Actual source code: ex10.c

      1: /*
      2:   This example implements the model described in

      4:     Rauenzahn, Mousseau, Knoll. "Temporal accuracy of the nonequilibrium radiation diffusion
      5:     equations employing a Saha ionization model" 2005.

      7:   The paper discusses three examples, the first two are nondimensional with a simple
      8:   ionization model.  The third example is fully dimensional and uses the Saha ionization
      9:   model with realistic parameters.
     10: */

     12: #include <petscts.h>
     13: #include <petscdm.h>
     14: #include <petscdmda.h>

     16: typedef enum {
     17:   BC_DIRICHLET,
     18:   BC_NEUMANN,
     19:   BC_ROBIN
     20: } BCType;
     21: static const char *const BCTypes[] = {"DIRICHLET", "NEUMANN", "ROBIN", "BCType", "BC_", 0};
     22: typedef enum {
     23:   JACOBIAN_ANALYTIC,
     24:   JACOBIAN_MATRIXFREE,
     25:   JACOBIAN_FD_COLORING,
     26:   JACOBIAN_FD_FULL
     27: } JacobianType;
     28: static const char *const JacobianTypes[] = {"ANALYTIC", "MATRIXFREE", "FD_COLORING", "FD_FULL", "JacobianType", "FD_", 0};
     29: typedef enum {
     30:   DISCRETIZATION_FD,
     31:   DISCRETIZATION_FE
     32: } DiscretizationType;
     33: static const char *const DiscretizationTypes[] = {"FD", "FE", "DiscretizationType", "DISCRETIZATION_", 0};
     34: typedef enum {
     35:   QUADRATURE_GAUSS1,
     36:   QUADRATURE_GAUSS2,
     37:   QUADRATURE_GAUSS3,
     38:   QUADRATURE_GAUSS4,
     39:   QUADRATURE_LOBATTO2,
     40:   QUADRATURE_LOBATTO3
     41: } QuadratureType;
     42: static const char *const QuadratureTypes[] = {"GAUSS1", "GAUSS2", "GAUSS3", "GAUSS4", "LOBATTO2", "LOBATTO3", "QuadratureType", "QUADRATURE_", 0};

     44: typedef struct {
     45:   PetscScalar E; /* radiation energy */
     46:   PetscScalar T; /* material temperature */
     47: } RDNode;

     49: typedef struct {
     50:   PetscReal meter, kilogram, second, Kelvin; /* Fundamental units */
     51:   PetscReal Joule, Watt;                     /* Derived units */
     52: } RDUnit;

     54: typedef struct _n_RD *RD;

     56: struct _n_RD {
     57:   void (*MaterialEnergy)(RD, const RDNode *, PetscScalar *, RDNode *);
     58:   DM                 da;
     59:   PetscBool          monitor_residual;
     60:   DiscretizationType discretization;
     61:   QuadratureType     quadrature;
     62:   JacobianType       jacobian;
     63:   PetscInt           initial;
     64:   BCType             leftbc;
     65:   PetscBool          view_draw;
     66:   char               view_binary[PETSC_MAX_PATH_LEN];
     67:   PetscBool          test_diff;
     68:   PetscBool          endpoint;
     69:   PetscBool          bclimit;
     70:   PetscBool          bcmidpoint;
     71:   RDUnit             unit;

     73:   /* model constants, see Table 2 and RDCreate() */
     74:   PetscReal rho, K_R, K_p, I_H, m_p, m_e, h, k, c, sigma_b, beta, gamma;

     76:   /* Domain and boundary conditions */
     77:   PetscReal Eapplied; /* Radiation flux from the left */
     78:   PetscReal L;        /* Length of domain */
     79:   PetscReal final_time;
     80: };

     82: static PetscErrorCode RDDestroy(RD *rd)
     83: {
     84:   PetscFunctionBeginUser;
     85:   PetscCall(DMDestroy(&(*rd)->da));
     86:   PetscCall(PetscFree(*rd));
     87:   PetscFunctionReturn(PETSC_SUCCESS);
     88: }

     90: /* The paper has a time derivative for material energy (Eq 2) which is a dependent variable (computable from temperature
     91:  * and density through an uninvertible relation).  Computing this derivative is trivial for trapezoid rule (used in the
     92:  * paper), but does not generalize nicely to higher order integrators.  Here we use the implicit form which provides
     93:  * time derivatives of the independent variables (radiation energy and temperature), so we must compute the time
     94:  * derivative of material energy ourselves (could be done using AD).
     95:  *
     96:  * There are multiple ionization models, this interface dispatches to the one currently in use.
     97:  */
     98: static void RDMaterialEnergy(RD rd, const RDNode *n, PetscScalar *Em, RDNode *dEm)
     99: {
    100:   rd->MaterialEnergy(rd, n, Em, dEm);
    101: }

    103: /* Solves a quadratic equation while propagating tangents */
    104: static void QuadraticSolve(PetscScalar a, PetscScalar a_t, PetscScalar b, PetscScalar b_t, PetscScalar c, PetscScalar c_t, PetscScalar *x, PetscScalar *x_t)
    105: {
    106:   PetscScalar disc = b * b - 4. * a * c, disc_t = 2. * b * b_t - 4. * a_t * c - 4. * a * c_t, num = -b + PetscSqrtScalar(disc), /* choose positive sign */
    107:     num_t = -b_t + 0.5 / PetscSqrtScalar(disc) * disc_t, den = 2. * a, den_t = 2. * a_t;
    108:   *x   = num / den;
    109:   *x_t = (num_t * den - num * den_t) / PetscSqr(den);
    110: }

    112: /* The primary model presented in the paper */
    113: static void RDMaterialEnergy_Saha(RD rd, const RDNode *n, PetscScalar *inEm, RDNode *dEm)
    114: {
    115:   PetscScalar Em, alpha, alpha_t, T = n->T, T_t = 1., chi = rd->I_H / (rd->k * T), chi_t = -chi / T * T_t, a = 1., a_t = 0, b = 4. * rd->m_p / rd->rho * PetscPowScalarReal(2. * PETSC_PI * rd->m_e * rd->I_H / PetscSqr(rd->h), 1.5) * PetscExpScalar(-chi) * PetscPowScalarReal(chi, 1.5), /* Eq 7 */
    116:     b_t = -b * chi_t + 1.5 * b / chi * chi_t, c = -b, c_t = -b_t;
    117:   QuadraticSolve(a, a_t, b, b_t, c, c_t, &alpha, &alpha_t);      /* Solve Eq 7 for alpha */
    118:   Em = rd->k * T / rd->m_p * (1.5 * (1. + alpha) + alpha * chi); /* Eq 6 */
    119:   if (inEm) *inEm = Em;
    120:   if (dEm) {
    121:     dEm->E = 0;
    122:     dEm->T = Em / T * T_t + rd->k * T / rd->m_p * (1.5 * alpha_t + alpha_t * chi + alpha * chi_t);
    123:   }
    124: }
    125: /* Reduced ionization model, Eq 30 */
    126: static void RDMaterialEnergy_Reduced(RD rd, const RDNode *n, PetscScalar *Em, RDNode *dEm)
    127: {
    128:   PetscScalar alpha, alpha_t, T = n->T, T_t = 1., chi = -0.3 / T, chi_t = -chi / T * T_t, a = 1., a_t = 0., b = PetscExpScalar(chi), b_t = b * chi_t, c = -b, c_t = -b_t;
    129:   QuadraticSolve(a, a_t, b, b_t, c, c_t, &alpha, &alpha_t);
    130:   if (Em) *Em = (1. + alpha) * T + 0.3 * alpha;
    131:   if (dEm) {
    132:     dEm->E = 0;
    133:     dEm->T = alpha_t * T + (1. + alpha) * T_t + 0.3 * alpha_t;
    134:   }
    135: }

    137: /* Eq 5 */
    138: static void RDSigma_R(RD rd, RDNode *n, PetscScalar *sigma_R, RDNode *dsigma_R)
    139: {
    140:   *sigma_R    = rd->K_R * rd->rho * PetscPowScalar(n->T, -rd->gamma);
    141:   dsigma_R->E = 0;
    142:   dsigma_R->T = -rd->gamma * (*sigma_R) / n->T;
    143: }

    145: /* Eq 4 */
    146: static void RDDiffusionCoefficient(RD rd, PetscBool limit, RDNode *n, RDNode *nx, PetscScalar *D_R, RDNode *dD_R, RDNode *dxD_R)
    147: {
    148:   PetscScalar sigma_R, denom;
    149:   RDNode      dsigma_R, ddenom, dxdenom;

