NAG Library Manual, Mark 27.3
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NAG CL Interface Introduction
Example description
/* nag_pde_dim1_parab_remesh_keller (d03prc) Example Program.
 *
 * Copyright 2021 Numerical Algorithms Group.
 *
 * Mark 27.3, 2021.
 */

#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[], const double[], Integer,
                            const double[], const double[], double[], Integer *,
                            Nag_Comm *);

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

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

#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) {
  const Integer npde = 2, npts = 61, ncode = 0, nxi = 0, nxfix = 0, nleft = 1;
  const Integer itype = 1, intpts = 5, neqn = npde * npts + ncode;
  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, tout, trmesh, ts, xratio;
  Integer exit_status = 0, 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);

  printf("nag_pde_dim1_parab_remesh_keller (d03prc) 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)) ||
      !(rsave = NAG_ALLOC(lrsave, double)) || !(rtol = NAG_ALLOC(1, double)) ||
      !(u = NAG_ALLOC(npde * npts + ncode, double)) ||
      !(ue = 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)) || !(xout = NAG_ALLOC(intpts, double)) ||
      !(isave = NAG_ALLOC(lisave, Integer))) {
    printf("Allocation failure\n");
    exit_status = 1;
    goto END;
  }

  itrace = 0;
  itol = 1;
  atol[0] = 5.0e-5;
  rtol[0] = atol[0];

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

  /* Set remesh parameters */

  remesh = Nag_TRUE;
  nrmesh = 3;
  dxmesh = 0.0;
  trmesh = 0.0;
  con = 5.0 / (npts - 1.0);
  xratio = 1.2;
  ipminf = 0;
  printf(" Remeshing every %3" NAG_IFMT " time steps\n\n", nrmesh);

  /* Initialize mesh */

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

  xout[0] = 0.0;
  xout[1] = 0.25;
  xout[2] = 0.5;
  xout[3] = 0.75;
  xout[4] = 1.0;
  printf(" x        ");

  for (i = 0; i < intpts; ++i) {
    printf("%10.4f", xout[i]);
    printf((i + 1) % 5 == 0 || i == 4 ? "\n" : " ");
  }
  printf("\n\n");

  xi[0] = 0.0;
  ind = 0;
  itask = 1;

  /* 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;
    algopt[5] = 2.0;
    algopt[6] = 1.0;
  }

  /* Loop over output value of t */

  ts = 0.0;
  for (it = 0; it < 5; ++it) {
    tout = 0.05 * (it + 1);

    /* nag_pde_dim1_parab_remesh_keller (d03prc).
     * General system of first-order PDEs, coupled DAEs, method
     * of lines, Keller box discretization, remeshing, one space
     * variable
     */
    nag_pde_dim1_parab_remesh_keller(
        npde, &ts, tout, pdedef, bndary, uvinit, u, npts, x, nleft, 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_keller (d03prc).\n%s\n",
             fail.message);
      exit_status = 1;
      goto END;
    }

    /* Interpolate at output points */

    /* nag_pde_dim1_parab_fd_interp (d03pzc). PDEs, spatial interpolation with
     * nag_pde_dim1_parab_remesh_keller (d03prc).
     */
    nag_pde_dim1_parab_fd_interp(npde, 0, 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, npde, intpts, xout, ue);

    printf(" t = %6.3f\n", ts);
    printf(" Approx u1");

    for (i = 1; i <= intpts; ++i) {
      printf("%10.4f", UOUT(1, i, 1));
      printf(i % 5 == 0 || i == 5 ? "\n" : "");
    }

    printf(" Exact  u1");

    for (i = 1; i <= 5; ++i) {
      printf("%10.4f", UE(1, i));
      printf(i % 5 == 0 || i == 5 ? "\n" : "");
    }

    printf(" Approx u2");

    for (i = 1; i <= 5; ++i) {
      printf("%10.4f", UOUT(2, i, 1));
      printf(i % 5 == 0 || i == 5 ? "\n" : "");
    }

    printf(" Exact  u2");

