1 static const char help[] = "Time-dependent Brusselator reaction-diffusion PDE in 1d formulated as a PDAE. Demonstrates solving PDEs with algebraic constraints (PDAE).\n"; 2 /* 3 u_t - alpha u_xx = A + u^2 v - (B+1) u 4 v_t - alpha v_xx = B u - u^2 v 5 0 < x < 1; 6 A = 1, B = 3, alpha = 1/50 7 8 Initial conditions: 9 u(x,0) = 1 + sin(2 pi x) 10 v(x,0) = 3 11 12 Boundary conditions: 13 u(0,t) = u(1,t) = 1 14 v(0,t) = v(1,t) = 3 15 */ 16 17 #include <petscdm.h> 18 #include <petscdmda.h> 19 #include <petscts.h> 20 21 typedef struct { 22 PetscScalar u,v; 23 } Field; 24 25 typedef struct _User *User; 26 struct _User { 27 PetscReal A,B; /* Reaction coefficients */ 28 PetscReal alpha; /* Diffusion coefficient */ 29 PetscReal uleft,uright; /* Dirichlet boundary conditions */ 30 PetscReal vleft,vright; /* Dirichlet boundary conditions */ 31 }; 32 33 static PetscErrorCode FormRHSFunction(TS,PetscReal,Vec,Vec,void*); 34 static PetscErrorCode FormIFunction(TS,PetscReal,Vec,Vec,Vec,void*); 35 static PetscErrorCode FormIJacobian(TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*); 36 static PetscErrorCode FormInitialSolution(TS,Vec,void*); 37 38 int main(int argc,char **argv) 39 { 40 TS ts; /* nonlinear solver */ 41 Vec X; /* solution, residual vectors */ 42 Mat J; /* Jacobian matrix */ 43 PetscInt steps,mx; 44 DM da; 45 PetscReal ftime,hx,dt; 46 struct _User user; /* user-defined work context */ 47 TSConvergedReason reason; 48 49 PetscFunctionBeginUser; 50 PetscCall(PetscInitialize(&argc,&argv,(char*)0,help)); 51 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 52 Create distributed array (DMDA) to manage parallel grid and vectors 53 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 54 PetscCall(DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,11,2,2,NULL,&da)); 55 PetscCall(DMSetFromOptions(da)); 56 PetscCall(DMSetUp(da)); 57 58 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 59 Extract global vectors from DMDA; 60 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 61 PetscCall(DMCreateGlobalVector(da,&X)); 62 63 /* Initialize user application context */ 64 PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Advection-reaction options",""); 65 { 66 user.A = 1; 67 user.B = 3; 68 user.alpha = 0.02; 69 user.uleft = 1; 70 user.uright = 1; 71 user.vleft = 3; 72 user.vright = 3; 73 PetscCall(PetscOptionsReal("-A","Reaction rate","",user.A,&user.A,NULL)); 74 PetscCall(PetscOptionsReal("-B","Reaction rate","",user.B,&user.B,NULL)); 75 PetscCall(PetscOptionsReal("-alpha","Diffusion coefficient","",user.alpha,&user.alpha,NULL)); 76 PetscCall(PetscOptionsReal("-uleft","Dirichlet boundary condition","",user.uleft,&user.uleft,NULL)); 77 PetscCall(PetscOptionsReal("-uright","Dirichlet boundary condition","",user.uright,&user.uright,NULL)); 78 PetscCall(PetscOptionsReal("-vleft","Dirichlet boundary condition","",user.vleft,&user.vleft,NULL)); 79 PetscCall(PetscOptionsReal("-vright","Dirichlet boundary condition","",user.vright,&user.vright,NULL)); 80 } 81 PetscOptionsEnd(); 82 83 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 84 Create timestepping solver context 85 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 86 PetscCall(TSCreate(PETSC_COMM_WORLD,&ts)); 87 PetscCall(TSSetDM(ts,da)); 88 PetscCall(TSSetType(ts,TSARKIMEX)); 89 PetscCall(TSSetEquationType(ts,TS_EQ_DAE_IMPLICIT_INDEX1)); 90 PetscCall(TSSetRHSFunction(ts,NULL,FormRHSFunction,&user)); 91 PetscCall(TSSetIFunction(ts,NULL,FormIFunction,&user)); 92 