    151:   RDSigma_R(rd, n, &sigma_R, &dsigma_R);
    152:   denom     = 3. * rd->rho * sigma_R + (int)limit * PetscAbsScalar(nx->E) / n->E;
    153:   ddenom.E  = -(int)limit * PetscAbsScalar(nx->E) / PetscSqr(n->E);
    154:   ddenom.T  = 3. * rd->rho * dsigma_R.T;
    155:   dxdenom.E = (int)limit * (PetscRealPart(nx->E) < 0 ? -1. : 1.) / n->E;
    156:   dxdenom.T = 0;
    157:   *D_R      = rd->c / denom;
    158:   if (dD_R) {
    159:     dD_R->E = -rd->c / PetscSqr(denom) * ddenom.E;
    160:     dD_R->T = -rd->c / PetscSqr(denom) * ddenom.T;
    161:   }
    162:   if (dxD_R) {
    163:     dxD_R->E = -rd->c / PetscSqr(denom) * dxdenom.E;
    164:     dxD_R->T = -rd->c / PetscSqr(denom) * dxdenom.T;
    165:   }
    166: }

    168: static PetscErrorCode RDStateView(RD rd, Vec X, Vec Xdot, Vec F)
    169: {
    170:   DMDALocalInfo info;
    171:   PetscInt      i;
    172:   const RDNode *x, *xdot, *f;
    173:   MPI_Comm      comm;

    175:   PetscFunctionBeginUser;
    176:   PetscCall(PetscObjectGetComm((PetscObject)rd->da, &comm));
    177:   PetscCall(DMDAGetLocalInfo(rd->da, &info));
    178:   PetscCall(DMDAVecGetArrayRead(rd->da, X, (void *)&x));
    179:   PetscCall(DMDAVecGetArrayRead(rd->da, Xdot, (void *)&xdot));
    180:   PetscCall(DMDAVecGetArrayRead(rd->da, F, (void *)&f));
    181:   for (i = info.xs; i < info.xs + info.xm; i++) {
    182:     PetscCall(PetscSynchronizedPrintf(comm, "x[%" PetscInt_FMT "] (%10.2G,%10.2G) (%10.2G,%10.2G) (%10.2G,%10.2G)\n", i, (double)PetscRealPart(x[i].E), (double)PetscRealPart(x[i].T), (double)PetscRealPart(xdot[i].E), (double)PetscRealPart(xdot[i].T),
    183:                                       (double)PetscRealPart(f[i].E), (double)PetscRealPart(f[i].T)));
    184:   }
    185:   PetscCall(DMDAVecRestoreArrayRead(rd->da, X, (void *)&x));
    186:   PetscCall(DMDAVecRestoreArrayRead(rd->da, Xdot, (void *)&xdot));
    187:   PetscCall(DMDAVecRestoreArrayRead(rd->da, F, (void *)&f));
    188:   PetscCall(PetscSynchronizedFlush(comm, PETSC_STDOUT));
    189:   PetscFunctionReturn(PETSC_SUCCESS);
    190: }

    192: static PetscScalar RDRadiation(RD rd, const RDNode *n, RDNode *dn)
    193: {
    194:   PetscScalar sigma_p = rd->K_p * rd->rho * PetscPowScalar(n->T, -rd->beta), sigma_p_T = -rd->beta * sigma_p / n->T, tmp = 4. * rd->sigma_b * PetscSqr(PetscSqr(n->T)) / rd->c - n->E, tmp_E = -1., tmp_T = 4. * rd->sigma_b * 4 * n->T * (PetscSqr(n->T)) / rd->c, rad = sigma_p * rd->c * rd->rho * tmp, rad_E = sigma_p * rd->c * rd->rho * tmp_E, rad_T = rd->c * rd->rho * (sigma_p_T * tmp + sigma_p * tmp_T);
    195:   if (dn) {
    196:     dn->E = rad_E;
    197:     dn->T = rad_T;
    198:   }
    199:   return rad;
    200: }

    202: static PetscScalar RDDiffusion(RD rd, PetscReal hx, const RDNode x[], PetscInt i, RDNode d[])
    203: {
    204:   PetscReal   ihx = 1. / hx;
    205:   RDNode      n_L, nx_L, n_R, nx_R, dD_L, dxD_L, dD_R, dxD_R, dfluxL[2], dfluxR[2];
    206:   PetscScalar D_L, D_R, fluxL, fluxR;

    208:   n_L.E  = 0.5 * (x[i - 1].E + x[i].E);
    209:   n_L.T  = 0.5 * (x[i - 1].T + x[i].T);
    210:   nx_L.E = (x[i].E - x[i - 1].E) / hx;
    211:   nx_L.T = (x[i].T - x[i - 1].T) / hx;
    212:   RDDiffusionCoefficient(rd, PETSC_TRUE, &n_L, &nx_L, &D_L, &dD_L, &dxD_L);
    213:   fluxL       = D_L * nx_L.E;
    214:   dfluxL[0].E = -ihx * D_L + (0.5 * dD_L.E - ihx * dxD_L.E) * nx_L.E;
    215:   dfluxL[1].E = +ihx * D_L + (0.5 * dD_L.E + ihx * dxD_L.E) * nx_L.E;
    216:   dfluxL[0].T = (0.5 * dD_L.T - ihx * dxD_L.T) * nx_L.E;
    217:   dfluxL[1].T = (0.5 * dD_L.T + ihx * dxD_L.T) * nx_L.E;

    219:   n_R.E  = 0.5 * (x[i].E + x[i + 1].E);
    220:   n_R.T  = 0.5 * (x[i].T + x[i + 1].T);
    221:   nx_R.E = (x[i + 1].E - x[i].E) / hx;
    222:   nx_R.T = (x[i + 1].T - x[i].T) / hx;
    223:   RDDiffusionCoefficient(rd, PETSC_TRUE, &n_R, &nx_R, &D_R, &dD_R, &dxD_R);
    224:   fluxR       = D_R * nx_R.E;
    225:   dfluxR[0].E = -ihx * D_R + (0.5 * dD_R.E - ihx * dxD_R.E) * nx_R.E;
    226:   dfluxR[1].E = +ihx * D_R + (0.5 * dD_R.E + ihx * dxD_R.E) * nx_R.E;
    227:   dfluxR[0].T = (0.5 * dD_R.T - ihx * dxD_R.T) * nx_R.E;
    228:   dfluxR[1].T = (0.5 * dD_R.T + ihx * dxD_R.T) * nx_R.E;

    230:   if (d) {
    231:     d[0].E = -ihx * dfluxL[0].E;
    232:     d[0].T = -ihx * dfluxL[0].T;
    233:     d[1].E = ihx * (dfluxR[0].E - dfluxL[1].E);
    234:     d[1].T = ihx * (dfluxR[0].T - dfluxL[1].T);
    235:     d[2].E = ihx * dfluxR[1].E;
    236:     d[2].T = ihx * dfluxR[1].T;
    237:   }
    238:   return ihx * (fluxR - fluxL);
    239: }

    241: static PetscErrorCode RDGetLocalArrays(RD rd, TS ts, Vec X, Vec Xdot, PetscReal *Theta, PetscReal *dt, Vec *X0loc, RDNode **x0, Vec *Xloc, RDNode **x, Vec *Xloc_t, RDNode **xdot)
    242: {
    243:   PetscBool istheta;

    245:   PetscFunctionBeginUser;
    246:   PetscCall(DMGetLocalVector(rd->da, X0loc));
    247:   PetscCall(DMGetLocalVector(rd->da, Xloc));
    248:   PetscCall(DMGetLocalVector(rd->da, Xloc_t));

    250:   PetscCall(DMGlobalToLocalBegin(rd->da, X, INSERT_VALUES, *Xloc));
    251:   PetscCall(DMGlobalToLocalEnd(rd->da, X, INSERT_VALUES, *Xloc));
    252:   PetscCall(DMGlobalToLocalBegin(rd->da, Xdot, INSERT_VALUES, *Xloc_t));
    253:   PetscCall(DMGlobalToLocalEnd(rd->da, Xdot, INSERT_VALUES, *Xloc_t));

    255:   /*
    256:     The following is a hack to subvert TSTHETA which is like an implicit midpoint method to behave more like a trapezoid
    257:     rule.  These methods have equivalent linear stability, but the nonlinear stability is somewhat different.  The
    258:     radiation system is inconvenient to write in explicit form because the ionization model is "on the left".
    259:    */
    260:   PetscCall(PetscObjectTypeCompare((PetscObject)ts, TSTHETA, &istheta));
    261:   if (istheta && rd->endpoint) {
    262:     PetscCall(TSThetaGetTheta(ts, Theta));
    263:   } else *Theta = 1.;

    265:   PetscCall(TSGetTimeStep(ts, dt));
    266:   PetscCall(VecWAXPY(*X0loc, -(*Theta) * (*dt), *Xloc_t, *Xloc)); /* back out the value at the start of this step */
    267:   if (rd->endpoint) { PetscCall(VecWAXPY(*Xloc, *dt, *Xloc_t, *X0loc)); /* move the abscissa to the end of the step */ }

