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) PetscCall(TSThetaGetTheta(ts, Theta));
262:   else *Theta = 1.;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

503: /*
504:  Various quadrature rules.  The nonlinear terms are non-polynomial so no standard quadrature will be exact.
505: */
506: static PetscErrorCode RDGetQuadrature(RD rd, PetscReal hx, PetscInt *nq, PetscReal weight[], PetscReal interp[][2], PetscReal deriv[][2])
507: {
508:   PetscInt         q, j;
509:   const PetscReal *refweight, (*refinterp)[2], (*refderiv)[2];

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

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

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

624: /*
625:  Finite element version
626: */
627: static PetscErrorCode RDIFunction_FE(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
628: {
629:   RD            rd = (RD)ctx;
630:   RDNode       *x, *x0, *xdot, *f;
631:   Vec           X0loc, Xloc, Xloc_t, Floc;
632:   PetscReal     hx, Theta, dt, weight[5], interp[5][2], deriv[5][2];
633:   DMDALocalInfo info;
634:   PetscInt      i, j, q, nq;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1020:     PetscCall(PetscOptionsBool("-rd_monitor_residual", "Display residuals every time they are evaluated", "", rd->monitor_residual, &rd->monitor_residual, NULL));
1021:     PetscCall(PetscOptionsEnum("-rd_discretization", "Discretization type", "", DiscretizationTypes, (PetscEnum)rd->discretization, (PetscEnum *)&rd->discretization, NULL));
1022:     if (rd->discretization == DISCRETIZATION_FE) {
1023:       rd->quadrature = QUADRATURE_GAUSS2;
1024:       PetscCall(PetscOptionsEnum("-rd_quadrature", "Finite element quadrature", "", QuadratureTypes, (PetscEnum)rd->quadrature, (PetscEnum *)&rd->quadrature, NULL));
1025:     }
1026:     PetscCall(PetscOptionsEnum("-rd_jacobian", "Type of finite difference Jacobian", "", JacobianTypes, (PetscEnum)rd->jacobian, (PetscEnum *)&rd->jacobian, NULL));
1027:     switch (rd->initial) {
1028:     case 1:
1029:       rd->leftbc     = BC_ROBIN;
1030:       rd->Eapplied   = 4 * rd->unit.Joule / PetscPowRealInt(rd->unit.meter, 3);
1031:       rd->L          = 1. * rd->unit.meter;
1032:       rd->beta       = 3.0;
1033:       rd->gamma      = 3.0;
1034:       rd->final_time = 3 * second;
1035:       break;
1036:     case 2:
1037:       rd->leftbc     = BC_NEUMANN;
1038:       rd->Eapplied   = 0.;
1039:       rd->L          = 1. * rd->unit.meter;
1040:       rd->beta       = 3.0;
1041:       rd->gamma      = 3.0;
1042:       rd->final_time = 1 * second;
1043:       break;
1044:     case 3:
1045:       rd->leftbc     = BC_ROBIN;
1046:       rd->Eapplied   = 7.503e6 * rd->unit.Joule / PetscPowRealInt(rd->unit.meter, 3);
1047:       rd->L          = 5. * rd->unit.meter;
1048:       rd->beta       = 3.5;
1049:       rd->gamma      = 3.5;
1050:       rd->final_time = 20e-9 * second;
1051:       break;
1052:     default:
1053:       SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Initial %" PetscInt_FMT, rd->initial);
1054:     }
1055:     PetscCall(PetscOptionsEnum("-rd_leftbc", "Left boundary condition", "", BCTypes, (PetscEnum)rd->leftbc, (PetscEnum *)&rd->leftbc, NULL));
1056:     PetscCall(PetscOptionsReal("-rd_E_applied", "Radiation flux at left end of domain", "", rd->Eapplied, &rd->Eapplied, NULL));
1057:     PetscCall(PetscOptionsReal("-rd_beta", "Thermal exponent for photon absorption", "", rd->beta, &rd->beta, NULL));
1058:     PetscCall(PetscOptionsReal("-rd_gamma", "Thermal exponent for diffusion coefficient", "", rd->gamma, &rd->gamma, NULL));
1059:     PetscCall(PetscOptionsBool("-rd_view_draw", "Draw final solution", "", rd->view_draw, &rd->view_draw, NULL));
1060:     PetscCall(PetscOptionsBool("-rd_endpoint", "Discretize using endpoints (like trapezoid rule) instead of midpoint", "", rd->endpoint, &rd->endpoint, NULL));
1061:     PetscCall(PetscOptionsBool("-rd_bcmidpoint", "Impose the boundary condition at the midpoint (Theta) of the interval", "", rd->bcmidpoint, &rd->bcmidpoint, NULL));
1062:     PetscCall(PetscOptionsBool("-rd_bclimit", "Limit diffusion coefficient in definition of Robin boundary condition", "", rd->bclimit, &rd->bclimit, NULL));
1063:     PetscCall(PetscOptionsBool("-rd_test_diff", "Test differentiation in constitutive relations", "", rd->test_diff, &rd->test_diff, NULL));
1064:     PetscCall(PetscOptionsString("-rd_view_binary", "File name to hold final solution", "", rd->view_binary, rd->view_binary, sizeof(rd->view_binary), NULL));
1065:   }
1066:   PetscOptionsEnd();

