/* nag_pde_dim1_parab_convdiff_dae (d03plc) Example Program.
*
* Copyright 2024 Numerical Algorithms Group.
*
* Mark 30.2, 2024.
*/
#include <math.h>
#include <nag.h>
#include <stdio.h>
/* Structure to communicate with user-supplied function arguments */
struct user {
double elo, ero, gamma, rlo, rro;
};
#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[],
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 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 *);
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 *);
#ifdef __cplusplus
}
#endif
static void init1(double, double *, Integer, double *, Integer, Integer);
static void init2(Integer, Integer, double *, double *, Nag_Comm *);
static void exact(double, double *, Integer, const double *, Integer);
static int ex1(void);
static int ex2(void);
#define P(I, J) p[npde * ((J)-1) + (I)-1]
#define UCP(I, J) ucp[npde * ((J)-1) + (I)-1]
#define U(I, J) u[npde * ((J)-1) + (I)-1]
#define UE(I, J) ue[npde * ((J)-1) + (I)-1]
int main(void) {
Integer exit_status_ex1 = 0;
Integer exit_status_ex2 = 0;
printf("nag_pde_dim1_parab_convdiff_dae (d03plc) 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) {
/* Constants */
const Integer print_statistics = 0;
const Integer npde = 2, npts = 201, nv = 2, nxi = 2;
const Integer neqn = npde * npts + nv;
static double ruser1[4] = {-1.0, -1.0, -1.0, -1.0};
/* Scalars */
double tout, ts, errmax, lerr, lwgt;
Integer exit_status = 0, i, ind, itask, itol, itrace, j;
Integer nsteps, nfuncs, njacs, niters, ierrmax;
Integer nwkres, lenode, lisave, lrsave;
/* Arrays */
double *algopt = 0, *atol = 0, *rsave = 0, *rtol = 0, *u = 0, *ue = 0;
double *x = 0, *xi = 0;
Integer *isave = 0;
/* Nag Types */
NagError fail;
Nag_Comm comm;
Nag_D03_Save saved;
INIT_FAIL(fail);
/* For communication with user-supplied functions: */
comm.user = ruser1;
printf("\n\nExample 1\n\n");
/* Allocate memory */
lisave = 25 * neqn + 24;
nwkres =
npde * (2 * npts + 6 * nxi + 3 * npde + 26) + nxi + nv + 7 * npts + 2;
lenode = 11 * neqn + 50;
lrsave = 4 * neqn + 11 * neqn / 2 + 1 + nwkres + lenode;
lisave = lisave * 4;
lrsave = lrsave * 4;
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(npde * npts, 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-5;
rtol[0] = 2.5e-4;
printf(" Method parameters:\n");
printf(" Number of mesh points used = %4" NAG_IFMT "\n", npts);
printf(" Relative tolerance used = %12.3e\n", rtol[0]);
printf(" Absolute tolerance used = %12.3e\n\n", atol[0]);
/* Initialize mesh */
for (i = 0; i < npts; ++i)
x[i] = i / (npts - 1.0);
xi[0] = 0.0;
xi[1] = 1.0;
/* Set initial values */
ts = 0.0;
init1(ts, u, npde, x, npts, nv);
ind = 0;
itask = 1;
for (i = 0; i < 30; ++i)
algopt[i] = 0.0;
/* BDF integration */
algopt[0] = 1.0;
/* Sparse matrix algebra parameters */
algopt[28] = 0.1;
algopt[29] = 1.1;
tout = 0.5;
/* nag_pde_dim1_parab_convdiff_dae (d03plc).