    269:   PetscCall(DMDAVecGetArray(rd->da, *X0loc, x0));
    270:   PetscCall(DMDAVecGetArray(rd->da, *Xloc, x));
    271:   PetscCall(DMDAVecGetArray(rd->da, *Xloc_t, xdot));
    272:   PetscFunctionReturn(PETSC_SUCCESS);
    273: }

    275: static PetscErrorCode RDRestoreLocalArrays(RD rd, Vec *X0loc, RDNode **x0, Vec *Xloc, RDNode **x, Vec *Xloc_t, RDNode **xdot)
    276: {
    277:   PetscFunctionBeginUser;
    278:   PetscCall(DMDAVecRestoreArray(rd->da, *X0loc, x0));
    279:   PetscCall(DMDAVecRestoreArray(rd->da, *Xloc, x));
    280:   PetscCall(DMDAVecRestoreArray(rd->da, *Xloc_t, xdot));
    281:   PetscCall(DMRestoreLocalVector(rd->da, X0loc));
    282:   PetscCall(DMRestoreLocalVector(rd->da, Xloc));
    283:   PetscCall(DMRestoreLocalVector(rd->da, Xloc_t));
    284:   PetscFunctionReturn(PETSC_SUCCESS);
    285: }

    287: static PetscErrorCode PETSC_UNUSED RDCheckDomain_Private(RD rd, TS ts, Vec X, PetscBool *in)
    288: {
    289:   PetscInt  minloc;
    290:   PetscReal min;

    292:   PetscFunctionBeginUser;
    293:   PetscCall(VecMin(X, &minloc, &min));
    294:   if (min < 0) {
    295:     SNES snes;
    296:     *in = PETSC_FALSE;
    297:     PetscCall(TSGetSNES(ts, &snes));
    298:     PetscCall(SNESSetFunctionDomainError(snes));
    299:     PetscCall(PetscInfo(ts, "Domain violation at %" PetscInt_FMT " field %" PetscInt_FMT " value %g\n", minloc / 2, minloc % 2, (double)min));
    300:   } else *in = PETSC_TRUE;
    301:   PetscFunctionReturn(PETSC_SUCCESS);
    302: }

    304: /* Energy and temperature must remain positive */
    305: #define RDCheckDomain(rd, ts, X) \
    306:   do { \
    307:     PetscBool _in; \
    308:     PetscCall(RDCheckDomain_Private(rd, ts, X, &_in)); \
    309:     if (!_in) PetscFunctionReturn(PETSC_SUCCESS); \
    310:   } while (0)

    312: static PetscErrorCode RDIFunction_FD(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
    313: {
    314:   RD            rd = (RD)ctx;
    315:   RDNode       *x, *x0, *xdot, *f;
    316:   Vec           X0loc, Xloc, Xloc_t;
    317:   PetscReal     hx, Theta, dt;
    318:   DMDALocalInfo info;
    319:   PetscInt      i;

    321:   PetscFunctionBeginUser;
    322:   PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
    323:   PetscCall(DMDAVecGetArray(rd->da, F, &f));
    324:   PetscCall(DMDAGetLocalInfo(rd->da, &info));
    325:   PetscCall(VecZeroEntries(F));

    327:   hx = rd->L / (info.mx - 1);

    329:   for (i = info.xs; i < info.xs + info.xm; i++) {
    330:     PetscReal   rho = rd->rho;
    331:     PetscScalar Em_t, rad;

    333:     rad = (1. - Theta) * RDRadiation(rd, &x0[i], 0) + Theta * RDRadiation(rd, &x[i], 0);
    334:     if (rd->endpoint) {
    335:       PetscScalar Em0, Em1;
    336:       RDMaterialEnergy(rd, &x0[i], &Em0, NULL);
    337:       RDMaterialEnergy(rd, &x[i], &Em1, NULL);
    338:       Em_t = (Em1 - Em0) / dt;
    339:     } else {
    340:       RDNode dEm;
    341:       RDMaterialEnergy(rd, &x[i], NULL, &dEm);
    342:       Em_t = dEm.E * xdot[i].E + dEm.T * xdot[i].T;
    343:     }
    344:     /* Residuals are multiplied by the volume element (hx).  */
    345:     /* The temperature equation does not have boundary conditions */
    346:     f[i].T = hx * (rho * Em_t + rad);

    348:     if (i == 0) { /* Left boundary condition */
    349:       PetscScalar D_R, bcTheta = rd->bcmidpoint ? Theta : 1.;
    350:       RDNode      n, nx;

    352:       n.E  = (1. - bcTheta) * x0[0].E + bcTheta * x[0].E;
    353:       n.T  = (1. - bcTheta) * x0[0].T + bcTheta * x[0].T;
    354:       nx.E = ((1. - bcTheta) * (x0[1].E - x0[0].E) + bcTheta * (x[1].E - x[0].E)) / hx;
    355:       nx.T = ((1. - bcTheta) * (x0[1].T - x0[0].T) + bcTheta * (x[1].T - x[0].T)) / hx;
    356:       switch (rd->leftbc) {
    357:       case BC_ROBIN:
    358:         RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D_R, 0, 0);
    359:         f[0].E = hx * (n.E - 2. * D_R * nx.E - rd->Eapplied);
    360:         break;
    361:       case BC_NEUMANN:
    362:         f[0].E = x[1].E - x[0].E;
    363:         break;
    364:       default:
    365:         SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
    366:       }
    367:     } else if (i == info.mx - 1) {  /* Right boundary */
    368:       f[i].E = x[i].E - x[i - 1].E; /* Homogeneous Neumann */
    369:     } else {
    370:       PetscScalar diff = (1. - Theta) * RDDiffusion(rd, hx, x0, i, 0) + Theta * RDDiffusion(rd, hx, x, i, 0);
    371:       f[i].E           = hx * (xdot[i].E - diff - rad);
    372:     }
    373:   }
    374:   PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
    375:   PetscCall(DMDAVecRestoreArray(rd->da, F, &f));
    376:   if (rd->monitor_residual) PetscCall(RDStateView(rd, X, Xdot, F));
    377:   PetscFunctionReturn(PETSC_SUCCESS);
    378: }

    380: static PetscErrorCode RDIJacobian_FD(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx)
    381: {
    382:   RD            rd = (RD)ctx;
    383:   RDNode       *x, *x0, *xdot;
    384:   Vec           X0loc, Xloc, Xloc_t;
    385:   PetscReal     hx, Theta, dt;
    386:   DMDALocalInfo info;
    387:   PetscInt      i;

    389:   PetscFunctionBeginUser;
    390:   PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
    391:   PetscCall(DMDAGetLocalInfo(rd->da, &info));
    392:   hx = rd->L / (info.mx - 1);
    393:   PetscCall(MatZeroEntries(B));

    395:   for (i = info.xs; i < info.xs + info.xm; i++) {
    396:     PetscInt                  col[3];
    397:     PetscReal                 rho = rd->rho;
    398:     PetscScalar /*Em_t,rad,*/ K[2][6];
    399:     RDNode                    dEm_t, drad;

    401:     /*rad = (1.-Theta)* */ RDRadiation(rd, &x0[i], 0); /* + Theta* */
    402:     RDRadiation(rd, &x[i], &drad);

    404:     if (rd->endpoint) {
    405:       PetscScalar Em0, Em1;
    406:       RDNode      dEm1;
    407:       RDMaterialEnergy(rd, &x0[i], &Em0, NULL);
    408:       RDMaterialEnergy(rd, &x[i], &Em1, &dEm1);
    409:       /*Em_t = (Em1 - Em0) / (Theta*dt);*/
    410:       dEm_t.E = dEm1.E / (Theta * dt);
    411:       dEm_t.T = dEm1.T / (Theta * dt);
    412:     } else {
    413:       const PetscScalar epsilon = x[i].T * PETSC_SQRT_MACHINE_EPSILON;
    414:       RDNode            n1;
    415:       RDNode            dEm, dEm1;
    416:       PetscScalar       Em_TT;

    418:       n1.E = x[i].E;
    419:       n1.T = x[i].T + epsilon;
    420:       RDMaterialEnergy(rd, &x[i], NULL, &dEm);
    421:       RDMaterialEnergy(rd, &n1, NULL, &dEm1);
    422:       /* The Jacobian needs another derivative.  We finite difference here instead of
    423:        * propagating second derivatives through the ionization model. */
    424:       Em_TT = (dEm1.T - dEm.T) / epsilon;
    425:       /*Em_t = dEm.E * xdot[i].E + dEm.T * xdot[i].T;*/
    426:       dEm_t.E = dEm.E * a;
    427:       dEm_t.T = dEm.T * a + Em_TT * xdot[i].T;
    428:     }

