static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\ Input parameters include:\n\ -m , where = number of grid points\n\ -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\ -debug : Activate debugging printouts\n\ -nox : Deactivate x-window graphics\n\n"; /* Concepts: TS^time-dependent linear problems Concepts: TS^heat equation Concepts: TS^diffusion equation Routines: TSCreate(); TSSetSolution(); TSSetRHSJacobian(), TSSetIJacobian(); Routines: TSSetInitialTimeStep(); TSSetDuration(); TSMonitorSet(); Routines: TSSetFromOptions(); TSStep(); TSDestroy(); Routines: TSSetTimeStep(); TSGetTimeStep(); Processors: 1 */ /* ------------------------------------------------------------------------ This program solves the one-dimensional heat equation (also called the diffusion equation), u_t = u_xx, on the domain 0 <= x <= 1, with the boundary conditions u(t,0) = 0, u(t,1) = 0, and the initial condition u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x). This is a linear, second-order, parabolic equation. We discretize the right-hand side using finite differences with uniform grid spacing h: u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2) We then demonstrate time evolution using the various TS methods by running the program via ex3 -ts_type We compare the approximate solution with the exact solution, given by u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) + 3*exp(-4*pi*pi*t) * sin(2*pi*x) Notes: This code demonstrates the TS solver interface to two variants of linear problems, u_t = f(u,t), namely - time-dependent f: f(u,t) is a function of t - time-independent f: f(u,t) is simply f(u) The parallel version of this code is ts/examples/tutorials/ex4.c ------------------------------------------------------------------------- */ /* Include "ts.h" so that we can use TS solvers. Note that this file automatically includes: petscsys.h - base PETSc routines vec.h - vectors sys.h - system routines mat.h - matrices is.h - index sets ksp.h - Krylov subspace methods viewer.h - viewers pc.h - preconditioners snes.h - nonlinear solvers */ #include #include /* User-defined application context - contains data needed by the application-provided call-back routines. */ typedef struct { Vec solution; /* global exact solution vector */ PetscInt m; /* total number of grid points */ PetscReal h; /* mesh width h = 1/(m-1) */ PetscBool debug; /* flag (1 indicates activation of debugging printouts) */ PetscViewer viewer1, viewer2; /* viewers for the solution and error */ PetscReal norm_2, norm_max; /* error norms */ } AppCtx; /* User-defined routines */ extern PetscErrorCode InitialConditions(Vec,AppCtx*); extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat,Mat,void*); extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*); extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*); extern PetscErrorCode MyBCRoutine(TS,PetscReal,Vec,void*); #undef __FUNCT__ #define __FUNCT__ "main" int main(int argc,char **argv) { AppCtx appctx; /* user-defined application context */ TS ts; /* timestepping context */ Mat A; /* matrix data structure */ Vec u; /* approximate solution vector */ PetscReal time_total_max = 100.0; /* default max total time */ PetscInt time_steps_max = 100; /* default max timesteps */ PetscDraw draw; /* drawing context */ PetscErrorCode ierr; PetscInt steps, m; PetscMPIInt size; PetscReal dt; PetscReal ftime; PetscBool flg; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Initialize program and set problem parameters - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ PetscInitialize(&argc,&argv,(char*)0,help); MPI_Comm_size(PETSC_COMM_WORLD,&size); if (size != 1) SETERRQ(PETSC_COMM_SELF,1,"This is a uniprocessor example only!"); m = 60; ierr = PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);CHKERRQ(ierr); ierr = PetscOptionsHasName(NULL,NULL,"-debug",&appctx.debug);CHKERRQ(ierr); appctx.m = m; appctx.h = 