xref: /petsc/src/snes/tutorials/ex5f90.F90 (revision c5e229c2f66f66995aed5443a26600af2aec4a3f)

!
!  Description: Solves a nonlinear system in parallel with SNES.
!  We solve the  Bratu (SFI - solid fuel ignition) problem in a 2D rectangular
!  domain, using distributed arrays (DMDAs) to partition the parallel grid.
!  The command line options include:
!    -par <parameter>, where <parameter> indicates the nonlinearity of the problem
!       problem SFI:  <parameter> = Bratu parameter (0 <= par <= 6.81)
!

!
!  --------------------------------------------------------------------------
!
!  Solid Fuel Ignition (SFI) problem.  This problem is modeled by
!  the partial differential equation
!
!          -Laplacian u - lambda*exp(u) = 0,  0 < x,y < 1,
!
!  with boundary conditions
!
!           u = 0  for  x = 0, x = 1, y = 0, y = 1.
!
!  A finite difference approximation with the usual 5-point stencil
!  is used to discretize the boundary value problem to obtain a nonlinear
!  system of equations.
!
!  The uniprocessor version of this code is snes/tutorials/ex4f.F
!
!  --------------------------------------------------------------------------
!  The following define must be used before including any PETSc include files
!  into a module or interface. This is because they can't handle declarations
!  in them
!
#include <petsc/finclude/petscsnes.h>
#include <petsc/finclude/petscdmda.h>
module ex5f90module
  use petscsnes
  use petscdmda
  type userctx
    PetscInt xs, xe, xm, gxs, gxe, gxm
    PetscInt ys, ye, ym, gys, gye, gym
    PetscInt mx, my
    PetscMPIInt rank
    PetscReal lambda
  end type userctx

contains
! ---------------------------------------------------------------------
!
!  FormFunction - Evaluates nonlinear function, F(x).
!
!  Input Parameters:
!  snes - the SNES context
!  X - input vector
!  dummy - optional user-defined context, as set by SNESSetFunction()
!          (not used here)
!
!  Output Parameter:
!  F - function vector
!
!  Notes:
!  This routine serves as a wrapper for the lower-level routine
!  "FormFunctionLocal", where the actual computations are
!  done using the standard Fortran style of treating the local
!  vector data as a multidimensional array over the local mesh.
!  This routine merely handles ghost point scatters and accesses
!  the local vector data via VecGetArray() and VecRestoreArray().
!
  subroutine FormFunction(snes, X, F, user, ierr)
    implicit none

!  Input/output variables:
    SNES snes
    Vec X, F
    PetscErrorCode ierr
    type(userctx) user
    DM da

!  Declarations for use with local arrays:
    PetscScalar, pointer :: lx_v(:), lf_v(:)
    Vec localX

!  Scatter ghost points to local vector, using the 2-step process
!     DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
!  By placing code between these two statements, computations can
!  be done while messages are in transition.
    PetscCall(SNESGetDM(snes, da, ierr))
    PetscCall(DMGetLocalVector(da, localX, ierr))
    PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX, ierr))
    PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX, 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 interface to VecGetArray() differs from VecGetArray().

    PetscCall(VecGetArray(localX, lx_v, ierr))
    PetscCall(VecGetArray(F, lf_v, ierr))

!  Compute function over the locally owned part of the grid
    PetscCall(FormFunctionLocal(lx_v, lf_v, user, ierr))

!  Restore vectors
    PetscCall(VecRestoreArray(localX, lx_v, ierr))
    PetscCall(VecRestoreArray(F, lf_v, ierr))

!  Insert values into global vector

    PetscCall(DMRestoreLocalVector(da, localX, ierr))
    PetscCall(PetscLogFlops(11.0d0*user%ym*user%xm, ierr))

!      PetscCallA(VecView(X,PETSC_VIEWER_STDOUT_WORLD,ierr))
!      PetscCallA(VecView(F,PETSC_VIEWER_STDOUT_WORLD,ierr))
  end subroutine formfunction
end module ex5f90module

module ex5f90moduleinterfaces
  use ex5f90module

  Interface SNESSetApplicationContext
    Subroutine SNESSetApplicationContext(snes, ctx, ierr)
      use ex5f90module
      SNES snes
      type(userctx) ctx
      PetscErrorCode ierr
    End Subroutine
  End Interface SNESSetApplicationContext

  Interface SNESGetApplicationContext
    Subroutine SNESGetApplicationContext(snes, ctx, ierr)
      use ex5f90module
      SNES snes
      type(userctx), pointer :: ctx
      PetscErrorCode ierr
    End Subroutine
  End Interface SNESGetApplicationContext
end module ex5f90moduleinterfaces

program main
  use ex5f90module
  use ex5f90moduleinterfaces
  implicit none
!