* 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, one space variable
*/
nag_pde_dim1_parab_convdiff_dae(
npde, &ts, tout, pdedef, nmflx1, bndry1, u, npts, x, nv, odedef, nxi, xi,
neqn, rtol, atol, itol, Nag_OneNorm, Nag_LinAlgSparse, algopt, rsave,
lrsave, isave, lisave, itask, itrace, 0, &ind, &comm, &saved, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_dim1_parab_convdiff_dae (d03plc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
/* Check against exact solution */
exact(tout, ue, npde, &x[0], npts);
errmax = 0.0;
for (i = 1; i < npts; i++) {
lerr = 0.0;
for (j = 0; j < npde; j++) {
lwgt = rtol[0] * fabs(ue[i * npde + j]) + atol[0];
lerr += fabs(u[i * npde + j] - ue[i * npde + j]) / lwgt;
}
lerr = lerr / (double)npde;
errmax = MAX(errmax, lerr);
}
ierrmax = 100 * (Integer)(errmax / 100.0) + 100;
printf("\n Integration Results:\n");
printf(" Global error is less than %4" NAG_IFMT ""
" times the local error tolerance.\n",
ierrmax);
/* Print integration statistics (reasonably rounded) */
if (print_statistics == 1) {
nsteps = 50 * ((isave[0] + 25) / 50);
nfuncs = 100 * ((isave[1] + 50) / 100);
njacs = 20 * ((isave[2] + 10) / 20);
niters = 100 * ((isave[4] + 50) / 100);
printf("\n Integration Statistics:\n");
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n", "Number of time steps",
50, nsteps);
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n",
"Number of function evaluations", 100, nfuncs);
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n",
"Number of Jacobian evaluations", 20, njacs);
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n", "Number of iterations",
100, niters);
}
END:
NAG_FREE(algopt);
NAG_FREE(atol);
NAG_FREE(rsave);
NAG_FREE(rtol);
NAG_FREE(u);
NAG_FREE(ue);
NAG_FREE(x);
NAG_FREE(xi);
NAG_FREE(isave);
return exit_status;
}
static void NAG_CALL pdedef(Integer npde, double t, double x, const double u[],
const double ux[], Integer nv, const double v[],
const double vdot[], double p[], double c[],
double d[], double s[], Integer *ires,
Nag_Comm *comm) {
Integer i, j;
if (comm->user[2] == -1.0) {
/* printf("(User-supplied callback pdedef, first invocation.)\n"); */
comm->user[2] = 0.0;
}
for (i = 1; i <= npde; ++i) {
c[i - 1] = 1.0;
d[i - 1] = 0.0;
s[i - 1] = 0.0;
for (j = 1; j <= npde; ++j) {
if (i == j) {
P(i, j) = 1.0;
} else {
P(i, j) = 0.0;
}
}
}
return;
}
static void NAG_CALL bndry1(Integer npde, Integer npts, double t,
const double x[], const double u[], Integer nv,
const double v[], const double vdot[], Integer ibnd,
double g[], Integer *ires, Nag_Comm *comm) {
double dudx;
double *ue = 0;
if (comm->user[0] == -1.0) {
/* printf("(User-supplied callback bndry1, first invocation.)\n"); */
comm->user[0] = 0.0;
}
/* Allocate memory */
if (!(ue = NAG_ALLOC(npde, double))) {
printf("Allocation failure\n");
goto END;
}
if (ibnd == 0) {
exact(t, ue, npde, &x[0], 1);
g[0] = U(1, 1) + U(2, 1) - UE(1, 1) - UE(2, 1);
dudx = (U(1, 2) - U(2, 2) - U(1, 1) + U(2, 1)) / (x[1] - x[0]);
g[1] = vdot[0] - dudx;
} else {
exact(t, ue, npde, &x[npts - 1], 1);
g[0] = U(1, npts) - U(2, npts) - UE(1, 1) + UE(2, 1);
dudx = (U(1, npts) + U(2, npts) - U(1, npts - 1) - U(2, npts - 1)) /
(x[npts - 1] - x[npts - 2]);
g[1] = vdot[1] + 3.0 * dudx;
}
END:
NAG_FREE(ue);
return;
}
static void NAG_CALL nmflx1(Integer npde, double t, double x, Integer nv,
const double v[], const double uleft[],
const double uright[], double flux[], Integer *ires,
Nag_Comm *comm, Nag_D03_Save *saved) {
if (comm->user[1] == -1.0) {
/* printf("(User-supplied callback nmflx1, first invocation.)\n"); */
comm->user[1] = 0.0;
}
flux[0] = 0.5 * (3.0 * uleft[0] - uright[0] + 3.0 * uleft[1] + uright[1]);
