NAG Library Manual, Mark 28.4
```/* nag_pde_dim1_parab_euler_exact (d03pxc) Example Program.
*
* Copyright 2022 Numerical Algorithms Group.
*
* Mark 28.4, 2022.
*/

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

/* Structure to communicate with user-supplied function arguments */

struct user {
double elo, ero, rlo, rro, ulo, uro, gamma;
};

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

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

#ifdef __cplusplus
}
#endif

#define U(I, J) u[npde * ((J)-1) + (I)-1]
#define UE(I, J) ue[npde * ((J)-1) + (I)-1]

int main(void) {
const Integer npde = 3, npts = 141, ncode = 0, nxi = 0;
const Integer neqn = npde * npts + ncode, lisave = neqn + 24, intpts = 9;
const Integer nwkres = npde * (2 * npts + 3 * npde + 32) + 7 * npts + 4,
lenode = 9 * neqn + 50;
const Integer mlu = 3 * npde - 1,
lrsave = (3 * mlu + 1) * neqn + nwkres + lenode;
double d, p, tout, ts, v;
Integer exit_status = 0, i, ind, itask, itol, itrace, k;
double *algopt = 0, *atol = 0, *rtol = 0, *u = 0, *ue = 0, *rsave = 0;
double *x = 0, *xi = 0;
Integer *isave = 0;
NagError fail;
Nag_Comm comm;
Nag_D03_Save saved;
struct user data;

INIT_FAIL(fail);

printf("nag_pde_dim1_parab_euler_exact (d03pxc) Example Program Results\n");

/* Allocate memory */

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

/* Skip heading in data file */
scanf("%*[^\n] ");

/* Problem parameters */

data.gamma = 1.4;
data.rlo = 5.99924;
data.rro = 5.99242;
data.ulo = 5.99924 * 19.5975;
data.uro = -5.99242 * 6.19633;
data.elo = 460.894 / (data.gamma - 1.0) + 0.5 * data.rlo * 19.5975 * 19.5975;
data.ero = 46.095 / (data.gamma - 1.0) + 0.5 * data.rro * 6.19633 * 6.19633;
comm.p = (Pointer)&data;

/* Initialize mesh */

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

/* Initial values */

for (i = 1; i <= npts; ++i) {
if (x[i - 1] < 0.5) {
U(1, i) = data.rlo;
U(2, i) = data.ulo;
U(3, i) = data.elo;
} else if (x[i - 1] == 0.5) {
U(1, i) = 0.5 * (data.rlo + data.rro);
U(2, i) = 0.5 * (data.ulo + data.uro);
U(3, i) = 0.5 * (data.elo + data.ero);
} else {
U(1, i) = data.rro;
U(2, i) = data.uro;
U(3, i) = data.ero;
}
}

itrace = 0;
itol = 1;
atol[0] = 0.005;
rtol[0] = 5e-4;
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] = 0.005;

ts = 0.0;
tout = 0.035;

/* 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, NULLFN, numflx, bndary, u, npts, x, ncode, 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;
}

printf("\n t = %6.3f\n\n", ts);
printf("%15s%18s%22s\n", "d", "v", "p");
printf("%3s%10s%9s%9s%9s%11s%11s\n", "x", "Approx", "Exact", "Approx",
"Exact", "Approx", "Exact");

/* Read exact data at output points */

for (i = 1; i <= intpts; ++i) {
scanf("%lf", &UE(1, i));
scanf("%lf", &UE(2, i));
scanf("%lf", &UE(3, i));
}

/* Calculate density, velocity and pressure */

k = 0;
for (i = 15; i <= 127; i += 14) {
++k;
d = U(1, i);
v = U(2, i) / d;
p = d * (data.gamma - 1.0) * (U(3, i) / d - 0.5 * v * v);
printf("%4.1f", x[i - 1]);
printf("%9.4f", d);
printf("%9.4f", UE(1, k));
printf("%9.4f", v);
printf("%9.4f", UE(2, k));
printf("%13.4e", p);
printf("%13.4e\n", UE(3, k));
}

printf("\n");
printf(" Number of time steps           = %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", isave[4]);

END:

NAG_FREE(algopt);
NAG_FREE(atol);
NAG_FREE(rtol);
NAG_FREE(u);
NAG_FREE(ue);
NAG_FREE(rsave);
NAG_FREE(x);
NAG_FREE(xi);
NAG_FREE(isave);

return exit_status;
}

static void NAG_CALL bndary(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) {
struct user *data = (struct user *)comm->p;

if (ibnd == 0) {
g[0] = U(1, 1) - data->rlo;
g[1] = U(2, 1) - data->ulo;
g[2] = U(3, 1) - data->elo;
} else {
g[0] = U(1, npts) - data->rro;
g[1] = U(2, npts) - data->uro;
g[2] = U(3, npts) - data->ero;
}
return;
}

static void NAG_CALL numflx(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) {
struct user *data = (struct user *)comm->p;
NagError fail;
Integer niter = 0;
double tol = 0.0;

INIT_FAIL(fail);

/* nag_pde_dim1_parab_euler_exact (d03pxc).
* Exact 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_exact(uleft, uright, data->gamma, tol, niter, flux,
saved, &fail);

if (fail.code != NE_NOERROR) {
printf("Error from nag_pde_dim1_parab_euler_exact (d03pxc).\n%s\n",
fail.message);
}

return;
}
```