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!                   Variable declarations
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
!  Variables:
!     snes        - nonlinear solver
!     x, r        - solution, residual vectors
!     J           - Jacobian matrix
!     its         - iterations for convergence
!     Nx, Ny      - number of preocessors in x- and y- directions
!     matrix_free - flag - 1 indicates matrix-free version
!
  SNES snes
  Vec x, r
  Mat J
  PetscErrorCode ierr
  PetscInt its
  PetscBool flg, matrix_free
  PetscInt ione, nfour
  PetscReal lambda_max, lambda_min
  type(userctx) user
  DM da

!  Note: Any user-defined Fortran routines (such as FormJacobian)
!  MUST be declared as external.
  external FormInitialGuess, FormJacobian

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Initialize program
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(PetscInitialize(ierr))
  PetscCallMPIA(MPI_Comm_rank(PETSC_COMM_WORLD, user%rank, ierr))

!  Initialize problem parameters
  lambda_max = 6.81
  lambda_min = 0.0
  user%lambda = 6.0
  ione = 1
  nfour = 4
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-par', user%lambda, flg, ierr))
  PetscCheckA(user%lambda < lambda_max .and. user%lambda > lambda_min, PETSC_COMM_SELF, PETSC_ERR_USER, 'Lambda provided with -par is out of range')

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create nonlinear solver context
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(SNESCreate(PETSC_COMM_WORLD, snes, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create vector data structures; set function evaluation routine
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

!  Create distributed array (DMDA) to manage parallel grid and vectors

! This really needs only the star-type stencil, but we use the box
! stencil temporarily.
  PetscCallA(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_BOX, nfour, nfour, PETSC_DECIDE, PETSC_DECIDE, ione, ione, PETSC_NULL_INTEGER_ARRAY, PETSC_NULL_INTEGER_ARRAY, da, ierr))
  PetscCallA(DMSetFromOptions(da, ierr))
  PetscCallA(DMSetUp(da, ierr))

  PetscCallA(DMDAGetInfo(da, PETSC_NULL_INTEGER, user%mx, user%my, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_DMBOUNDARYTYPE, PETSC_NULL_DMBOUNDARYTYPE, PETSC_NULL_DMBOUNDARYTYPE, PETSC_NULL_DMDASTENCILTYPE, ierr))

!
!   Visualize the distribution of the array across the processors
!
!     PetscCallA(DMView(da,PETSC_VIEWER_DRAW_WORLD,ierr))

!  Extract global and local vectors from DMDA; then duplicate for remaining
!  vectors that are the same types
  PetscCallA(DMCreateGlobalVector(da, x, ierr))
  PetscCallA(VecDuplicate(x, r, ierr))

!  Get local grid boundaries (for 2-dimensional DMDA)
  PetscCallA(DMDAGetCorners(da, user%xs, user%ys, PETSC_NULL_INTEGER, user%xm, user%ym, PETSC_NULL_INTEGER, ierr))
  PetscCallA(DMDAGetGhostCorners(da, user%gxs, user%gys, PETSC_NULL_INTEGER, user%gxm, user%gym, PETSC_NULL_INTEGER, ierr))

!  Here we shift the starting indices up by one so that we can easily
!  use the Fortran convention of 1-based indices (rather 0-based indices).
  user%xs = user%xs + 1
  user%ys = user%ys + 1
  user%gxs = user%gxs + 1
  user%gys = user%gys + 1

  user%ye = user%ys + user%ym - 1
  user%xe = user%xs + user%xm - 1
  user%gye = user%gys + user%gym - 1
  user%gxe = user%gxs + user%gxm - 1

  PetscCallA(SNESSetApplicationContext(snes, user, ierr))