flux[1] = 0.5 * (3.0 * uleft[0] + uright[0] + 3.0 * uleft[1] - uright[1]);
return;
}
static void NAG_CALL odedef(Integer npde, double t, Integer nv,
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) {
if (comm->user[3] == -1.0) {
/* printf("(User-supplied callback odedef, first invocation.)\n"); */
comm->user[3] = 0.0;
}
if (*ires == -1) {
r[0] = 0.0;
r[1] = 0.0;
} else {
r[0] = v[0] - UCP(1, 1) + UCP(2, 1);
r[1] = v[1] - UCP(1, 2) - UCP(2, 2);
}
return;
}
static void exact(double t, double *u, Integer npde, const double *x,
Integer npts) {
/* Exact solution (for comparison and b.c. purposes) */
double f, g, x1, x2;
Integer i;
for (i = 0; i < npts; ++i) {
x1 = x[i] - 3.0 * t;
x2 = x[i] + t;
f = exp(nag_math_pi * x1) * sin(2.0 * nag_math_pi * x1);
g = exp(-2.0 * nag_math_pi * x2) * cos(2.0 * nag_math_pi * x2);
u[npde * i] = f + g;
u[npde * i + 1] = f - g;
}
return;
}
static void init1(double t, double *u, Integer npde, double *x, Integer npts,
Integer nv) {
/* Initial solution */
double f, g, x1, x2;
Integer i, neqn;
neqn = npde * npts + nv;
for (i = 0; i < npts; ++i) {
x1 = x[i] - 3.0 * t;
x2 = x[i] + t;
f = exp(nag_math_pi * x1) * sin(2.0 * nag_math_pi * x1);
g = exp(-2.0 * nag_math_pi * x2) * cos(2.0 * nag_math_pi * x2);
u[npde * i] = f + g;
u[npde * i + 1] = f - g;
}
u[neqn - 2] = u[0] - u[1];
u[neqn - 1] = u[neqn - 3] + u[neqn - 4];
return;
}
int ex2(void) {
const Integer print_statistics = 0;
const Integer npde = 3, npts = 141, nv = 0, nxi = 0;
const Integer neqn = npde * npts + nv, lisave = neqn + 24;
const Integer lrsave = 16392;
static double ruser[2] = {-1.0, -1.0};
double d, p, tout, ts, v;
Integer exit_status = 0, i, ind, it, itask, itol, itrace;
Integer nsteps, nfuncs, njacs, niters;
double *algopt = 0, *atol = 0, *rsave = 0, *rtol = 0, *u = 0;
double *x = 0, *xi = 0;
Integer *isave = 0;
NagError fail;
Nag_Comm comm;
Nag_D03_Save saved;
struct user data;
INIT_FAIL(fail);
/* For communication with user-supplied functions: */
comm.user = ruser;
printf("\n\nExample 2\n\n");
/* 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)) || !(x = NAG_ALLOC(npts, double)) ||
!(xi = NAG_ALLOC(1, double)) || !(isave = NAG_ALLOC(447, Integer))) {
printf("Allocation failure\n");
exit_status = -1;
goto END;
}
/* Problem parameters */
data.elo = 2.5;
data.ero = 0.25;
data.gamma = 1.4;
data.rlo = 1.0;
data.rro = 0.125;
comm.p = (Pointer)&data;
itrace = 0;
itol = 1;
atol[0] = 0.005;
rtol[0] = 5e-4;
printf(" Problem parameters and initial conditions:\n");
printf(" gamma = %5.3f\n", data.gamma);
printf(" e(x<0.5,0) = %5.3f", data.elo);
printf(" e(x>0.5,0) = %5.3f\n", data.ero);
printf(" rho(x<0.5,0) = %5.3f", data.rlo);
printf(" rho(x>0.5,0) = %5.3f\n\n", data.rro);
printf(" Method parameters:\n");
printf(" Number of mesh points used = %4" NAG_IFMT "\n", npts);
printf(" Relative tolerance used = %12.3e\n", rtol[0]);
printf(" Absolute tolerance used = %12.3e\n\n", atol[0]);
/* Initialize mesh */
for (i = 0; i < npts; ++i)
x[i] = i / (npts - 1.0);
/* Initial values of variables */
init2(npde, npts, x, u, &comm);
xi[0] = 0.0;
ind = 0;
itask = 1;
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] = 0.005;
ts = 0.0;
printf(" Solution\n%4s%9s%9s%9s%9s\n", "t", "x", "d", "v", "p");
for (it = 0; it < 2; ++it) {
tout = 0.1 * (it + 1);
/* nag_pde_dim1_parab_convdiff_dae (d03plc), see above. */
nag_pde_dim1_parab_convdiff_dae(
npde, &ts, tout, NULLFN, nmflx2, bndry2, u, npts, x, nv, NULLFN, nxi,
xi, neqn, rtol, atol, itol, Nag_TwoNorm, Nag_LinAlgBand, algopt, rsave,
lrsave, isave, lisave, itask, itrace, 0, &ind, &comm, &saved, &fail);