1068:   switch (rd->initial) {
1069:   case 1:
1070:   case 2:
1071:     rd->rho            = 1.;
1072:     rd->c              = 1.;
1073:     rd->K_R            = 1.;
1074:     rd->K_p            = 1.;
1075:     rd->sigma_b        = 0.25;
1076:     rd->MaterialEnergy = RDMaterialEnergy_Reduced;
1077:     break;
1078:   case 3:
1079:     /* Table 2 */
1080:     rd->rho            = 1.17e-3 * kilogram / (meter * meter * meter); /* density */
1081:     rd->K_R            = 7.44e18 * PetscPowRealInt(meter, 5) * PetscPowReal(Kelvin, 3.5) * PetscPowRealInt(kilogram, -2);
1082:     rd->K_p            = 2.33e20 * PetscPowRealInt(meter, 5) * PetscPowReal(Kelvin, 3.5) * PetscPowRealInt(kilogram, -2);
1083:     rd->I_H            = 2.179e-18 * Joule;                                                         /* Hydrogen ionization potential */
1084:     rd->m_p            = 1.673e-27 * kilogram;                                                      /* proton mass */
1085:     rd->m_e            = 9.109e-31 * kilogram;                                                      /* electron mass */
1086:     rd->h              = 6.626e-34 * Joule * second;                                                /* Planck's constant */
1087:     rd->k              = 1.381e-23 * Joule / Kelvin;                                                /* Boltzman constant */
1088:     rd->c              = 3.00e8 * meter / second;                                                   /* speed of light */
1089:     rd->sigma_b        = 5.67e-8 * Watt * PetscPowRealInt(meter, -2) * PetscPowRealInt(Kelvin, -4); /* Stefan-Boltzman constant */
1090:     rd->MaterialEnergy = RDMaterialEnergy_Saha;
1091:     break;
1092:   }

1094:   PetscCall(DMDACreate1d(comm, DM_BOUNDARY_NONE, 20, sizeof(RDNode) / sizeof(PetscScalar), 1, NULL, &rd->da));
1095:   PetscCall(DMSetFromOptions(rd->da));
1096:   PetscCall(DMSetUp(rd->da));
1097:   PetscCall(DMDASetFieldName(rd->da, 0, "E"));
1098:   PetscCall(DMDASetFieldName(rd->da, 1, "T"));
1099:   PetscCall(DMDASetUniformCoordinates(rd->da, 0., 1., 0., 0., 0., 0.));

1101:   *inrd = rd;
1102:   PetscFunctionReturn(PETSC_SUCCESS);
1103: }

1105: int main(int argc, char *argv[])
1106: {
1107:   RD        rd;
1108:   TS        ts;
1109:   SNES      snes;
1110:   Vec       X;
1111:   Mat       A, B;
1112:   PetscInt  steps;
1113:   PetscReal ftime;