!  Set function evaluation routine and vector
  PetscCallA(SNESSetFunction(snes, r, FormFunction, user, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create matrix data structure; set Jacobian evaluation routine
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

!  Set Jacobian matrix data structure and default Jacobian evaluation
!  routine. User can override with:
!     -snes_fd : default finite differencing approximation of Jacobian
!     -snes_mf : matrix-free Newton-Krylov method with no preconditioning
!                (unless user explicitly sets preconditioner)
!     -snes_mf_operator : form matrix used to construct the preconditioner as set by the user,
!                         but use matrix-free approx for Jacobian-vector
!                         products within Newton-Krylov method
!
!  Note:  For the parallel case, vectors and matrices MUST be partitioned
!     accordingly.  When using distributed arrays (DMDAs) to create vectors,
!     the DMDAs determine the problem partitioning.  We must explicitly
!     specify the local matrix dimensions upon its creation for compatibility
!     with the vector distribution.  Thus, the generic MatCreate() routine
!     is NOT sufficient when working with distributed arrays.
!
!     Note: Here we only approximately preallocate storage space for the
!     Jacobian.  See the users manual for a discussion of better techniques
!     for preallocating matrix memory.

  PetscCallA(PetscOptionsHasName(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-snes_mf', matrix_free, ierr))
  if (.not. matrix_free) then
    PetscCallA(DMSetMatType(da, MATAIJ, ierr))
    PetscCallA(DMCreateMatrix(da, J, ierr))
    PetscCallA(SNESSetJacobian(snes, J, J, FormJacobian, user, ierr))
  end if

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Customize nonlinear solver; set runtime options
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Set runtime options (e.g., -snes_monitor -snes_rtol <rtol> -ksp_type <type>)
  PetscCallA(SNESSetDM(snes, da, ierr))
  PetscCallA(SNESSetFromOptions(snes, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Evaluate initial guess; then solve nonlinear system.
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Note: The user should initialize the vector, x, with the initial guess
!  for the nonlinear solver prior to calling SNESSolve().  In particular,
!  to employ an initial guess of zero, the user should explicitly set
!  this vector to zero by calling VecSet().

  PetscCallA(FormInitialGuess(snes, x, ierr))
  PetscCallA(SNESSolve(snes, PETSC_NULL_VEC, x, ierr))
  PetscCallA(SNESGetIterationNumber(snes, its, ierr))
  if (user%rank == 0) then
    write (6, 100) its
  end if
100 format('Number of SNES iterations = ', i5)

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Free work space.  All PETSc objects should be destroyed when they
!  are no longer needed.
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  if (.not. matrix_free) PetscCallA(MatDestroy(J, ierr))
  PetscCallA(VecDestroy(x, ierr))
  PetscCallA(VecDestroy(r, ierr))
  PetscCallA(SNESDestroy(snes, ierr))
  PetscCallA(DMDestroy(da, ierr))

  PetscCallA(PetscFinalize(ierr))
end

! ---------------------------------------------------------------------
!
!  FormInitialGuess - Forms initial approximation.
!
!  Input Parameters:
!  X - vector
!
!  Output Parameter:
!  X - vector
!
!  Notes:
!  This routine serves as a wrapper for the lower-level routine
!  "InitialGuessLocal", where the actual computations are
!  done using the standard Fortran style of treating the local
!  vector data as a multidimensional array over the local mesh.
!  This routine merely handles ghost point scatters and accesses
!  the local vector data via VecGetArray() and VecRestoreArray().
!
subroutine FormInitialGuess(snes, X, ierr)
  use ex5f90module
  use ex5f90moduleinterfaces
  implicit none

!  Input/output variables:
  SNES snes
  type(userctx), pointer:: puser
  Vec X
  PetscErrorCode ierr
  DM da

!  Declarations for use with local arrays:
  PetscScalar, pointer :: lx_v(:)

  ierr = 0
  PetscCallA(SNESGetDM(snes, da, ierr))
  PetscCallA(SNESGetApplicationContext(snes, puser, 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 interface to VecGetArray() differs from VecGetArray().