    430:     PetscCall(PetscMemzero(K, sizeof(K)));
    431:     /* Residuals are multiplied by the volume element (hx).  */
    432:     if (i == 0) {
    433:       PetscScalar D, bcTheta = rd->bcmidpoint ? Theta : 1.;
    434:       RDNode      n, nx;
    435:       RDNode      dD, dxD;

    437:       n.E  = (1. - bcTheta) * x0[0].E + bcTheta * x[0].E;
    438:       n.T  = (1. - bcTheta) * x0[0].T + bcTheta * x[0].T;
    439:       nx.E = ((1. - bcTheta) * (x0[1].E - x0[0].E) + bcTheta * (x[1].E - x[0].E)) / hx;
    440:       nx.T = ((1. - bcTheta) * (x0[1].T - x0[0].T) + bcTheta * (x[1].T - x[0].T)) / hx;
    441:       switch (rd->leftbc) {
    442:       case BC_ROBIN:
    443:         RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, &dD, &dxD);
    444:         K[0][1 * 2 + 0] = (bcTheta / Theta) * hx * (1. - 2. * D * (-1. / hx) - 2. * nx.E * dD.E + 2. * nx.E * dxD.E / hx);
    445:         K[0][1 * 2 + 1] = (bcTheta / Theta) * hx * (-2. * nx.E * dD.T);
    446:         K[0][2 * 2 + 0] = (bcTheta / Theta) * hx * (-2. * D * (1. / hx) - 2. * nx.E * dD.E - 2. * nx.E * dxD.E / hx);
    447:         break;
    448:       case BC_NEUMANN:
    449:         K[0][1 * 2 + 0] = -1. / Theta;
    450:         K[0][2 * 2 + 0] = 1. / Theta;
    451:         break;
    452:       default:
    453:         SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
    454:       }
    455:     } else if (i == info.mx - 1) {
    456:       K[0][0 * 2 + 0] = -1. / Theta;
    457:       K[0][1 * 2 + 0] = 1. / Theta;
    458:     } else {
    459:       /*PetscScalar diff;*/
    460:       RDNode ddiff[3];
    461:       /*diff = (1.-Theta)*RDDiffusion(rd,hx,x0,i,0) + Theta* */ RDDiffusion(rd, hx, x, i, ddiff);
    462:       K[0][0 * 2 + 0] = -hx * ddiff[0].E;
    463:       K[0][0 * 2 + 1] = -hx * ddiff[0].T;
    464:       K[0][1 * 2 + 0] = hx * (a - ddiff[1].E - drad.E);
    465:       K[0][1 * 2 + 1] = hx * (-ddiff[1].T - drad.T);
    466:       K[0][2 * 2 + 0] = -hx * ddiff[2].E;
    467:       K[0][2 * 2 + 1] = -hx * ddiff[2].T;
    468:     }

    470:     K[1][1 * 2 + 0] = hx * (rho * dEm_t.E + drad.E);
    471:     K[1][1 * 2 + 1] = hx * (rho * dEm_t.T + drad.T);

    473:     col[0] = i - 1;
    474:     col[1] = i;
    475:     col[2] = i + 1 < info.mx ? i + 1 : -1;
    476:     PetscCall(MatSetValuesBlocked(B, 1, &i, 3, col, &K[0][0], INSERT_VALUES));
    477:   }
    478:   PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
    479:   PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    480:   PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
    481:   if (A != B) {
    482:     PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    483:     PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
    484:   }
    485:   PetscFunctionReturn(PETSC_SUCCESS);
    486: }

    488: /* Evaluate interpolants and derivatives at a select quadrature point */
    489: static void RDEvaluate(PetscReal interp[][2], PetscReal deriv[][2], PetscInt q, const RDNode x[], PetscInt i, RDNode *n, RDNode *nx)
    490: {
    491:   PetscInt j;
    492:   n->E  = 0;
    493:   n->T  = 0;
    494:   nx->E = 0;
    495:   nx->T = 0;
    496:   for (j = 0; j < 2; j++) {
    497:     n->E += interp[q][j] * x[i + j].E;
    498:     n->T += interp[q][j] * x[i + j].T;
    499:     nx->E += deriv[q][j] * x[i + j].E;
    500:     nx->T += deriv[q][j] * x[i + j].T;
    501:   }
    502: }

    504: /*
    505:  Various quadrature rules.  The nonlinear terms are non-polynomial so no standard quadrature will be exact.
    506: */
    507: static PetscErrorCode RDGetQuadrature(RD rd, PetscReal hx, PetscInt *nq, PetscReal weight[], PetscReal interp[][2], PetscReal deriv[][2])
    508: {
    509:   PetscInt         q, j;
    510:   const PetscReal *refweight, (*refinterp)[2], (*refderiv)[2];

    512:   PetscFunctionBeginUser;
    513:   switch (rd->quadrature) {
    514:   case QUADRATURE_GAUSS1: {
    515:     static const PetscReal ww[1]    = {1.};
    516:     static const PetscReal ii[1][2] = {
    517:       {0.5, 0.5}
    518:     };
    519:     static const PetscReal dd[1][2] = {
    520:       {-1., 1.}
    521:     };
    522:     *nq       = 1;
    523:     refweight = ww;
    524:     refinterp = ii;
    525:     refderiv  = dd;
    526:   } break;
    527:   case QUADRATURE_GAUSS2: {
    528:     static const PetscReal ii[2][2] = {
    529:       {0.78867513459481287, 0.21132486540518713},
    530:       {0.21132486540518713, 0.78867513459481287}
    531:     };
    532:     static const PetscReal dd[2][2] = {
    533:       {-1., 1.},
    534:       {-1., 1.}
    535:     };
    536:     static const PetscReal ww[2] = {0.5, 0.5};
    537:     *nq                          = 2;
    538:     refweight                    = ww;
    539:     refinterp                    = ii;
    540:     refderiv                     = dd;
    541:   } break;
    542:   case QUADRATURE_GAUSS3: {
    543:     static const PetscReal ii[3][2] = {
    544:       {0.8872983346207417, 0.1127016653792583},
    545:       {0.5,                0.5               },
    546:       {0.1127016653792583, 0.8872983346207417}
    547:     };
    548:     static const PetscReal dd[3][2] = {
    549:       {-1, 1},
    550:       {-1, 1},
    551:       {-1, 1}
    552:     };
    553:     static const PetscReal ww[3] = {5. / 18, 8. / 18, 5. / 18};
    554:     *nq                          = 3;
    555:     refweight                    = ww;
    556:     refinterp                    = ii;
    557:     refderiv                     = dd;
    558:   } break;
    559:   case QUADRATURE_GAUSS4: {
    560:     static const PetscReal ii[][2] = {
    561:       {0.93056815579702623,  0.069431844202973658},
    562:       {0.66999052179242813,  0.33000947820757187 },
    563:       {0.33000947820757187,  0.66999052179242813 },
    564:       {0.069431844202973658, 0.93056815579702623 }
    565:     };
    566:     static const PetscReal dd[][2] = {
    567:       {-1, 1},
    568:       {-1, 1},
    569:       {-1, 1},
    570:       {-1, 1}
    571:     };
    572:     static const PetscReal ww[] = {0.17392742256872692, 0.3260725774312731, 0.3260725774312731, 0.17392742256872692};

    574:     *nq       = 4;
    575:     refweight = ww;
    576:     refinterp = ii;
    577:     refderiv  = dd;
    578:   } break;
    579:   case QUADRATURE_LOBATTO2: {
    580:     static const PetscReal ii[2][2] = {
    581:       {1., 0.},
    582:       {0., 1.}
    583:     };
    584:     static const PetscReal dd[2][2] = {
    585:       {-1., 1.},
    586:       {-1., 1.}
    587:     };
    588:     static const PetscReal ww[2] = {0.5, 0.5};
    589:     *nq                          = 2;
    590:     refweight                    = ww;
    591:     refinterp                    = ii;
    592:     refderiv                     = dd;
    593:   } break;
    594:   case QUADRATURE_LOBATTO3: {
    595:     static const PetscReal ii[3][2] = {
    596:       {1,   0  },
    597:       {0.5, 0.5},
    598:       {0,   1  }
    599:     };
    600:     static const PetscReal dd[3][2] = {
    601:       {-1, 1},
    602:       {-1, 1},
    603:       {-1, 1}
    604:     };
    605:     static const PetscReal ww[3] = {1. / 6, 4. / 6, 1. / 6};
    606:     *nq                          = 3;
    607:     refweight                    = ww;
    608:     refinterp                    = ii;
    609:     refderiv                     = dd;
    610:   } break;
    611:   default:
    612:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Unknown quadrature %d", (int)rd->quadrature);
    613:   }

