/* nag_pde_parab_1d_cd_ode_remesh (d03psc) Example Program. * * Copyright 2001 Numerical Algorithms Group. * * Mark 7, 2001. */ #include #include #include #include #include #include int ex1(void), ex2(void); static void uvin1(Integer, Integer, Integer, const double[], const double[], double[],Integer, double[], Nag_Comm *); static void uvin2(Integer, Integer, Integer, const double[], const double[], double[],Integer, double[], Nag_Comm *); static void pdef1(Integer, double, double, const double[], const double[], Integer, const double[], const double[], double[], double[], double[], double[], Integer *, Nag_Comm *); static void pdef2(Integer, double, double, const double[], const double[], Integer, const double[], const double[], double[], double[], double[], double[], Integer *, Nag_Comm *); static void bndry1(Integer, Integer, double, const double[], const double[], Integer, const double[], const double[], Integer, double[], Integer *, Nag_Comm *); static void bndry2(Integer, Integer, double, const double[], const double[], Integer, const double[], const double[], Integer, double[], Integer *, Nag_Comm *); static void monit1(double, Integer, Integer, const double[], const double[], double[], Nag_Comm *); static void monit2(double, Integer, Integer, const double[], const double[], double[], Nag_Comm *); static void nmflx1(Integer, double, double, Integer, const double[], const double[], const double[], double[], Integer *, Nag_Comm *, Nag_D03_Save *); static void nmflx2(Integer, double, double, Integer, const double[], const double[], const double[], double[], Integer *, Nag_Comm *, Nag_D03_Save *); 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) { Vprintf("d03psc Example Program Results\n"); ex1(); ex2(); return 0; } int ex1(void) { const Integer npde=1, npts=61, ncode=0, nxi=0, nxfix=0, itype=1, neqn=npde*npts+ncode, intpts=7, lisave=25+nxfix+neqn, nwkres=npde*(3*npts+3*npde+32)+7*npts+3, lenode=11*neqn+50, mlu=3*npde-1, lrsave=(3*mlu+1)*neqn+nwkres+lenode; static double xout[7] = { .2,.3,.4,.5,.6,.7,.8 }; double con, dxmesh, tout, trmesh, ts, xratio; Integer exit_status, i, ind, ipminf, it, itask, itol, itrace, m, nrmesh; Boolean remesh; double *algopt=0, *atol=0, *rsave=0, *rtol=0, *u=0, *uout=0, *x=0, *xfix=0, *xi=0; Integer *isave=0; NagError fail; Nag_Comm comm; Nag_D03_Save saved; INIT_FAIL(fail); exit_status = 0; /* 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)) ) { Vprintf("Allocation failure\n"); exit_status = 1; goto END; } Vprintf("\n\nExample 1\n\n"); itrace = 0; itol = 1; atol[0] = 1.0e-4; rtol[0] = 1.0e-4; Vprintf(" npts = %4ld", npts); Vprintf(" atol = %10.3e", atol[0]); Vprintf(" rtol = %10.3e\n\n", rtol[0]); /* Initialise mesh */ for (i = 0; i < npts; ++i) x[i] = i/(npts-1.0); xfix[0] = 0.0; /* Set remesh parameters */ remesh = 1; 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; itask = 1; 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; tout = 0.0; for (it = 0; it < 3; ++it) { tout = 0.1*(it+1); d03psc(npde, &ts, tout, pdef1, nmflx1, bndry1, uvin1, u, npts, x, ncode, d03pek, nxi, xi, neqn, rtol, atol, itol, Nag_OneNorm, Nag_LinAlgBand, algopt, remesh, nxfix, xfix, nrmesh, dxmesh, trmesh, ipminf, xratio, con, monit1, rsave, lrsave, isave, lisave, itask, itrace, 0, &ind, &comm, &saved, &fail); if (fail.code != NE_NOERROR) { Vprintf("Error from d03psc.\n%s\n", fail.message); exit_status = 1; goto END; } Vprintf(" t = %6.3f\n", ts); Vprintf(" x "); for (i = 1; i <= intpts; ++i) { Vprintf("%9.4f", xout[i-1]); Vprintf(i%7 == 0 || i == 7 ?"\n":""); } /* Interpolate at output points */ m = 0; d03pzc(npde, m, u, npts, x, xout, intpts, itype, uout, &fail); if (fail.code != NE_NOERROR) { Vprintf("Error from d03pzc.\n%s\n", fail.message); exit_status = 1; goto END; } Vprintf(" Approx u "); for (i = 1; i <= intpts; ++i) { Vprintf("%9.4f", UOUT(1,i,1)); Vprintf(i%7 == 0 || i == 7 ?"