(1, i) - U(1, i - 1)) / h1);
if (ux > uxmax) {
uxmax = ux;
xmax = x[i - 1];
}
}

xa = 7.0/60.0;

xl = xmax - xa;
xr = xmax + xa;

/* Assign monitor function */

for (i = 0; i < npts; ++i) {
if (x[i] > xl && x[i] < xr) {
fmon[i] = 1.0 + cos(nag_pi * (x[i] - xmax) / xa);
}
else {
fmon[i] = 0.0;
}
}
return;
}

static void exact(double t, double *u, const double *x, Integer npde,
Integer npts)
{
/* Exact solution (for comparison and b.c. purposes) */

double del, psi, rm, rn, s;
Integer i;
s = 0.1;
del = 0.01;
rm = -1.0 / del;
rn = s / del + 1.0;

for (i = 1; i <= npts; ++i) {
psi = x[i - 1] - t;
if (psi < s) {
U(1, i) = 1.0;
}
else if (psi > del + s) {
U(1, i) = 0.0;
}
else {
U(1, i) = rm * psi + rn;
}
}
return;
}
```