    615:   for (q = 0; q < *nq; q++) {
    616:     weight[q] = refweight[q] * hx;
    617:     for (j = 0; j < 2; j++) {
    618:       interp[q][j] = refinterp[q][j];
    619:       deriv[q][j]  = refderiv[q][j] / hx;
    620:     }
    621:   }
    622:   PetscFunctionReturn(PETSC_SUCCESS);
    623: }

    625: /*
    626:  Finite element version
    627: */
    628: static PetscErrorCode RDIFunction_FE(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
    629: {
    630:   RD            rd = (RD)ctx;
    631:   RDNode       *x, *x0, *xdot, *f;
    632:   Vec           X0loc, Xloc, Xloc_t, Floc;
    633:   PetscReal     hx, Theta, dt, weight[5], interp[5][2], deriv[5][2];
    634:   DMDALocalInfo info;
    635:   PetscInt      i, j, q, nq;

    637:   PetscFunctionBeginUser;
    638:   PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));

    640:   PetscCall(DMGetLocalVector(rd->da, &Floc));
    641:   PetscCall(VecZeroEntries(Floc));
    642:   PetscCall(DMDAVecGetArray(rd->da, Floc, &f));
    643:   PetscCall(DMDAGetLocalInfo(rd->da, &info));

    645:   /* Set up shape functions and quadrature for elements (assumes a uniform grid) */
    646:   hx = rd->L / (info.mx - 1);
    647:   PetscCall(RDGetQuadrature(rd, hx, &nq, weight, interp, deriv));

    649:   for (i = info.xs; i < PetscMin(info.xs + info.xm, info.mx - 1); i++) {
    650:     for (q = 0; q < nq; q++) {
    651:       PetscReal   rho = rd->rho;
    652:       PetscScalar Em_t, rad, D_R, D0_R;
    653:       RDNode      n, n0, nx, n0x, nt, ntx;
    654:       RDEvaluate(interp, deriv, q, x, i, &n, &nx);
    655:       RDEvaluate(interp, deriv, q, x0, i, &n0, &n0x);
    656:       RDEvaluate(interp, deriv, q, xdot, i, &nt, &ntx);

    658:       rad = (1. - Theta) * RDRadiation(rd, &n0, 0) + Theta * RDRadiation(rd, &n, 0);
    659:       if (rd->endpoint) {
    660:         PetscScalar Em0, Em1;
    661:         RDMaterialEnergy(rd, &n0, &Em0, NULL);
    662:         RDMaterialEnergy(rd, &n, &Em1, NULL);
    663:         Em_t = (Em1 - Em0) / dt;
    664:       } else {
    665:         RDNode dEm;
    666:         RDMaterialEnergy(rd, &n, NULL, &dEm);
    667:         Em_t = dEm.E * nt.E + dEm.T * nt.T;
    668:       }
    669:       RDDiffusionCoefficient(rd, PETSC_TRUE, &n0, &n0x, &D0_R, 0, 0);
    670:       RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, 0, 0);
    671:       for (j = 0; j < 2; j++) {
    672:         f[i + j].E += (deriv[q][j] * weight[q] * ((1. - Theta) * D0_R * n0x.E + Theta * D_R * nx.E) + interp[q][j] * weight[q] * (nt.E - rad));
    673:         f[i + j].T += interp[q][j] * weight[q] * (rho * Em_t + rad);
    674:       }
    675:     }
    676:   }
    677:   if (info.xs == 0) {
    678:     switch (rd->leftbc) {
    679:     case BC_ROBIN: {
    680:       PetscScalar D_R, D_R_bc;
    681:       PetscReal   ratio, bcTheta = rd->bcmidpoint ? Theta : 1.;
    682:       RDNode      n, nx;

    684:       n.E  = (1 - bcTheta) * x0[0].E + bcTheta * x[0].E;
    685:       n.T  = (1 - bcTheta) * x0[0].T + bcTheta * x[0].T;
    686:       nx.E = (x[1].E - x[0].E) / hx;
    687:       nx.T = (x[1].T - x[0].T) / hx;
    688:       RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, 0, 0);
    689:       RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D_R_bc, 0, 0);
    690:       ratio = PetscRealPart(D_R / D_R_bc);
    691:       PetscCheck(ratio <= 1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Limited diffusivity is greater than unlimited");
    692:       PetscCheck(ratio >= 1e-3, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Heavily limited diffusivity");
    693:       f[0].E += -ratio * 0.5 * (rd->Eapplied - n.E);
    694:     } break;
    695:     case BC_NEUMANN:
    696:       /* homogeneous Neumann is the natural condition */
    697:       break;
    698:     default:
    699:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
    700:     }
    701:   }

    703:   PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
    704:   PetscCall(DMDAVecRestoreArray(rd->da, Floc, &f));
    705:   PetscCall(VecZeroEntries(F));
    706:   PetscCall(DMLocalToGlobalBegin(rd->da, Floc, ADD_VALUES, F));
    707:   PetscCall(DMLocalToGlobalEnd(rd->da, Floc, ADD_VALUES, F));
    708:   PetscCall(DMRestoreLocalVector(rd->da, &Floc));

    710:   if (rd->monitor_residual) PetscCall(RDStateView(rd, X, Xdot, F));
    711:   PetscFunctionReturn(PETSC_SUCCESS);
    712: }

    714: static PetscErrorCode RDIJacobian_FE(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx)
    715: {
    716:   RD            rd = (RD)ctx;
    717:   RDNode       *x, *x0, *xdot;
    718:   Vec           X0loc, Xloc, Xloc_t;
    719:   PetscReal     hx, Theta, dt, weight[5], interp[5][2], deriv[5][2];
    720:   DMDALocalInfo info;
    721:   PetscInt      i, j, k, q, nq;
    722:   PetscScalar   K[4][4];

    724:   PetscFunctionBeginUser;
    725:   PetscCall(RDGetLocalArrays(rd, ts, X, Xdot, &Theta, &dt, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
    726:   PetscCall(DMDAGetLocalInfo(rd->da, &info));
    727:   hx = rd->L / (info.mx - 1);
    728:   PetscCall(RDGetQuadrature(rd, hx, &nq, weight, interp, deriv));
    729:   PetscCall(MatZeroEntries(B));
    730:   for (i = info.xs; i < PetscMin(info.xs + info.xm, info.mx - 1); i++) {
    731:     PetscInt rc[2];

    733:     rc[0] = i;
    734:     rc[1] = i + 1;
    735:     PetscCall(PetscMemzero(K, sizeof(K)));
    736:     for (q = 0; q < nq; q++) {
    737:       PetscScalar              D_R;
    738:       PETSC_UNUSED PetscScalar rad;
    739:       RDNode                   n, nx, nt, ntx, drad, dD_R, dxD_R, dEm;
    740:       RDEvaluate(interp, deriv, q, x, i, &n, &nx);
    741:       RDEvaluate(interp, deriv, q, xdot, i, &nt, &ntx);
    742:       rad = RDRadiation(rd, &n, &drad);
    743:       RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, &dD_R, &dxD_R);
    744:       RDMaterialEnergy(rd, &n, NULL, &dEm);
    745:       for (j = 0; j < 2; j++) {
    746:         for (k = 0; k < 2; k++) {
    747:           K[j * 2 + 0][k * 2 + 0] += (+interp[q][j] * weight[q] * (a - drad.E) * interp[q][k] + deriv[q][j] * weight[q] * ((D_R + dxD_R.E * nx.E) * deriv[q][k] + dD_R.E * nx.E * interp[q][k]));
    748:           K[j * 2 + 0][k * 2 + 1] += (+interp[q][j] * weight[q] * (-drad.T * interp[q][k]) + deriv[q][j] * weight[q] * (dxD_R.T * deriv[q][k] + dD_R.T * interp[q][k]) * nx.E);
    749:           K[j * 2 + 1][k * 2 + 0] += interp[q][j] * weight[q] * drad.E * interp[q][k];
    750:           K[j * 2 + 1][k * 2 + 1] += interp[q][j] * weight[q] * (a * rd->rho * dEm.T + drad.T) * interp[q][k];
    751:         }
    752:       }
    753:     }
    754:     PetscCall(MatSetValuesBlocked(B, 2, rc, 2, rc, &K[0][0], ADD_VALUES));
    755:   }
    756:   if (info.xs == 0) {
    757:     switch (rd->leftbc) {
    758:     case BC_ROBIN: {
    759:       PetscScalar D_R, D_R_bc;
    760:       PetscReal   ratio;
    761:       RDNode      n, nx;

