Some questions about external_petsc_problem1.0?

[Wang-Yihu]

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Question

Q1: What does the problem the \modules\external_petsc_solver\test\tests\external_petsc_problem\petsc_transient_as_sub.i describe?
Q2: What makes the Picard iteration? What's the meaning of the Picard residual in the moose_as_parent.i and petsc_transient_as_sub.i?

Recently, I still study the code in \modules\external_petsc_solver\. I encounter some puzzles...
1.
It seems that the moose_as_parent.i tells a story of

∂u/∂t - ∇·∇u – 10000v = 0

And v comes from the transferred value from the u in petsc_transient_as_sub.i. But what does the petsc_transient_as_sub.i say? From the code comment of PETScDiffusionFDM.C, it seems that the govern function is

∂u/∂t = ∂2u/∂x2 + ∂2u/∂y2

However, I have two questions from its other comments.
1.1 the comment says At t=0: u(x,y) = exp(crrr), if r=PetscSqrtReal((x-.5)(x-.5) + (y-.5)*(y-.5)) < .125. If that is right, at t = 0, x = 0.5, y = 0.5, the u must be 1, right? However, the initial condition of u at the center point from the *.e file(moose_as_parent_out_sub_app0.e) generated by MOOSE is 0 not 1... What's the wrong with it?
1.2 the comment also says that

Boundary conditions:
Drichlet BC:
At x=0, x=1, y=0, y=1: u = 0.0

Neumann BC:
At x=0, x=1: du(x,y,t)/dx = 0
At y=0, y=1: du(x,y,t)/dy = 0

I also find similar things in the code PETScDiffusionFDM.C

 for (j = ys; j < ys + ym; j++)
  {
    for (i = xs; i < xs + xm; i++)
    {
      nc = 0;
      row.j = j;
      row.i = i;
      if (PETSC_TRUE && (i == 0 || i == Mx - 1 || j == 0 || j == My - 1))
      {
        col[nc].j = j;
        col[nc].i = i;
        vals[nc++] = 1.0;
      }
      else if (PETSC_FALSE && i == 0)
      { /* Left Neumann */
        col[nc].j = j;
        col[nc].i = i;
        vals[nc++] = 1.0;
        col[nc].j = j;
        col[nc].i = i + 1;
        vals[nc++] = -1.0;
      }
      else if (PETSC_FALSE && i == Mx - 1)
      { /* Right Neumann */
        col[nc].j = j;
        col[nc].i = i;
        vals[nc++] = 1.0;
        col[nc].j = j;
        col[nc].i = i - 1;
        vals[nc++] = -1.0;
      }
      else if (PETSC_FALSE && j == 0)
      { /* Bottom Neumann */
        col[nc].j = j;
        col[nc].i = i;
        vals[nc++] = 1.0;
        col[nc].j = j + 1;
        col[nc].i = i;
        vals[nc++] = -1.0;
      }
      else if (PETSC_FALSE && j == My - 1)
      { /* Top Neumann */
        col[nc].j = j;
        col[nc].i = i;
        vals[nc++] = 1.0;
        col[nc].j = j - 1;
        col[nc].i = i;
        vals[nc++] = -1.0;
      }
      else
      { /* Interior */
        col[nc].j = j - 1;
        col[nc].i = i;
        vals[nc++] = -sy;
        col[nc].j = j;
        col[nc].i = i - 1;
        vals[nc++] = -sx;
        col[nc].j = j;
        col[nc].i = i;
        vals[nc++] = 2.0 * (sx + sy) + a;
        col[nc].j = j;
        col[nc].i = i + 1;
        vals[nc++] = -sx;
        col[nc].j = j + 1;
        col[nc].i = i;
        vals[nc++] = -sy;
      }
      ierr = MatSetValuesStencil(Jpre, 1, &row, nc, col, vals, INSERT_VALUES);
      CHKERRQ(ierr);
    }
  }

However, I find there is a PETSC_FALSE near these Neumaan comment, it seems that program statements of else if (PETSC_FALSE && i == 0), else if (PETSC_FALSE && i == Mx - 1), else if (PETSC_FALSE && j == 0), else if (PETSC_FALSE && j == My - 1), will not be executed by default... is this a paradox???
I will show Q2 in another discusssion...

[GiudGiud]

Q1: What does the problem the \modules\external_petsc_solver\test\tests\external_petsc_problem\petsc_transient_as_sub.i describe?

You have to look into the ExternalPetscProblem class to see what it's doing. It's doing implicit Euler for time integration.
then there is a call to externalPETScDiffusionFDMSolve which is defined here
https://github.com/idaholab/moose/blob/next/modules/external_petsc_solver/src/petscsolver/PETScDiffusionFDM.C

It's a finite difference diffusion problem.

What makes the Picard iteration? What's the meaning of the Picard residual in the moose_as_parent.i and petsc_transient_as_sub.i?

See the other post. Picard iteration is from the FixedPointSolve created in the main app

he comment says At t=0: u(x,y) = exp(crrr), if r=PetscSqrtReal((x-.5)(x-.5) + (y-.5)*(y-.5)) < .125. If that is right, at t = 0, x = 0.5, y = 0.5, the u must be 1, right? However, the initial condition of u at the center point from the *.e file(moose_as_parent_out_sub_app0.e) generated by MOOSE is 0 not 1... What's the wrong with it?

Dont read too much into the comments. The code might have evolved without updating the comments. This is one of the main flaws of commenting code.