PetscCall(DMSetMatType(da,MATAIJ)); 93 PetscCall(DMCreateMatrix(da,&J)); 94 PetscCall(TSSetIJacobian(ts,J,J,FormIJacobian,&user)); 95 96 ftime = 10.0; 97 PetscCall(TSSetMaxTime(ts,ftime)); 98 PetscCall(TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER)); 99 100 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 101 Set initial conditions 102 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 103 PetscCall(FormInitialSolution(ts,X,&user)); 104 PetscCall(TSSetSolution(ts,X)); 105 PetscCall(VecGetSize(X,&mx)); 106 hx = 1.0/(PetscReal)(mx/2-1); 107 dt = 0.4 * PetscSqr(hx) / user.alpha; /* Diffusive stability limit */ 108 PetscCall(TSSetTimeStep(ts,dt)); 109 110 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 111 Set runtime options 112 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 113 PetscCall(TSSetFromOptions(ts)); 114 115 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 116 Solve nonlinear system 117 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 118 PetscCall(TSSolve(ts,X)); 119 PetscCall(TSGetSolveTime(ts,&ftime)); 120 PetscCall(TSGetStepNumber(ts,&steps)); 121 PetscCall(TSGetConvergedReason(ts,&reason)); 122 PetscCall(PetscPrintf(PETSC_COMM_WORLD,"%s at time %g after %" PetscInt_FMT " steps\n",TSConvergedReasons[reason],(double)ftime,steps)); 123 124 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 125 Free work space. 126 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 127 PetscCall(MatDestroy(&J)); 128 PetscCall(VecDestroy(&X)); 129 PetscCall(TSDestroy(&ts)); 130 PetscCall(DMDestroy(&da)); 131 PetscCall(PetscFinalize()); 132 return 0; 133 } 134 135 static PetscErrorCode FormIFunction(TS ts,PetscReal t,Vec X,Vec Xdot,Vec F,void *ptr) 136 { 137 User user = (User)ptr; 138 DM da; 139 DMDALocalInfo info; 140 PetscInt i; 141 Field *x,*xdot,*f; 142 PetscReal hx; 143 Vec Xloc; 144 145 PetscFunctionBeginUser; 146 PetscCall(TSGetDM(ts,&da)); 147 PetscCall(DMDAGetLocalInfo(da,&info)); 148 hx = 1.0/(PetscReal)(info.mx-1); 149 150 /* 151 Scatter ghost points to local vector,using the 2-step process 152 DMGlobalToLocalBegin(),DMGlobalToLocalEnd(). 153 By placing code between these two statements, computations can be 154 done while messages are in transition. 155 */ 156 PetscCall(DMGetLocalVector(da,&Xloc)); 157 PetscCall(DMGlobalToLocalBegin(da,X,INSERT_VALUES,Xloc)); 158 PetscCall(DMGlobalToLocalEnd(da,X,INSERT_VALUES,Xloc)); 159 160 /* Get pointers to vector data */ 161 PetscCall(DMDAVecGetArrayRead(da,Xloc,&x)); 162 PetscCall(DMDAVecGetArrayRead(da,Xdot,&xdot)); 163 PetscCall(DMDAVecGetArray(da,F,&f)); 164 165 /* Compute function over the locally owned part of the grid */ 166 for (i=info.xs; i<info.xs+info.xm; i++) { 167 if (i == 0) { 168 f[i].u = hx * (x[i].u - user->uleft); 169 f[i].v = hx * (x[i].v - user->vleft); 170 } else if (i == info.mx-1) { 171 f[i].u = hx * (x[i].u - user->uright); 172 f[i].v = hx * (x[i].v - user->vright); 173 } else { 174 f[i].u = hx * xdot[i].u - user->alpha * (x[i-1].u - 2.*x[i].u + x[i+1].u) / hx; 175 f[i].v = hx * xdot[i].v - user->alpha * (x[i-1].v - 2.