    763:       n.E  = (1 - Theta) * x0[0].E + Theta * x[0].E;
    764:       n.T  = (1 - Theta) * x0[0].T + Theta * x[0].T;
    765:       nx.E = (x[1].E - x[0].E) / hx;
    766:       nx.T = (x[1].T - x[0].T) / hx;
    767:       RDDiffusionCoefficient(rd, PETSC_TRUE, &n, &nx, &D_R, 0, 0);
    768:       RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D_R_bc, 0, 0);
    769:       ratio = PetscRealPart(D_R / D_R_bc);
    770:       PetscCall(MatSetValue(B, 0, 0, ratio * 0.5, ADD_VALUES));
    771:     } break;
    772:     case BC_NEUMANN:
    773:       /* homogeneous Neumann is the natural condition */
    774:       break;
    775:     default:
    776:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Case %" PetscInt_FMT, rd->initial);
    777:     }
    778:   }

    780:   PetscCall(RDRestoreLocalArrays(rd, &X0loc, &x0, &Xloc, &x, &Xloc_t, &xdot));
    781:   PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    782:   PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
    783:   if (A != B) {
    784:     PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    785:     PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
    786:   }
    787:   PetscFunctionReturn(PETSC_SUCCESS);
    788: }

    790: /* Temperature that is in equilibrium with the radiation density */
    791: static PetscScalar RDRadiationTemperature(RD rd, PetscScalar E)
    792: {
    793:   return PetscPowScalar(E * rd->c / (4. * rd->sigma_b), 0.25);
    794: }

    796: static PetscErrorCode RDInitialState(RD rd, Vec X)
    797: {
    798:   DMDALocalInfo info;
    799:   PetscInt      i;
    800:   RDNode       *x;

    802:   PetscFunctionBeginUser;
    803:   PetscCall(DMDAGetLocalInfo(rd->da, &info));
    804:   PetscCall(DMDAVecGetArray(rd->da, X, &x));
    805:   for (i = info.xs; i < info.xs + info.xm; i++) {
    806:     PetscReal coord = i * rd->L / (info.mx - 1);
    807:     switch (rd->initial) {
    808:     case 1:
    809:       x[i].E = 0.001;
    810:       x[i].T = RDRadiationTemperature(rd, x[i].E);
    811:       break;
    812:     case 2:
    813:       x[i].E = 0.001 + 100. * PetscExpReal(-PetscSqr(coord / 0.1));
    814:       x[i].T = RDRadiationTemperature(rd, x[i].E);
    815:       break;
    816:     case 3:
    817:       x[i].E = 7.56e-2 * rd->unit.Joule / PetscPowScalarInt(rd->unit.meter, 3);
    818:       x[i].T = RDRadiationTemperature(rd, x[i].E);
    819:       break;
    820:     default:
    821:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "No initial state %" PetscInt_FMT, rd->initial);
    822:     }
    823:   }
    824:   PetscCall(DMDAVecRestoreArray(rd->da, X, &x));
    825:   PetscFunctionReturn(PETSC_SUCCESS);
    826: }

    828: static PetscErrorCode RDView(RD rd, Vec X, PetscViewer viewer)
    829: {
    830:   Vec             Y;
    831:   const RDNode   *x;
    832:   PetscScalar    *y;
    833:   PetscInt        i, m, M;
    834:   const PetscInt *lx;
    835:   DM              da;
    836:   MPI_Comm        comm;

    838:   PetscFunctionBeginUser;
    839:   /*
    840:     Create a DMDA (one dof per node, zero stencil width, same layout) to hold Trad
    841:     (radiation temperature).  It is not necessary to create a DMDA for this, but this way
    842:     output and visualization will have meaningful variable names and correct scales.
    843:   */
    844:   PetscCall(DMDAGetInfo(rd->da, 0, &M, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
    845:   PetscCall(DMDAGetOwnershipRanges(rd->da, &lx, 0, 0));
    846:   PetscCall(PetscObjectGetComm((PetscObject)rd->da, &comm));
    847:   PetscCall(DMDACreate1d(comm, DM_BOUNDARY_NONE, M, 1, 0, lx, &da));
    848:   PetscCall(DMSetFromOptions(da));
    849:   PetscCall(DMSetUp(da));
    850:   PetscCall(DMDASetUniformCoordinates(da, 0., rd->L, 0., 0., 0., 0.));
    851:   PetscCall(DMDASetFieldName(da, 0, "T_rad"));
    852:   PetscCall(DMCreateGlobalVector(da, &Y));

    854:   /* Compute the radiation temperature from the solution at each node */
    855:   PetscCall(VecGetLocalSize(Y, &m));
    856:   PetscCall(VecGetArrayRead(X, (const PetscScalar **)&x));
    857:   PetscCall(VecGetArray(Y, &y));
    858:   for (i = 0; i < m; i++) y[i] = RDRadiationTemperature(rd, x[i].E);
    859:   PetscCall(VecRestoreArrayRead(X, (const PetscScalar **)&x));
    860:   PetscCall(VecRestoreArray(Y, &y));

    862:   PetscCall(VecView(Y, viewer));
    863:   PetscCall(VecDestroy(&Y));
    864:   PetscCall(DMDestroy(&da));
    865:   PetscFunctionReturn(PETSC_SUCCESS);
    866: }

    868: static PetscErrorCode RDTestDifferentiation(RD rd)
    869: {
    870:   MPI_Comm    comm;
    871:   RDNode      n, nx;
    872:   PetscScalar epsilon;

    874:   PetscFunctionBeginUser;
    875:   PetscCall(PetscObjectGetComm((PetscObject)rd->da, &comm));
    876:   epsilon = 1e-8;
    877:   {
    878:     RDNode      dEm, fdEm;
    879:     PetscScalar T0 = 1000., T1 = T0 * (1. + epsilon), Em0, Em1;
    880:     n.E = 1.;
    881:     n.T = T0;
    882:     rd->MaterialEnergy(rd, &n, &Em0, &dEm);
    883:     n.E = 1. + epsilon;
    884:     n.T = T0;
    885:     rd->MaterialEnergy(rd, &n, &Em1, 0);
    886:     fdEm.E = (Em1 - Em0) / epsilon;
    887:     n.E    = 1.;
    888:     n.T    = T1;
    889:     rd->MaterialEnergy(rd, &n, &Em1, 0);
    890:     fdEm.T = (Em1 - Em0) / (T0 * epsilon);
    891:     PetscCall(PetscPrintf(comm, "dEm {%g,%g}, fdEm {%g,%g}, diff {%g,%g}\n", (double)PetscRealPart(dEm.E), (double)PetscRealPart(dEm.T), (double)PetscRealPart(fdEm.E), (double)PetscRealPart(fdEm.T), (double)PetscRealPart(dEm.E - fdEm.E),
    892:                           (double)PetscRealPart(dEm.T - fdEm.T)));
    893:   }
    894:   {
    895:     PetscScalar D0, D;
    896:     RDNode      dD, dxD, fdD, fdxD;
    897:     n.E  = 1.;
    898:     n.T  = 1.;
    899:     nx.E = 1.;
    900:     n.T  = 1.;
    901:     RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D0, &dD, &dxD);
    902:     n.E  = 1. + epsilon;
    903:     n.T  = 1.;
    904:     nx.E = 1.;
    905:     n.T  = 1.;
    906:     RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
    907:     fdD.E = (D - D0) / epsilon;
    908:     n.E   = 1;
    909:     n.T   = 1. + epsilon;
    910:     nx.E  = 1.;
    911:     n.T   = 1.;
    912:     RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
    913:     fdD.T = (D - D0) / epsilon;
    914:     n.E   = 1;
    915:     n.T   = 1.;
    916:     nx.E  = 1. + epsilon;
    917:     n.T   = 1.;
    918:     RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
    919:     fdxD.E = (D - D0) / epsilon;
    920:     n.E    = 1;
    921:     n.T    = 1.;
    922:     nx.E   = 1.;
    923:     n.T    = 1. + epsilon;
    924:     RDDiffusionCoefficient(rd, rd->bclimit, &n, &nx, &D, 0, 0);
    925:     fdxD.T = (D - D0) / epsilon;
    926:     PetscCall(PetscPrintf(comm, "dD {%g,%g}, fdD {%g,%g}, diff {%g,%g}\n", (double)PetscRealPart(dD.E), (double)PetscRealPart(dD.T), (double)PetscRealPart(fdD.E), (double)PetscRealPart(fdD.T), (double)PetscRealPart(dD.E - fdD.E),
    927:                           (double)PetscRealPart(dD.T - fdD.T)));
    928:     PetscCall(PetscPrintf(comm, "dxD {%g,%g}, fdxD {%g,%g}, diffx {%g,%g}\n", (double)PetscRealPart(dxD.E), (double)PetscRealPart(dxD.T), (double)PetscRealPart(fdxD.E), (double)PetscRealPart(fdxD.T), (double)PetscRealPart(dxD.E - fdxD.E),
    929:                           (double)PetscRealPart(dxD.T - fdxD.T)));
    930:   }
    931:   {
    932:     PetscInt    i;
    933:     PetscReal   hx = 1.;
    934:     PetscScalar a0;
    935:     RDNode      n0[3], n1[3], d[3], fd[3];