if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_dim1_parab_convdiff_dae (d03plc).\n%s\n",
fail.message);
exit_status = 1;
goto END;
}
/* Calculate density, velocity and pressure */
for (i = 1; i <= npts; i += 14) {
d = U(1, i);
v = U(2, i) / d;
p = d * (data.gamma - 1.0) * (U(3, i) / d - 0.5 * v * v);
if (i == 1) {
printf("%6.3f %7.4f %7.4f %7.4f %7.4f\n", ts, x[i - 1], d, v, p);
} else {
printf("%6s %7.4f %7.4f %7.4f %7.4f\n", "", x[i - 1], d, v, p);
}
}
printf("\n");
}
/* Print integration statistics (reasonably rounded) */
if (print_statistics == 1) {
nsteps = 50 * ((isave[0] + 25) / 50);
nfuncs = 50 * ((isave[1] + 25) / 50);
njacs = isave[2];
niters = isave[4];
printf("\n Integration Statistics:\n");
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n", "Number of time steps",
50, nsteps);
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n",
"Number of function evaluations", 50, nfuncs);
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n",
"Number of Jacobian evaluations", 1, njacs);
printf(" %-30s (nearest %3d) = %6" NAG_IFMT "\n", "Number of iterations",
1, niters);
}
END:
NAG_FREE(algopt);
NAG_FREE(atol);
NAG_FREE(rsave);
NAG_FREE(rtol);
NAG_FREE(u);
NAG_FREE(x);
NAG_FREE(xi);
NAG_FREE(isave);
return exit_status;
}
static void init2(Integer npde, Integer npts, double *x, double *u,
Nag_Comm *comm) {
Integer i, j;
struct user *data = (struct user *)comm->p;
j = 0;
for (i = 0; i < npts; ++i) {
if (x[i] < 0.5) {
u[j] = data->rlo;
u[j + 1] = 0.0;
u[j + 2] = data->elo;
} else if (x[i] == 0.5) {
u[j] = 0.5 * (data->rlo + data->rro);
u[j + 1] = 0.0;
u[j + 2] = 0.5 * (data->elo + data->ero);
} else {
u[j] = data->rro;
u[j + 1] = 0.0;
u[j + 2] = data->ero;
}
j += 3;
}
return;
}
static void NAG_CALL bndry2(Integer npde, Integer npts, double t,
const double x[], const double u[], Integer nv,
const double v[], const double vdot[], Integer ibnd,
double g[], Integer *ires, Nag_Comm *comm) {
struct user *data = (struct user *)comm->p;
if (comm->user[0] == -1.0) {
/* printf("(User-supplied callback bndry2, first invocation.)\n"); */
comm->user[0] = 0.0;
}
if (ibnd == 0) {
g[0] = U(1, 1) - data->rlo;
g[1] = U(2, 1);
g[2] = U(3, 1) - data->elo;
} else {
g[0] = U(1, npts) - data->rro;
g[1] = U(2, npts);
g[2] = U(3, npts) - data->ero;
}
return;
}
static void NAG_CALL nmflx2(Integer npde, double t, double x, Integer nv,
const double v[], const double uleft[],
const double uright[], double flux[], Integer *ires,
Nag_Comm *comm, Nag_D03_Save *saved) {
char solver;
NagError fail;
struct user *data = (struct user *)comm->p;
if (comm->user[1] == -1.0) {
/* printf("(User-supplied callback nmflx2, first invocation.)\n"); */
comm->user[1] = 0.0;
}
INIT_FAIL(fail);
solver = 'R';
if (solver == 'R') {
/* ROE SCHEME */
/* nag_pde_dim1_parab_euler_roe (d03puc).
* Roe's approximate Riemann solver for Euler equations in
* conservative form, for use with nag_pde_dim1_parab_convdiff
* (d03pfc), nag_pde_dim1_parab_convdiff_dae (d03plc) and
* nag_pde_dim1_parab_convdiff_remesh (d03psc)
*/
nag_pde_dim1_parab_euler_roe(uleft, uright, data->gamma, flux, saved,
&fail);
} else {
/* OSHER SCHEME */
/* nag_pde_dim1_parab_euler_osher (d03pvc).
* Osher's approximate Riemann solver for Euler equations in
* conservative form, for use with nag_pde_dim1_parab_convdiff
* (d03pfc), nag_pde_dim1_parab_convdiff_dae (d03plc) and
* nag_pde_dim1_parab_convdiff_remesh (d03psc)
*/
nag_pde_dim1_parab_euler_osher(uleft, uright, data->gamma,
Nag_OsherPhysical, flux, saved, &fail);
}
if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_dim1_parab_euler_osher (d03pvc).\n%s\n",
fail.message);
}
return;
}