1 2 static char help[] = "Solves the motion of spring.\n\ 3 Input parameters include:\n"; 4 5 /* ------------------------------------------------------------------------ 6 7 This program solves the motion of spring by Hooke's law 8 x' = f(t,v) = v 9 v' = g(t,x) = -omega^2*x 10 on the interval 0 <= t <= 0.1, with the initial conditions 11 x(0) = 0.2, x'(0) = v(0) = 0, 12 and 13 omega = 64. 14 The exact solution is 15 x(t) = A*sin(t*omega) + B*cos(t*omega) 16 where A and B are constants that can be determined from the initial conditions. 17 In this case, B=0.2, A=0. 18 19 Notes: 20 This code demonstrates the TS solver interface to solve a separable Hamiltonian 21 system, which can be split into two subsystems involving two coupling components, 22 named generailized momentum and generailized position respectively. 23 Using a symplectic intergrator can preserve energy 24 E = (v^2+omega^2*x^2-omega^2*h*v*x)/2 25 ------------------------------------------------------------------------- */ 26 27 #include <petscts.h> 28 #include <petscvec.h> 29 30 typedef struct _n_User *User; 31 struct _n_User { 32 PetscReal omega; 33 PetscInt nts; /* print the energy at each nts time steps */ 34 }; 35 36 /* 37 User-defined routines. 38 The first RHS function provides f(t,x), the residual for the generalized momentum, 39 and the second one provides g(t,v), the residual for the generalized position. 40 */ 41 static PetscErrorCode RHSFunction2(TS ts, PetscReal t, Vec X, Vec Vres, void *ctx) 42 { 43 User user = (User)ctx; 44 const PetscScalar *x; 45 PetscScalar *vres; 46 47 PetscFunctionBeginUser; 48 PetscCall(VecGetArrayRead(X, &x)); 49 PetscCall(VecGetArray(Vres, &vres)); 50 vres[0] = -user->omega * user->omega * x[0]; 51 PetscCall(VecRestoreArray(Vres, &vres)); 52 PetscCall(VecRestoreArrayRead(X, &x)); 53 PetscFunctionReturn(PETSC_SUCCESS); 54 } 55 56 static PetscErrorCode RHSFunction1(TS ts, PetscReal t, Vec V, Vec Xres, void *ctx) 57 { 58 const PetscScalar *v; 59 PetscScalar *xres; 60 61 PetscFunctionBeginUser; 62 PetscCall(VecGetArray(Xres, &xres)); 63 PetscCall(VecGetArrayRead(V, &v)); 64 xres[0] = v[0]; 65 PetscCall(VecRestoreArrayRead(V, &v)); 66 PetscCall(VecRestoreArray(Xres, &xres)); 67 PetscFunctionReturn(PETSC_SUCCESS); 68 } 69 70 static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec U, Vec R, void *ctx) 71 { 72 User user = (User)ctx; 73 const PetscScalar *u; 74 PetscScalar *r; 75 76 PetscFunctionBeginUser; 77 PetscCall(VecGetArrayRead(U, &u)); 78 PetscCall(VecGetArray(R, &r)); 79 r[0] = u[1]; 80 r[1] = -user->omega * user->omega * u[0]; 81 PetscCall(VecRestoreArrayRead(U, &u)); 82 PetscCall(VecRestoreArray(R, &r)); 83 PetscFunctionReturn(PETSC_SUCCESS); 84 } 85 86 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */ 87 static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec U, void *ctx) 88 { 89 const PetscScalar *u; 90 PetscReal dt; 91 PetscScalar energy, menergy; 92 User user = (User)ctx; 93 94 PetscFunctionBeginUser; 95 if (step % user->nts == 0) { 96 PetscCall(TSGetTimeStep(ts, &dt)); 97 PetscCall(VecGetArrayRead(U, &u)); 98 menergy = (u[1] * u[1] + user->omega * user->omega * u[0] * u[0] - user->omega * user->omega * dt * u[0] * u[1]) / 2.; 99 energy = (u[1] * u[1] + user->omega * user->omega * u[0] * u[0]) / 2.; 100 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "At time %.6lf, Energy = %8g, Modified Energy = %8g\n", (double)t, (double)energy, (double)menergy)); 101 PetscCall(VecRestoreArrayRead(U, &u)); 102 } 103 PetscFunctionReturn(PETSC_SUCCESS); 104 } 105 106 int main(int argc, char **argv) 107 { 108 TS ts; /* nonlinear solver */ 109 Vec U; /* solution, residual vectors */ 110 IS is1, is2; 111 PetscInt nindices[1]; 112 PetscReal ftime = 0.1; 113 PetscBool monitor = PETSC_FALSE; 114 PetscScalar *u_ptr; 115 PetscMPIInt size; 116 struct _n_User user; 117 118 