    937:     n0[0].E = 1.;
    938:     n0[0].T = 1.;
    939:     n0[1].E = 5.;
    940:     n0[1].T = 3.;
    941:     n0[2].E = 4.;
    942:     n0[2].T = 2.;
    943:     a0      = RDDiffusion(rd, hx, n0, 1, d);
    944:     for (i = 0; i < 3; i++) {
    945:       PetscCall(PetscMemcpy(n1, n0, sizeof(n0)));
    946:       n1[i].E += epsilon;
    947:       fd[i].E = (RDDiffusion(rd, hx, n1, 1, 0) - a0) / epsilon;
    948:       PetscCall(PetscMemcpy(n1, n0, sizeof(n0)));
    949:       n1[i].T += epsilon;
    950:       fd[i].T = (RDDiffusion(rd, hx, n1, 1, 0) - a0) / epsilon;
    951:       PetscCall(PetscPrintf(comm, "ddiff[%" PetscInt_FMT "] {%g,%g}, fd {%g %g}, diff {%g,%g}\n", i, (double)PetscRealPart(d[i].E), (double)PetscRealPart(d[i].T), (double)PetscRealPart(fd[i].E), (double)PetscRealPart(fd[i].T),
    952:                             (double)PetscRealPart(d[i].E - fd[i].E), (double)PetscRealPart(d[i].T - fd[i].T)));
    953:     }
    954:   }
    955:   {
    956:     PetscScalar rad0, rad;
    957:     RDNode      drad, fdrad;
    958:     n.E     = 1.;
    959:     n.T     = 1.;
    960:     rad0    = RDRadiation(rd, &n, &drad);
    961:     n.E     = 1. + epsilon;
    962:     n.T     = 1.;
    963:     rad     = RDRadiation(rd, &n, 0);
    964:     fdrad.E = (rad - rad0) / epsilon;
    965:     n.E     = 1.;
    966:     n.T     = 1. + epsilon;
    967:     rad     = RDRadiation(rd, &n, 0);
    968:     fdrad.T = (rad - rad0) / epsilon;
    969:     PetscCall(PetscPrintf(comm, "drad {%g,%g}, fdrad {%g,%g}, diff {%g,%g}\n", (double)PetscRealPart(drad.E), (double)PetscRealPart(drad.T), (double)PetscRealPart(fdrad.E), (double)PetscRealPart(fdrad.T), (double)PetscRealPart(drad.E - drad.E),
    970:                           (double)PetscRealPart(drad.T - fdrad.T)));
    971:   }
    972:   PetscFunctionReturn(PETSC_SUCCESS);
    973: }

    975: static PetscErrorCode RDCreate(MPI_Comm comm, RD *inrd)
    976: {
    977:   RD        rd;
    978:   PetscReal meter = 0, kilogram = 0, second = 0, Kelvin = 0, Joule = 0, Watt = 0;

    980:   PetscFunctionBeginUser;
    981:   *inrd = 0;
    982:   PetscCall(PetscNew(&rd));

    984:   PetscOptionsBegin(comm, NULL, "Options for nonequilibrium radiation-diffusion with RD ionization", NULL);
    985:   {
    986:     rd->initial = 1;
    987:     PetscCall(PetscOptionsInt("-rd_initial", "Initial condition (1=Marshak, 2=Blast, 3=Marshak+)", "", rd->initial, &rd->initial, 0));
    988:     switch (rd->initial) {
    989:     case 1:
    990:     case 2:
    991:       rd->unit.kilogram = 1.;
    992:       rd->unit.meter    = 1.;
    993:       rd->unit.second   = 1.;
    994:       rd->unit.Kelvin   = 1.;
    995:       break;
    996:     case 3:
    997:       rd->unit.kilogram = 1.e12;
    998:       rd->unit.meter    = 1.;
    999:       rd->unit.second   = 1.e9;
    1000:       rd->unit.Kelvin   = 1.;
    1001:       break;
    1002:     default:
    1003:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Unknown initial condition %" PetscInt_FMT, rd->initial);
    1004:     }
    1005:     /* Fundamental units */
    1006:     PetscCall(PetscOptionsReal("-rd_unit_meter", "Length of 1 meter in nondimensional units", "", rd->unit.meter, &rd->unit.meter, 0));
    1007:     PetscCall(PetscOptionsReal("-rd_unit_kilogram", "Mass of 1 kilogram in nondimensional units", "", rd->unit.kilogram, &rd->unit.kilogram, 0));
    1008:     PetscCall(PetscOptionsReal("-rd_unit_second", "Time of a second in nondimensional units", "", rd->unit.second, &rd->unit.second, 0));
    1009:     PetscCall(PetscOptionsReal("-rd_unit_Kelvin", "Temperature of a Kelvin in nondimensional units", "", rd->unit.Kelvin, &rd->unit.Kelvin, 0));
    1010:     /* Derived units */
    1011:     rd->unit.Joule = rd->unit.kilogram * PetscSqr(rd->unit.meter / rd->unit.second);
    1012:     rd->unit.Watt  = rd->unit.Joule / rd->unit.second;
    1013:     /* Local aliases */
    1014:     meter    = rd->unit.meter;
    1015:     kilogram = rd->unit.kilogram;
    1016:     second   = rd->unit.second;
    1017:     Kelvin   = rd->unit.Kelvin;
    1018:     Joule    = rd->unit.Joule;
    1019:     Watt     = rd->unit.Watt;

    1021:     PetscCall(PetscOptionsBool("-rd_monitor_residual", "Display residuals every time they are evaluated", "", rd->monitor_residual, &rd->monitor_residual, NULL));
    1022:     PetscCall(PetscOptionsEnum("-rd_discretization", "Discretization type", "", DiscretizationTypes, (PetscEnum)rd->discretization, (PetscEnum *)&rd->discretization, NULL));
    1023:     if (rd->discretization == DISCRETIZATION_FE) {
    1024:       rd->quadrature = QUADRATURE_GAUSS2;
    1025:       PetscCall(PetscOptionsEnum("-rd_quadrature", "Finite element quadrature", "", QuadratureTypes, (PetscEnum)rd->quadrature, (PetscEnum *)&rd->quadrature, NULL));
    1026:     }
    1027:     PetscCall(PetscOptionsEnum("-rd_jacobian", "Type of finite difference Jacobian", "", JacobianTypes, (PetscEnum)rd->jacobian, (PetscEnum *)&rd->jacobian, NULL));
    1028:     switch (rd->initial) {
    1029:     case 1:
    1030:       rd->leftbc     = BC_ROBIN;
    1031:       rd->Eapplied   = 4 * rd->unit.Joule / PetscPowRealInt(rd->unit.meter, 3);
    1032:       rd->L          = 1. * rd->unit.meter;
    1033:       rd->beta       = 3.0;
    1034:       rd->gamma      = 3.0;
    1035:       rd->final_time = 3 * second;
    1036:       break;
    1037:     case 2:
    1038:       rd->leftbc     = BC_NEUMANN;
    1039:       rd->Eapplied   = 0.;
    1040:       rd->L          = 1. * rd->unit.meter;
    1041:       rd->beta       = 3.0;
    1042:       rd->gamma      = 3.0;
    1043:       rd->final_time = 1 * second;
    1044:       break;
    1045:     case 3:
    1046:       rd->leftbc     = BC_ROBIN;
    1047:       rd->Eapplied   = 7.503e6 * rd->unit.Joule / PetscPowRealInt(rd->unit.meter, 3);
    1048:       rd->L          = 5. * rd->unit.meter;
    1049:       rd->beta       = 3.5;
    1050:       rd->gamma      = 3.5;
    1051:       rd->final_time = 20e-9 * second;
    1052:       break;
    1053:     default:
    1054:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Initial %" PetscInt_FMT, rd->initial);
    1055:     }
    1056:     PetscCall(PetscOptionsEnum("-rd_leftbc", "Left boundary condition", "", BCTypes, (PetscEnum)rd->leftbc, (PetscEnum *)&rd->leftbc, NULL));
    1057:     PetscCall(PetscOptionsReal("-rd_E_applied", "Radiation flux at left end of domain", "", rd->Eapplied, &rd->Eapplied, NULL));
    1058:     PetscCall(PetscOptionsReal("-rd_beta", "Thermal exponent for photon absorption", "", rd->beta, &rd->beta, NULL));
    1059:     PetscCall(PetscOptionsReal("-rd_gamma", "Thermal exponent for diffusion coefficient", "", rd->gamma, &rd->gamma, NULL));
    1060:     PetscCall(PetscOptionsBool("-rd_view_draw", "Draw final solution", "", rd->view_draw, &rd->view_draw, NULL));
    1061:     PetscCall(PetscOptionsBool("-rd_endpoint", "Discretize using endpoints (like trapezoid rule) instead of midpoint", "", rd->endpoint, &rd->endpoint, NULL));
    1062:     PetscCall(PetscOptionsBool("-rd_bcmidpoint", "Impose the boundary condition at the midpoint (Theta) of the interval", "", rd->bcmidpoint, &rd->bcmidpoint, NULL));
    1063:     PetscCall(PetscOptionsBool("-rd_bclimit", "Limit diffusion coefficient in definition of Robin boundary condition", "", rd->bclimit, &rd->bclimit, NULL));
    1064:     PetscCall(PetscOptionsBool("-rd_test_diff", "Test differentiation in constitutive relations", "", rd->test_diff, &rd->test_diff, NULL));
    1065:     PetscCall(PetscOptionsString("-rd_view_binary", "File name to hold final solution", "", rd->view_binary, rd->view_binary, sizeof(rd->view_binary), NULL));
    1066:   }
    1067:   PetscOptionsEnd();