1.0/(m-1.0); appctx.norm_2 = 0.0; appctx.norm_max = 0.0; ierr = PetscPrintf(PETSC_COMM_SELF,"Solving a linear TS problem on 1 processor\n");CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create vector data structures - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* Create vector data structures for approximate and exact solutions */ ierr = VecCreateSeq(PETSC_COMM_SELF,m,&u);CHKERRQ(ierr); ierr = VecDuplicate(u,&appctx.solution);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set up displays to show graphs of the solution and error - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,380,400,160,&appctx.viewer1);CHKERRQ(ierr); ierr = PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);CHKERRQ(ierr); ierr = PetscDrawSetDoubleBuffer(draw);CHKERRQ(ierr); ierr = PetscViewerDrawOpen(PETSC_COMM_SELF,0,"",80,0,400,160,&appctx.viewer2);CHKERRQ(ierr); ierr = PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);CHKERRQ(ierr); ierr = PetscDrawSetDoubleBuffer(draw);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create timestepping solver context - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSCreate(PETSC_COMM_SELF,&ts);CHKERRQ(ierr); ierr = TSSetProblemType(ts,TS_LINEAR);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set optional user-defined monitoring routine - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSMonitorSet(ts,Monitor,&appctx,NULL);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Create matrix data structure; set matrix evaluation routine. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = MatCreate(PETSC_COMM_SELF,&A);CHKERRQ(ierr); ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);CHKERRQ(ierr); ierr = MatSetFromOptions(A);CHKERRQ(ierr); ierr = MatSetUp(A);CHKERRQ(ierr); ierr = PetscOptionsHasName(NULL,NULL,"-time_dependent_rhs",&flg);CHKERRQ(ierr); if (flg) { /* For linear problems with a time-dependent f(u,t) in the equation u_t = f(u,t), the user provides the discretized right-hand-side as a time-dependent matrix. */ ierr = TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);CHKERRQ(ierr); ierr = TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);CHKERRQ(ierr); } else { /* For linear problems with a time-independent f(u) in the equation u_t = f(u), the user provides the discretized right-hand-side as a matrix only once, and then sets a null matrix evaluation routine. */ ierr = RHSMatrixHeat(ts,0.0,u,A,A,&appctx);CHKERRQ(ierr); ierr = TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,&appctx);CHKERRQ(ierr); ierr = TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);CHKERRQ(ierr); } /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Set solution vector and initial timestep - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ dt = appctx.h*appctx.h/2.0; ierr = TSSetInitialTimeStep(ts,0.0,dt);CHKERRQ(ierr); ierr = TSSetSolution(ts,u);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Customize timestepping solver: - Set the solution method to be the Backward Euler method. - Set timestepping duration info Then set runtime options, which can override these defaults. For example, -ts_max_steps -ts_final_time to override the defaults set by TSSetDuration(). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSSetDuration(ts,time_steps_max,time_total_max);CHKERRQ(ierr); ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);CHKERRQ(ierr); ierr = TSSetFromOptions(ts);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solve the problem - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* Evaluate initial conditions */ ierr = InitialConditions(u,&appctx);CHKERRQ(ierr); /* Run the timestepping solver */ ierr = TSSolve(ts,u);CHKERRQ(ierr); ierr = TSGetSolveTime(ts,&ftime);CHKERRQ(ierr); ierr = TSGetTimeStepNumber(ts,&steps);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - View timestepping solver info - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = PetscPrintf(PETSC_COMM_SELF,"avg. error (2 norm) = %g, avg. error (max norm) = %g\n",(double)(appctx.norm_2/steps),(double)(appctx.norm_max/steps));CHKERRQ(ierr); ierr = TSView(ts,PETSC_VIEWER_STDOUT_SELF);CHKERRQ(ierr); /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Free work space. All PETSc objects should be destroyed when they are no longer needed. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ ierr = TSDestroy(&ts);CHKERRQ(ierr); ierr = MatDestroy(&A);CHKERRQ(ierr); ierr = VecDestroy(&u);CHKERRQ(ierr); ierr = PetscViewerDestroy(&appctx.viewer1);CHKERRQ(ierr); ierr = PetscViewerDestroy(&appctx.viewer2);CHKERRQ(ierr); ierr = VecDestroy(&appctx.solution);CHKERRQ(ierr); /* Always call PetscFinalize() before exiting a program. This routine - finalizes the PETSc libraries as well as MPI - provides summary and diagnostic information if certain runtime options are chosen (e.g., -log_summary). */ PetscFinalize(); return 0; } /* --------------------------------------------------------------------- */ #undef __FUNCT__ #define __FUNCT__ "InitialConditions" /* InitialConditions - Computes the solution at the initial time. Input Parameter: u - uninitialized solution vector (global) appctx - user-defined application context Output Parameter: u - vector with solution at initial time (global) */ PetscErrorCode InitialConditions(Vec u,AppCtx *appctx) { PetscScalar *u_localptr; PetscInt i; PetscErrorCode ierr; /* Get a pointer to vector data. - For default PETSc vectors, VecGetArray() returns a pointer to the data array. Otherwise, the routine is implementation dependent. - You MUST call VecRestoreArray() when you no longer need access to the array. - Note that the Fortran interface to VecGetArray() differs from the C version. See the users manual for details. */ ierr = VecGetArray(u,&u_localptr);CHKERRQ(ierr); /* We initialize the solution array by simply writing the solution directly into the array locations. Alternatively, we could use VecSetValues() or VecSetValuesLocal(). */ for (i=0; im; i++) u_localptr[i] = PetscSinReal(PETSC_PI*i*6.*appctx->h) + 3.*PetscSinReal(PETSC_PI*i*2.*appctx->h); /* Restore vector */ ierr = VecRestoreArray(u,&u_localptr);CHKERRQ(ierr); /* Print debugging information if desired */ if (appctx->debug) { ierr = VecView(u,PETSC_VIEWER_STDOUT_SELF);CHKERRQ(ierr); } return 0; } /* --------------------------------------------------------------------- */ #undef __FUNCT__ #define __FUNCT__ "ExactSolution" /* ExactSolution - Computes the exact solution at a given time. Input Parameters: t - current time solution - vector in which exact solution will be computed appctx - user-defined application context Output Parameter: solution - vector with the newly computed exact solution */ PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx) { PetscScalar *s_localptr, h = appctx->h, ex1, ex2, sc1, sc2; PetscInt i; PetscErrorCode ierr; /* Get a pointer to vector data. */ ierr = VecGetArray(solution,&s_localptr);CHKERRQ(ierr); /* Simply write the solution directly into the array locations. Alternatively, we culd use VecSetValues() or VecSetValuesLocal(). */ ex1 = PetscExpReal(-36.*PETSC_PI*PETSC_PI*t); ex2 = PetscExpReal(-4.*PETSC_PI*PETSC_PI*t); sc1 = PETSC_PI*6.*h; sc2 = PETSC_PI*2.*h; for (i=0; im; i++) s_localptr[i] = PetscSinReal(PetscRealPart(sc1)*(PetscReal)i)*ex1 + 3.