1115:   PetscFunctionBeginUser;
1116:   PetscCall(PetscInitialize(&argc, &argv, 0, NULL));
1117:   PetscCall(RDCreate(PETSC_COMM_WORLD, &rd));
1118:   PetscCall(DMCreateGlobalVector(rd->da, &X));
1119:   PetscCall(DMSetMatType(rd->da, MATAIJ));
1120:   PetscCall(DMCreateMatrix(rd->da, &B));
1121:   PetscCall(RDInitialState(rd, X));

1123:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
1124:   PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
1125:   PetscCall(TSSetType(ts, TSTHETA));
1126:   PetscCall(TSSetDM(ts, rd->da));
1127:   switch (rd->discretization) {
1128:   case DISCRETIZATION_FD:
1129:     PetscCall(TSSetIFunction(ts, NULL, RDIFunction_FD, rd));
1130:     if (rd->jacobian == JACOBIAN_ANALYTIC) PetscCall(TSSetIJacobian(ts, B, B, RDIJacobian_FD, rd));
1131:     break;
1132:   case DISCRETIZATION_FE:
1133:     PetscCall(TSSetIFunction(ts, NULL, RDIFunction_FE, rd));
1134:     if (rd->jacobian == JACOBIAN_ANALYTIC) PetscCall(TSSetIJacobian(ts, B, B, RDIJacobian_FE, rd));
1135:     break;
1136:   }
1137:   PetscCall(TSSetMaxTime(ts, rd->final_time));
1138:   PetscCall(TSSetTimeStep(ts, 1e-3));
1139:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
1140:   PetscCall(TSSetFromOptions(ts));

1142:   A = B;
1143:   PetscCall(TSGetSNES(ts, &snes));
1144:   switch (rd->jacobian) {
1145:   case JACOBIAN_ANALYTIC:
1146:     break;
1147:   case JACOBIAN_MATRIXFREE:
1148:     break;
1149:   case JACOBIAN_FD_COLORING: {
1150:     PetscCall(SNESSetJacobian(snes, A, B, SNESComputeJacobianDefaultColor, 0));
1151:   } break;
1152:   case JACOBIAN_FD_FULL:
1153:     PetscCall(SNESSetJacobian(snes, A, B, SNESComputeJacobianDefault, ts));
1154:     break;
1155:   }

1157:   if (rd->test_diff) PetscCall(RDTestDifferentiation(rd));
1158:   PetscCall(TSSolve(ts, X));
1159:   PetscCall(TSGetSolveTime(ts, &ftime));
1160:   PetscCall(TSGetStepNumber(ts, &steps));
1161:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Steps %" PetscInt_FMT "  final time %g\n", steps, (double)ftime));
1162:   if (rd->view_draw) PetscCall(RDView(rd, X, PETSC_VIEWER_DRAW_WORLD));
1163:   if (rd->view_binary[0]) {
1164:     PetscViewer viewer;
1165:     PetscCall(PetscViewerBinaryOpen(PETSC_COMM_WORLD, rd->view_binary, FILE_MODE_WRITE, &viewer));
1166:     PetscCall(RDView(rd, X, viewer));
1167:     PetscCall(PetscViewerDestroy(&viewer));
1168:   }
1169:   PetscCall(VecDestroy(&X));
1170:   PetscCall(MatDestroy(&B));
1171:   PetscCall(RDDestroy(&rd));
1172:   PetscCall(TSDestroy(&ts));
1173:   PetscCall(PetscFinalize());
1174:   return 0;
1175: }
1176: /*TEST

1178:     test:
1179:       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
1180:       requires: !single

1182:     test:
1183:       suffix: 2
1184:       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
1185:       requires: !single

1187:     test:
1188:       suffix: 3
1189:       nsize: 2
1190:       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
1191:       requires: !single

1193: TEST*/