\n":""); } Vprintf("\n"); } Vprintf(" Number of integration steps in time = %6ld\n", isave[0]); Vprintf(" Number of function evaluations = %6ld\n", isave[1]); Vprintf(" Number of Jacobian evaluations =%6ld\n", isave[2]); Vprintf(" Number of iterations = %6ld\n\n", isave[4]); END: if (algopt) NAG_FREE(algopt); if (atol) NAG_FREE(atol); if (rsave) NAG_FREE(rsave); if (rtol) NAG_FREE(rtol); if (u) NAG_FREE(u); if (uout) NAG_FREE(uout); if (x) NAG_FREE(x); if (xfix) NAG_FREE(xfix); if (xi) NAG_FREE(xi); if (isave) NAG_FREE(isave); return exit_status; } static void uvin1(Integer npde, Integer npts, Integer nxi, const double x[], const double xi[], double u[],Integer ncode, double v[], Nag_Comm *comm) { Integer i; 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 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) { P(1, 1) = 1.0; c[0] = 0.002; d[0] = ux[0]; s[0] = 0.0; return; } static void 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 (ibnd == 0) { g[0] = U(1, 1); } else { g[0] = U(1, npts); } return; } static void monit1(double t, Integer npts, Integer npde, const double x[], const double u[], double fmon[], Nag_Comm *comm) { double h1, h2, h3; Integer i; 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 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) { flux[0] = uleft[0]; return; } int ex2(void) { const Integer npde=1, npts=61, ncode=0, nxi=0, nxfix=0, itype=1, neqn=npde*npts+ncode, intpts=7, lisave=25+nxfix+neqn, nwkres=npde*(3*npts+3*npde+32)+7*npts+3, lenode=11*neqn+50, mlu=3*npde-1, lrsave=(3*mlu+1)*neqn+nwkres+lenode; static double xout[7] = { 0.,.3,.4,.5,.6,.7,1. }; double con, dxmesh, tout, trmesh, ts, xratio; Integer exit_status, i, ind, ipminf, it, itask, itol, itrace, m, nrmesh; Boolean remesh; double *algopt=0, *atol=0, *rsave=0, *rtol=0, *u=0, *ue=0, *uout=0, *x=0, *xfix=0, *xi=0; Integer *isave=0; NagError fail; Nag_Comm comm; Nag_D03_Save saved; INIT_FAIL(fail); exit_status = 0; /* 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)) ) { Vprintf("Allocation failure\n"); exit_status = 1; goto END; } Vprintf("\n\nExample 2\n\n"); itrace = 0; itol = 1; atol[0] = 5e-4; rtol[0] = 0.05; Vprintf(" npts = %4ld", npts); Vprintf(" atol = %10.3e", atol[0]); Vprintf(" rtol = %10.3e\n\n", rtol[0]); /* Initialise mesh */ for (i = 0; i < npts; ++i) x[i] = i/(npts-1.0); xfix[0] = 0.0; /* Set remesh parameters */ remesh = 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; 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] = 0.0025; ts = 0.0; tout = 0.0; for (it = 0; it < 2; ++it) { tout = 0.2*(it+1); d03psc(npde, &ts, tout, pdef2, nmflx2, bndry2, uvin2, u, npts, x, ncode, d03pek, nxi, xi, neqn, rtol, atol, itol, Nag_OneNorm, Nag_LinAlgBand, algopt, remesh, nxfix, xfix, nrmesh, dxmesh, trmesh, ipminf, xratio, con, monit2, rsave, lrsave, isave, lisave, itask, itrace, 0, &ind, &comm, &saved, &fail); if (fail.code != NE_NOERROR) { Vprintf("Error from d03psc.\n%s\n", fail.message); exit_status = 1; goto END; } Vprintf(" t = %6.3f\n", ts); Vprintf(" x Approx u Exact u\n\n"); /* Interpolate at output points */ m = 0; d03pzc(npde, m, u, npts, x, xout, intpts, itype, uout, &fail); if (fail.code != NE_NOERROR) { Vprintf("Error from d03pzc.\n%s\n", fail.message); exit_status = 1; goto END; } /* Check against exact solution */ exact(tout, ue, xout, npde, intpts); for (i = 1; i <= intpts; ++i) { Vprintf(" %9.4f", xout[i-1]); Vprintf(" %9.4f", UOUT(1,i,1)); Vprintf(" %9.4f\n", UE(1,i)); } } Vprintf(" Number of integration steps in time = %6ld\n", isave[0]); Vprintf(" Number of function evaluations = %6ld\n", isave[1]); Vprintf(" Number of Jacobian evaluations =%6ld\n", isave[2]); Vprintf(" Number of iterations = %6ld\n\n", isave[4]); END: if (algopt) NAG_FREE(algopt); if (atol) NAG_FREE(atol); if (rsave) NAG_FREE(rsave); if (rtol) NAG_FREE(rtol); if (u) NAG_FREE(u); if (ue) NAG_FREE(ue); if (uout) NAG_FREE(uout); if (x) NAG_FREE(x); if (xfix) NAG_FREE(xfix); if (xi) NAG_FREE(xi); if (isave) NAG_FREE(isave); return exit_status; } static void uvin2(Integer npde, Integer npts, Integer nxi, const double x[], const double xi[], double u[],Integer ncode, double v[], Nag_Comm *comm) { double t; t = 0.0; exact(t, u, x, npde, npts); return; } static void 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) { 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 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 (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 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) { flux[0] = uleft[0]; return; } static void monit2(double t, Integer npts, Integer npde, const double x[], const double u[], double fmon[], Nag_Comm *comm) { static double xa = 0.0; static Integer icount = 0; double h1, ux, uxmax, xl, xleft, xmax, xr, xright; Integer i; /* 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]; } } /* Assign width (on first call only) */ if (icount == 0) { icount = 1; xleft = xmax - x[0]; xright = x[npts-1] - xmax; if (xleft > xright) { xa = xright; } else { xa = xleft; } } 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; }