    1069:   switch (rd->initial) {
    1070:   case 1:
    1071:   case 2:
    1072:     rd->rho            = 1.;
    1073:     rd->c              = 1.;
    1074:     rd->K_R            = 1.;
    1075:     rd->K_p            = 1.;
    1076:     rd->sigma_b        = 0.25;
    1077:     rd->MaterialEnergy = RDMaterialEnergy_Reduced;
    1078:     break;
    1079:   case 3:
    1080:     /* Table 2 */
    1081:     rd->rho            = 1.17e-3 * kilogram / (meter * meter * meter); /* density */
    1082:     rd->K_R            = 7.44e18 * PetscPowRealInt(meter, 5) * PetscPowReal(Kelvin, 3.5) * PetscPowRealInt(kilogram, -2);
    1083:     rd->K_p            = 2.33e20 * PetscPowRealInt(meter, 5) * PetscPowReal(Kelvin, 3.5) * PetscPowRealInt(kilogram, -2);
    1084:     rd->I_H            = 2.179e-18 * Joule;                                                         /* Hydrogen ionization potential */
    1085:     rd->m_p            = 1.673e-27 * kilogram;                                                      /* proton mass */
    1086:     rd->m_e            = 9.109e-31 * kilogram;                                                      /* electron mass */
    1087:     rd->h              = 6.626e-34 * Joule * second;                                                /* Planck's constant */
    1088:     rd->k              = 1.381e-23 * Joule / Kelvin;                                                /* Boltzman constant */
    1089:     rd->c              = 3.00e8 * meter / second;                                                   /* speed of light */
    1090:     rd->sigma_b        = 5.67e-8 * Watt * PetscPowRealInt(meter, -2) * PetscPowRealInt(Kelvin, -4); /* Stefan-Boltzman constant */
    1091:     rd->MaterialEnergy = RDMaterialEnergy_Saha;
    1092:     break;
    1093:   }

    1095:   PetscCall(DMDACreate1d(comm, DM_BOUNDARY_NONE, 20, sizeof(RDNode) / sizeof(PetscScalar), 1, NULL, &rd->da));
    1096:   PetscCall(DMSetFromOptions(rd->da));
    1097:   PetscCall(DMSetUp(rd->da));
    1098:   PetscCall(DMDASetFieldName(rd->da, 0, "E"));
    1099:   PetscCall(DMDASetFieldName(rd->da, 1, "T"));
    1100:   PetscCall(DMDASetUniformCoordinates(rd->da, 0., 1., 0., 0., 0., 0.));

    1102:   *inrd = rd;
    1103:   PetscFunctionReturn(PETSC_SUCCESS);
    1104: }

    1106: int main(int argc, char *argv[])
    1107: {
    1108:   RD        rd;
    1109:   TS        ts;
    1110:   SNES      snes;
    1111:   Vec       X;
    1112:   Mat       A, B;
    1113:   PetscInt  steps;
    1114:   PetscReal ftime;

    1116:   PetscFunctionBeginUser;
    1117:   PetscCall(PetscInitialize(&argc, &argv, 0, NULL));
    1118:   PetscCall(RDCreate(PETSC_COMM_WORLD, &rd));
    1119:   PetscCall(DMCreateGlobalVector(rd->da, &X));
    1120:   PetscCall(DMSetMatType(rd->da, MATAIJ));
    1121:   PetscCall(DMCreateMatrix(rd->da, &B));
    1122:   PetscCall(RDInitialState(rd, X));

    1124:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
    1125:   PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
    1126:   PetscCall(TSSetType(ts, TSTHETA));
    1127:   PetscCall(TSSetDM(ts, rd->da));
    1128:   switch (rd->discretization) {
    1129:   case DISCRETIZATION_FD:
    1130:     PetscCall(TSSetIFunction(ts, NULL, RDIFunction_FD, rd));
    1131:     if (rd->jacobian == JACOBIAN_ANALYTIC) PetscCall(TSSetIJacobian(ts, B, B, RDIJacobian_FD, rd));
    1132:     break;
    1133:   case DISCRETIZATION_FE:
    1134:     PetscCall(TSSetIFunction(ts, NULL, RDIFunction_FE, rd));
    1135:     if (rd->jacobian == JACOBIAN_ANALYTIC) PetscCall(TSSetIJacobian(ts, B, B, RDIJacobian_FE, rd));
    1136:     break;
    1137:   }
    1138:   PetscCall(TSSetMaxTime(ts, rd->final_time));
    1139:   PetscCall(TSSetTimeStep(ts, 1e-3));
    1140:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
    1141:   PetscCall(TSSetFromOptions(ts));

    1143:   A = B;
    1144:   PetscCall(TSGetSNES(ts, &snes));
    1145:   switch (rd->jacobian) {
    1146:   case JACOBIAN_ANALYTIC:
    1147:     break;
    1148:   case JACOBIAN_MATRIXFREE:
    1149:     break;
    1150:   case JACOBIAN_FD_COLORING: {
    1151:     PetscCall(SNESSetJacobian(snes, A, B, SNESComputeJacobianDefaultColor, 0));
    1152:   } break;
    1153:   case JACOBIAN_FD_FULL:
    1154:     PetscCall(SNESSetJacobian(snes, A, B, SNESComputeJacobianDefault, ts));
    1155:     break;
    1156:   }

    1158:   if (rd->test_diff) PetscCall(RDTestDifferentiation(rd));
    1159:   PetscCall(TSSolve(ts, X));
    1160:   PetscCall(TSGetSolveTime(ts, &ftime));
    1161:   PetscCall(TSGetStepNumber(ts, &steps));
    1162:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Steps %" PetscInt_FMT "  final time %g\n", steps, (double)ftime));
    1163:   if (rd->view_draw) PetscCall(RDView(rd, X, PETSC_VIEWER_DRAW_WORLD));
    1164:   if (rd->view_binary[0]) {
    1165:     PetscViewer viewer;
    1166:     PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD, rd->view_binary, FILE_MODE_WRITE, &viewer));
    1167:     PetscCall(RDView(rd, X, viewer));
    1168:     PetscCall(PetscViewerDestroy(&viewer));
    1169:   }
    1170:   PetscCall(VecDestroy(&X));
    1171:   PetscCall(MatDestroy(&B));
    1172:   PetscCall(RDDestroy(&rd));
    1173:   PetscCall(TSDestroy(&ts));
    1174:   PetscCall(PetscFinalize());
    1175:   return 0;
    1176: }
    1177: /*TEST

    1179:     test:
    1180:       args: -da_grid_x 20 -rd_initial 1 -rd_discretization fd -rd_jacobian fd_coloring -rd_endpoint -ts_max_time 1 -ts_dt 2e-1 -ts_theta_initial_guess_extrapolate 0 -ts_monitor -snes_monitor_short -ksp_monitor_short
    1181:       requires: !single

    1183:     test:
    1184:       suffix: 2
    1185:       args: -da_grid_x 20 -rd_initial 1 -rd_discretization fe -rd_quadrature lobatto2 -rd_jacobian fd_coloring -rd_endpoint -ts_max_time 1 -ts_dt 2e-1 -ts_theta_initial_guess_extrapolate 0 -ts_monitor -snes_monitor_short -ksp_monitor_short
    1186:       requires: !single

    1188:     test:
    1189:       suffix: 3
    1190:       nsize: 2
    1191:       args: -da_grid_x 20 -rd_initial 1 -rd_discretization fd -rd_jacobian analytic -rd_endpoint -ts_max_time 3 -ts_dt 1e-1 -ts_theta_initial_guess_extrapolate 0 -ts_monitor -snes_monitor_short -ksp_monitor_short
    1192:       requires: !single

    1194: TEST*/