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 119 Initialize program 120 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 121 PetscFunctionBeginUser; 122 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 123 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size)); 124 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!"); 125 126 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 127 Set runtime options 128 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 129 user.omega = 64.; 130 user.nts = 100; 131 PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL)); 132 PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Physical parameters", NULL); 133 PetscCall(PetscOptionsReal("-omega", "parameter", "<64>", user.omega, &user.omega, NULL)); 134 PetscCall(PetscOptionsInt("-next_output", "time steps for next output point", "<100>", user.nts, &user.nts, NULL)); 135 PetscOptionsEnd(); 136 137 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 138 Create necessary matrix and vectors, solve same ODE on every process 139 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 140 PetscCall(VecCreateSeq(PETSC_COMM_SELF, 2, &U)); 141 nindices[0] = 0; 142 PetscCall(ISCreateGeneral(PETSC_COMM_SELF, 1, nindices, PETSC_COPY_VALUES, &is1)); 143 nindices[0] = 1; 144 PetscCall(ISCreateGeneral(PETSC_COMM_SELF, 1, nindices, PETSC_COPY_VALUES, &is2)); 145 146 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 147 Create timestepping solver context 148 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 149 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts)); 150 PetscCall(TSSetType(ts, TSBASICSYMPLECTIC)); 151 PetscCall(TSRHSSplitSetIS(ts, "position", is1)); 152 PetscCall(TSRHSSplitSetIS(ts, "momentum", is2)); 153 PetscCall(TSRHSSplitSetRHSFunction(ts, "position", NULL, RHSFunction1, &user)); 154 PetscCall(TSRHSSplitSetRHSFunction(ts, "momentum", NULL, RHSFunction2, &user)); 155 PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &user)); 156 157 PetscCall(TSSetMaxTime(ts, ftime)); 158 PetscCall(TSSetTimeStep(ts, 0.0001)); 159 PetscCall(TSSetMaxSteps(ts, 1000)); 160 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP)); 161 if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL)); 162 163 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 164 Set initial conditions 165 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 166 PetscCall(VecGetArray(U, &u_ptr)); 167 u_ptr[0] = 0.2; 168 u_ptr[1] = 0.0; 169 PetscCall(VecRestoreArray(U, &u_ptr)); 170 171 PetscCall(TSSetTime(ts, 0.0)); 172 173 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 174 Set runtime options 175 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 176 PetscCall(TSSetFromOptions(ts)); 177 178 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 179 Solve nonlinear system 180 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 181 PetscCall(TSSolve(ts, U)); 182 PetscCall(TSGetSolveTime(ts, &ftime)); 183 PetscCall(VecView(U, PETSC_VIEWER_STDOUT_WORLD)); 184 185 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "The exact solution at time %.6lf is [%g %g]\n", (double)ftime, (double)(0.2 * PetscCosReal(user.omega * ftime)), (double)(-0.2 * user.omega * PetscSinReal(user.omega * ftime)))); 186 187 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 188 Free work space. All PETSc objects should be destroyed when they 189 are no longer needed. 190 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ 191 PetscCall(VecDestroy(&U)); 192 PetscCall(TSDestroy(&ts)); 193 PetscCall(ISDestroy(&is1)); 194 PetscCall(ISDestroy(&is2)); 195 PetscCall(PetscFinalize()); 196 return 0; 197 } 198 199 /*TEST 200 build: 201 requires: !single !complex 202 203 test: 204 args: -ts_basicsymplectic_type 1 -monitor 205 206 test: 207 suffix: 2 208 args: -ts_basicsymplectic_type 2 -monitor 209 210 TEST*/ 211