*x[i].v + x[i+1].v) / hx; 176 } 177 } 178 179 /* Restore vectors */ 180 PetscCall(DMDAVecRestoreArrayRead(da,Xloc,&x)); 181 PetscCall(DMDAVecRestoreArrayRead(da,Xdot,&xdot)); 182 PetscCall(DMDAVecRestoreArray(da,F,&f)); 183 PetscCall(DMRestoreLocalVector(da,&Xloc)); 184 PetscFunctionReturn(0); 185 } 186 187 static PetscErrorCode FormRHSFunction(TS ts,PetscReal t,Vec X,Vec F,void *ptr) 188 { 189 User user = (User)ptr; 190 DM da; 191 DMDALocalInfo info; 192 PetscInt i; 193 PetscReal hx; 194 Field *x,*f; 195 196 PetscFunctionBeginUser; 197 PetscCall(TSGetDM(ts,&da)); 198 PetscCall(DMDAGetLocalInfo(da,&info)); 199 hx = 1.0/(PetscReal)(info.mx-1); 200 201 /* Get pointers to vector data */ 202 PetscCall(DMDAVecGetArrayRead(da,X,&x)); 203 PetscCall(DMDAVecGetArray(da,F,&f)); 204 205 /* Compute function over the locally owned part of the grid */ 206 for (i=info.xs; i<info.xs+info.xm; i++) { 207 PetscScalar u = x[i].u,v = x[i].v; 208 f[i].u = hx*(user->A + u*u*v - (user->B+1)*u); 209 f[i].v = hx*(user->B*u - u*u*v); 210 } 211 212 /* Restore vectors */ 213 PetscCall(DMDAVecRestoreArrayRead(da,X,&x)); 214 PetscCall(DMDAVecRestoreArray(da,F,&f)); 215 PetscFunctionReturn(0); 216 } 217 218 /* --------------------------------------------------------------------- */ 219 /* 220 IJacobian - Compute IJacobian = dF/dU + a dF/dUdot 221 */ 222 PetscErrorCode FormIJacobian(TS ts,PetscReal t,Vec X,Vec Xdot,PetscReal a,Mat J,Mat Jpre,void *ptr) 223 { 224 User user = (User)ptr; 225 DMDALocalInfo info; 226 PetscInt i; 227 PetscReal hx; 228 DM da; 229 Field *x,*xdot; 230 231 PetscFunctionBeginUser; 232 PetscCall(TSGetDM(ts,&da)); 233 PetscCall(DMDAGetLocalInfo(da,&info)); 234 hx = 1.0/(PetscReal)(info.mx-1); 235 236 /* Get pointers to vector data */ 237 PetscCall(DMDAVecGetArrayRead(da,X,&x)); 238 PetscCall(DMDAVecGetArrayRead(da,Xdot,&xdot)); 239 240 /* Compute function over the locally owned part of the grid */ 241 for (i=info.xs; i<info.xs+info.xm; i++) { 242 if (i == 0 || i == info.mx-1) { 243 const PetscInt row = i,col = i; 244 const PetscScalar vals[2][2] = {{hx,0},{0,hx}}; 245 PetscCall(MatSetValuesBlocked(Jpre,1,&row,1,&col,&vals[0][0],INSERT_VALUES)); 246 } else { 247 const PetscInt row = i,col[] = {i-1,i,i+1}; 248 const PetscScalar dxxL = -user->alpha/hx,dxx0 = 2.*user->alpha/hx,dxxR = -user->alpha/hx; 249 const PetscScalar vals[2][3][2] = {{{dxxL,0},{a *hx+dxx0,0},{dxxR,0}}, 250 {{0,dxxL},{0,a*hx+dxx0},{0,dxxR}}}; 251 PetscCall(MatSetValuesBlocked(Jpre,1,&row,3,col,&vals[0][0][0],INSERT_VALUES)); 252 } 253 } 254 255 /* Restore vectors */ 256 PetscCall(DMDAVecRestoreArrayRead(da,X,&x)); 257 PetscCall(DMDAVecRestoreArrayRead(da,Xdot,&xdot)); 258 259 PetscCall(MatAssemblyBegin(Jpre,MAT_FINAL_ASSEMBLY)); 260 PetscCall(MatAssemblyEnd(Jpre,MAT_FINAL_ASSEMBLY)); 261 if (J != Jpre) { 262 PetscCall(MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY)); 263 PetscCall(MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY)); 264 } 265 PetscFunctionReturn(0); 266 } 267 268 PetscErrorCode FormInitialSolution(TS ts,Vec X,void *ctx) 269 { 270 User user = (User)ctx; 271 DM da; 272 PetscInt i; 273 DMDALocalInfo info; 274 Field *x; 275 PetscReal hx; 276 277 PetscFunctionBeginUser; 278 PetscCall(TSGetDM(ts,&da)); 279 PetscCall(DMDAGetLocalInfo(da,&info)); 280 hx = 1.0/(PetscReal)(info.mx-1); 281 282 /* Get pointers to vector data */ 283 PetscCall(DMDAVecGetArray(da,X,&x)); 284 285 /* Compute function over the locally owned part of the grid */ 286 for (i=info.xs; i<info.xs+info.xm; i++) { 287 PetscReal xi = i*hx; 288 x[i].u = user->uleft*(1.-xi) + user->uright*xi + PetscSinReal(2.*PETSC_PI*xi); 289 x[i].v = user->vleft*(1.-xi) + user->vright*xi; 290 } 291 PetscCall(DMDAVecRestoreArray(da,X,&x)); 292 PetscFunctionReturn(0); 293 } 294 295 /*TEST 296 297 test: 298 args: -nox -da_grid_x 20 -ts_monitor_draw_solution -ts_type rosw -ts_rosw_type 2p -ts_dt 5e-2 -ts_adapt_type none 299 300 TEST*/ 301