*PetscSinReal(PetscRealPart(sc2)*(PetscReal)i)*ex2; /* Restore vector */ ierr = VecRestoreArray(solution,&s_localptr);CHKERRQ(ierr); return 0; } /* --------------------------------------------------------------------- */ #undef __FUNCT__ #define __FUNCT__ "Monitor" /* Monitor - User-provided routine to monitor the solution computed at each timestep. This example plots the solution and computes the error in two different norms. This example also demonstrates changing the timestep via TSSetTimeStep(). Input Parameters: ts - the timestep context step - the count of the current step (with 0 meaning the initial condition) crtime - the current time u - the solution at this timestep ctx - the user-provided context for this monitoring routine. In this case we use the application context which contains information about the problem size, workspace and the exact solution. */ PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal crtime,Vec u,void *ctx) { AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */ PetscErrorCode ierr; PetscReal norm_2, norm_max, dt, dttol; PetscBool flg; /* View a graph of the current iterate */ ierr = VecView(u,appctx->viewer2);CHKERRQ(ierr); /* Compute the exact solution */ ierr = ExactSolution(crtime,appctx->solution,appctx);CHKERRQ(ierr); /* Print debugging information if desired */ if (appctx->debug) { ierr = PetscPrintf(PETSC_COMM_SELF,"Computed solution vector\n");CHKERRQ(ierr); ierr = VecView(u,PETSC_VIEWER_STDOUT_SELF);CHKERRQ(ierr); ierr = PetscPrintf(PETSC_COMM_SELF,"Exact solution vector\n");CHKERRQ(ierr); ierr = VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);CHKERRQ(ierr); } /* Compute the 2-norm and max-norm of the error */ ierr = VecAXPY(appctx->solution,-1.0,u);CHKERRQ(ierr); ierr = VecNorm(appctx->solution,NORM_2,&norm_2);CHKERRQ(ierr); norm_2 = PetscSqrtReal(appctx->h)*norm_2; ierr = VecNorm(appctx->solution,NORM_MAX,&norm_max);CHKERRQ(ierr); ierr = TSGetTimeStep(ts,&dt);CHKERRQ(ierr); if (norm_2 > 1.e-2) { ierr = PetscPrintf(PETSC_COMM_SELF,"Timestep %D: step size = %g, time = %g, 2-norm error = %g, max norm error = %g\n",step,(double)dt,(double)crtime,(double)norm_2,(double)norm_max);CHKERRQ(ierr); } appctx->norm_2 += norm_2; appctx->norm_max += norm_max; dttol = .0001; ierr = PetscOptionsGetReal(NULL,NULL,"-dttol",&dttol,&flg);CHKERRQ(ierr); if (dt < dttol) { dt *= .999; ierr = TSSetTimeStep(ts,dt);CHKERRQ(ierr); } /* View a graph of the error */ ierr = VecView(appctx->solution,appctx->viewer1);CHKERRQ(ierr); /* Print debugging information if desired */ if (appctx->debug) { ierr = PetscPrintf(PETSC_COMM_SELF,"Error vector\n");CHKERRQ(ierr); ierr = VecView(appctx->solution,PETSC_VIEWER_STDOUT_SELF);CHKERRQ(ierr); } return 0; } /* --------------------------------------------------------------------- */ #undef __FUNCT__ #define __FUNCT__ "RHSMatrixHeat" /* RHSMatrixHeat - User-provided routine to compute the right-hand-side matrix for the heat equation. Input Parameters: ts - the TS context t - current time global_in - global input vector dummy - optional user-defined context, as set by TSetRHSJacobian() Output Parameters: AA - Jacobian matrix BB - optionally different preconditioning matrix str - flag indicating matrix structure Notes: Recall that MatSetValues() uses 0-based row and column numbers in Fortran as well as in C. */ PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat BB,void *ctx) { Mat A = AA; /* Jacobian matrix */ AppCtx *appctx = (AppCtx*) ctx; /* user-defined application context */ PetscInt mstart = 0; PetscInt mend = appctx->m; PetscErrorCode ierr; PetscInt i, idx[3]; PetscScalar v[3], stwo = -2./(appctx->h*appctx->h), sone = -.5*stwo; /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Compute entries for the locally owned part of the matrix - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */ /* Set matrix rows corresponding to boundary data */ mstart = 0; v[0] = 1.0; ierr = MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);CHKERRQ(ierr); mstart++; mend--; v[0] = 1.0; ierr = MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);CHKERRQ(ierr); /* Set matrix rows corresponding to interior data. We construct the matrix one row at a time. */ v[0] = sone; v[1] = stwo; v[2] = sone; for (i=mstart; im; PetscScalar *fa; ierr = VecGetArray(f,&fa);CHKERRQ(ierr); fa[0] = 0.0; fa[m-1] = 1.0; ierr = VecRestoreArray(f,&fa);CHKERRQ(ierr); ierr = PetscPrintf(PETSC_COMM_SELF,"t=%g\n",(double)t);CHKERRQ(ierr); return 0; }