[petsc-users] How to apply boundary conditions when using TS?

[zhfreewill]

Thank you for your reply. This is a good FD example for me and I think I
understood it at least for FD discretization.
Best,
Qw

Matthew Knepley <knepley at gmail.com> 于2021年11月16日周二 下午7:11写道：

> On Mon, Nov 15, 2021 at 10:24 PM zhfreewill <zhfreewill at gmail.com> wrote:
>
>> Hi,
>> I'm new to PETSc. I have been confused about how to apply boundary
>> conditions (BCs) (e.g., the Dirichlet and the Neumann type) when using the
>> time-stepping (TS) object of PETSc.
>>
>> I have read the example codes (
>> https://petsc.org/release/src/ts/tutorials/index.html) and vaguely found
>> that BCs seems to be set  as follows:
>>
>> 1. when evaluating a matrix of the right hand side function g(x) (e.g.,
>> using a function like RHSMatrixHeat)
>> 2. when evaluating a matrix of the Jacobian matrix of g'(x) (e.g., using
>> a function like FormJacobian)
>>
>> For instance, in the ex1.4 (
>> https://petsc.org/release/src/ts/tutorials/ex4.c.html), with comments at
>> the beginning states
>>
>>  17: /*
>> ------------------------------------------------------------------------
>>  19:    This program solves the one-dimensional heat equation (also
>> called the
>>  20:    diffusion equation),
>>  21:        u_t = u_xx,
>>  22:    on the domain 0 <= x <= 1, with the boundary conditions
>>  23:        u(t,0) = 0, u(t,1) = 0,
>>  24:    and the initial condition
>>  25:        u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
>>  26:    This is a linear, second-order, parabolic equation.
>>
>>  28:    We discretize the right-hand side using finite differences with
>>  29:    uniform grid spacing h:
>>  30:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
>> ...
>>  47:
>> ------------------------------------------------------------------------- */
>>
>>  and Dirichlet BCs, u(t,0) = 0 and u(t,1) = 0, are to set on the both
>> side of the discretized domain, the corresponding codes can be found on the
>> following lines in a function RHSMatrixHeat:
>>
>> /*
>> ------------------------------------------------------------------------- */
>> 463: RHSMatrixHeat - User-provided routine to compute the right-hand-side
>> matrix for the heat equation. */
>> 491: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat AA,Mat
>> BB,void *ctx)
>> 492: {
>> ...
>> 505:   /* Set matrix rows corresponding to boundary data */
>> 509:   if (mstart == 0) {  /* first processor only */
>> 510:     v[0] = 1.0;
>> 511:     MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
>> 512:     mstart++;
>> 513:   }
>> 515:   if (mend == appctx->m) { /* last processor only */
>> 516:     mend--;
>> 517:     v[0] = 1.0;
>> 518:     MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
>> 519:   }
>>
>> 521:   /* Set matrix rows corresponding to interior data.  We construct
>> the matrix one row at a time. */
>> 525:   v[0] = sone; v[1] = stwo; v[2] = sone;
>> 526:   for (i=mstart; i<mend; i++) {
>> 527:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
>> 528:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
>> 529:   }
>> ...
>> 550: }
>> /*
>> ------------------------------------------------------------------------- */
>>
>>
>> My questions are:
>>
>> 1. How do these lines of code embody the Dirichlet BCs u(t,0) = 0 and
>> u(t,1) = 0.
>>
>
> This is the finite difference version of BC. It is just moving the term
> for a known value to the forcing. However, since
> the BC is 0, then we just do not compute the term. If the BC were nonzero,
> you would also see the term in the
> residual. For example,
>
>
> https://gitlab.com/petsc/petsc/-/blob/main/src/snes/tutorials/ex19.c#L271
>
>
>> 2. Similarly, if I want to apply a Neumann type BCs in these lines, How
>> can I do?
>>
>
> For finite differences, you usually just approximate the derivative as the
> one-sided difference of the last two values,
> and set this equal to the known value. For finite elements, Neumann
> conditions are just an addition weak form
> integrated over the boundary.
>
>   Thanks,
>
>      Matt
>
>
>> Any suggestions or documents?
>>
>> Best,
>> Qw
>>
>
>
> --
> What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which their
> experiments lead.
> -- Norbert Wiener
>
> https://www.cse.buffalo.edu/~knepley/
> <http://www.cse.buffalo.edu/~knepley/>
>
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