    for (i = 1; i <= 5; ++i) {
      printf("%10.4f", UE(2, i));
      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) {
  Integer i;

  if (comm->user[0] == -1.0) {
    printf("(User-supplied callback uvinit, first invocation.)\n");
    comm->user[0] = 0.0;
  }
  for (i = 1; i <= npts; ++i) {
    U(1, i) = exp(x[i - 1]);
    U(2, i) =
        x[i - 1] * x[i - 1] + sin(2.0 * nag_math_pi * (x[i - 1] * x[i - 1]));
  }
  return;
}

static void NAG_CALL pdedef(Integer npde, double t, double x, const double u[],
                            const double udot[], 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] = udot[0];
    res[1] = udot[1];
  } else {
    res[0] = udot[0] + ux[0] + ux[1];
    res[1] = udot[1] + 4.0 * ux[0] + ux[1];
  }
  return;
}

static void NAG_CALL bndary(Integer npde, double t, Integer ibnd, Integer nobc,
                            const double u[], const double udot[],
                            Integer ncode, const double v[],
                            const double vdot[], double res[], Integer *ires,
                            Nag_Comm *comm) {
  double pp;

  if (comm->user[2] == -1.0) {
    printf("(User-supplied callback bndary, first invocation.)\n");
    comm->user[2] = 0.0;
  }

  pp = 2.0 * nag_math_pi;

  if (ibnd == 0) {
    if (*ires == -1) {
      res[0] = 0.0;
    } else {
      res[0] = u[0] - 0.5 * (exp(t) + exp(-3.0 * t)) -
               0.25 * (sin(9.0 * pp * t * t) - sin(pp * t * t)) - 2.0 * t * t;
    }
  } else {
    if (*ires == -1) {
      res[0] = 0.0;
    } else {
      res[0] = u[1] - (exp(1.0 - 3.0 * t) - exp(1.0 + t) +
                       0.5 * sin(pp * (1.0 - 3.0 * t) * (1.0 - 3.0 * t)) +
                       0.5 * sin(pp * (1.0 + t) * (1.0 + t)) + 1.0 +
                       5.0 * t * t - 2.0 * t);
    }
  }
  return;
}

static void NAG_CALL monitf(double t, Integer npts, Integer npde,
                            const double x[], const double u[], double fmon[],
                            Nag_Comm *comm) {
  double d2x1, d2x2, h1, h2, h3;
  Integer i;

  if (comm->user[3] == -1.0) {
    printf("(User-supplied callback monitf, 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 */

    d2x1 = fabs(((U(1, i + 1) - U(1, i)) / h2 - (U(1, i) - U(1, i - 1)) / h1) /
                h3);
    d2x2 = fabs(((U(2, i + 1) - U(2, i)) / h2 - (U(2, i) - U(2, i - 1)) / h1) /
                h3);
    fmon[i - 1] = d2x1;
    if (d2x2 > d2x1)
      fmon[i - 1] = d2x2;
  }
  fmon[0] = fmon[1];
  fmon[npts - 1] = fmon[npts - 2];

  return;
}

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

  double pp;
  Integer i;

  pp = 2.0 * nag_math_pi;
  for (i = 1; i <= npts; ++i) {
    U(1, i) = 0.5 * (exp(x[i - 1] + t) + exp(x[i - 1] - 3.0 * t)) +
              0.25 * (sin(pp * (x[i - 1] - 3.0 * t) * (x[i - 1] - 3.0 * t)) -
                      sin(pp * (x[i - 1] + t) * (x[i - 1] + t))) +
              2.0 * t * t - 2.0 * x[i - 1] * t;

    U(2, i) = exp(x[i - 1] - 3.0 * t) - exp(x[i - 1] + t) +
              0.5 * (sin(pp * ((x[i - 1] - 3.0 * t) * (x[i - 1] - 3.0 * t))) +
                     sin(pp * ((x[i - 1] + t) * (x[i - 1] + t)))) +
              x[i - 1] * x[i - 1] + 5.0 * t * t - 2.0 * x[i - 1] * t;
  }
  return;
}