保存型のソース項を用いた対流・拡散偏微分方程式

nag_pde_parab_1d_cd_ode (d03plc) Example Program Results


Example 1

 Method parameters:
  Number of mesh points used =  201
  Relative tolerance used    =    2.500e-04
  Absolute tolerance used    =    1.000e-05


 Integration Results:
  Global error is less than  100 times the local error tolerance.

 Integration Statistics:
  Number of time steps           (nearest  50) =    150
  Number of function evaluations (nearest 100) =   1400
  Number of Jacobian evaluations (nearest  20) =     20
  Number of iterations           (nearest 100) =    400


Example 2

 Problem parameters and initial conditions:
  gamma          = 1.400
      e(x<0.5,0) = 2.500      e(x>0.5,0) = 0.250
    rho(x<0.5,0) = 1.000    rho(x>0.5,0) = 0.125

 Method parameters:
  Number of mesh points used =  141
  Relative tolerance used    =    5.000e-04
  Absolute tolerance used    =    5.000e-03

 Solution
   t        x        d        v        p
 0.100   0.0000   1.0000   0.0000   1.0000
         0.1000   1.0000  -0.0000   1.0000
         0.2000   1.0000  -0.0000   1.0000
         0.3000   1.0000  -0.0000   1.0000
         0.4000   0.8668   0.1665   0.8188
         0.5000   0.4299   0.9182   0.3071
         0.6000   0.2969   0.9274   0.3028
         0.7000   0.1250   0.0000   0.1000
         0.8000   0.1250  -0.0000   0.1000
         0.9000   0.1250  -0.0000   0.1000
         1.0000   0.1250   0.0000   0.1000

 0.200   0.0000   1.0000   0.0000   1.0000
         0.1000   1.0000  -0.0000   1.0000
         0.2000   1.0000  -0.0000   1.0000
         0.3000   0.8718   0.1601   0.8253
         0.4000   0.6113   0.5543   0.5022
         0.5000   0.4245   0.9314   0.3014
         0.6000   0.4259   0.9277   0.3030
         0.7000   0.2772   0.9272   0.3031
         0.8000   0.2657   0.9276   0.3032
         0.9000   0.1250  -0.0000   0.1000
         1.0000   0.1250   0.0000   0.1000


 Integration Statistics:
  Number of time steps           (nearest  50) =    150
  Number of function evaluations (nearest  50) =    400
  Number of Jacobian evaluations (nearest   1) =      1
  Number of iterations           (nearest   1) =      2

/* nag_pde_parab_1d_cd_ode (d03plc) Example Program.
 *
 * CLL6I261D/CLL6I261DL Version.
 *
 * Copyright 2017 Numerical Algorithms Group.
 *
 * Mark 26.1, 2017.
 */

#include <stdio.h>
#include <nag.h>
#include <nag_stdlib.h>
#include <nagd03.h>
#include <nagx01.h>
#include <math.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_parab_1d_cd_ode (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 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 = %4ld\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_parab_1d_cd_ode (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_parab_1d_cd_ode(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_parab_1d_cd_ode (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 %4ld"
         " times the local error tolerance.\n", ierrmax);

  /* Print integration statistics (reasonably rounded) */
  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) = %6ld\n",
         "Number of time steps", 50, nsteps);
  printf("  %-30s (nearest %3d) = %6ld\n",
         "Number of function evaluations", 100, nfuncs);
  printf("  %-30s (nearest %3d) = %6ld\n",
         "Number of Jacobian evaluations", 20, njacs);
  printf("  %-30s (nearest %3d) = %6ld\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_pi * x1) * sin(2.0 * nag_pi * x1);
    g = exp(-2.0 * nag_pi * x2) * cos(2.0 * nag_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_pi * x1) * sin(2.0 * nag_pi * x1);
    g = exp(-2.0 * nag_pi * x2) * cos(2.0 * nag_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 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 = %4ld\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_parab_1d_cd_ode (d03plc), see above. */
    nag_pde_parab_1d_cd_ode(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_parab_1d_cd_ode (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) */
  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 %3ld) = %6ld\n",
         "Number of time steps", 50, nsteps);
  printf("  %-30s (nearest %3ld) = %6ld\n",
         "Number of function evaluations", 50, nfuncs);
  printf("  %-30s (nearest %3ld) = %6ld\n",
         "Number of Jacobian evaluations", 1, njacs);
  printf("  %-30s (nearest %3ld) = %6ld\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_parab_1d_euler_roe (d03puc).
     * Roe's approximate Riemann solver for Euler equations in
     * conservative form, for use with nag_pde_parab_1d_cd
     * (d03pfc), nag_pde_parab_1d_cd_ode (d03plc) and
     * nag_pde_parab_1d_cd_ode_remesh (d03psc)
     */
    nag_pde_parab_1d_euler_roe(uleft, uright, data->gamma, flux, saved,
                               &fail);
  }
  else {
    /* OSHER SCHEME */

    /* nag_pde_parab_1d_euler_osher (d03pvc).
     * Osher's approximate Riemann solver for Euler equations in
     * conservative form, for use with nag_pde_parab_1d_cd
     * (d03pfc), nag_pde_parab_1d_cd_ode (d03plc) and
     * nag_pde_parab_1d_cd_ode_remesh (d03psc)
     */
    nag_pde_parab_1d_euler_osher(uleft, uright, data->gamma,
                                 Nag_OsherPhysical, flux, saved, &fail);
  }

  if (fail.code != NE_NOERROR) {
    printf("Error from nag_pde_parab_1d_euler_osher (d03pvc).\n%s\n",
           fail.message);
  }

  return;
}