  PetscCallA(VecGetArray(X, lx_v, ierr))

!  Compute initial guess over the locally owned part of the grid
  PetscCallA(InitialGuessLocal(puser, lx_v, ierr))

!  Restore vector
  PetscCallA(VecRestoreArray(X, lx_v, ierr))

!  Insert values into global vector

end

! ---------------------------------------------------------------------
!
!  InitialGuessLocal - Computes initial approximation, called by
!  the higher level routine FormInitialGuess().
!
!  Input Parameter:
!  x - local vector data
!
!  Output Parameters:
!  x - local vector data
!  ierr - error code
!
!  Notes:
!  This routine uses standard Fortran-style computations over a 2-dim array.
!
subroutine InitialGuessLocal(user, x, ierr)
  use ex5f90module
  implicit none

!  Input/output variables:
  type(userctx) user
  PetscScalar x(user%xs:user%xe, user%ys:user%ye)
  PetscErrorCode ierr

!  Local variables:
  PetscInt i, j
  PetscReal temp1, temp, hx, hy
  PetscReal one

!  Set parameters

  ierr = 0
  one = 1.0
  hx = one/(user%mx - 1)
  hy = one/(user%my - 1)
  temp1 = user%lambda/(user%lambda + one)

  do 20 j = user%ys, user%ye
    temp = min(j - 1, user%my - j)*hy
    do 10 i = user%xs, user%xe
      if (i == 1 .or. j == 1 .or. i == user%mx .or. j == user%my) then
        x(i, j) = 0.0
      else
        x(i, j) = temp1*sqrt(min(hx*min(i - 1, user%mx - i), temp))
      end if
10    continue
20    continue

    end

! ---------------------------------------------------------------------
!
!  FormFunctionLocal - Computes nonlinear function, called by
!  the higher level routine FormFunction().
!
!  Input Parameter:
!  x - local vector data
!
!  Output Parameters:
!  f - local vector data, f(x)
!  ierr - error code
!
!  Notes:
!  This routine uses standard Fortran-style computations over a 2-dim array.
!
    subroutine FormFunctionLocal(x, f, user, ierr)
      use ex5f90module

      implicit none

!  Input/output variables:
      type(userctx) user
      PetscScalar x(user%gxs:user%gxe, user%gys:user%gye)
      PetscScalar f(user%xs:user%xe, user%ys:user%ye)
      PetscErrorCode ierr

!  Local variables:
      PetscScalar two, one, hx, hy, hxdhy, hydhx, sc
      PetscScalar u, uxx, uyy
      PetscInt i, j

      one = 1.0
      two = 2.0
      hx = one/(user%mx - 1)
      hy = one/(user%my - 1)
      sc = hx*hy*user%lambda
      hxdhy = hx/hy
      hydhx = hy/hx

!  Compute function over the locally owned part of the grid

      do 20 j = user%ys, user%ye
        do 10 i = user%xs, user%xe
          if (i == 1 .or. j == 1 .or. i == user%mx .or. j == user%my) then
            f(i, j) = x(i, j)
          else
            u = x(i, j)
            uxx = hydhx*(two*u - x(i - 1, j) - x(i + 1, j))
            uyy = hxdhy*(two*u - x(i, j - 1) - x(i, j + 1))
            f(i, j) = uxx + uyy - sc*exp(u)
          end if
10        continue
20        continue

        end

! ---------------------------------------------------------------------
!
!  FormJacobian - Evaluates Jacobian matrix.
!
!  Input Parameters:
!  snes     - the SNES context
!  x        - input vector
!  dummy    - optional user-defined context, as set by SNESSetJacobian()
!             (not used here)
!
!  Output Parameters:
!  jac      - Jacobian matrix
!  jac_prec - optionally different matrix used to construct the preconditioner (not used here)
!
!  Notes:
!  This routine serves as a wrapper for the lower-level routine
!  "FormJacobianLocal", where the actual computations are
!  done using the standard Fortran style of treating the local
!  vector data as a multidimensional array over the local mesh.
!  This routine merely accesses the local vector data via
!  VecGetArray() and VecRestoreArray().
!
!  Notes:
!  Due to grid point reordering with DMDAs, we must always work
!  with the local grid points, and then transform them to the new
!  global numbering with the "ltog" mapping
!  We cannot work directly with the global numbers for the original
!  uniprocessor grid!
!
!  Two methods are available for imposing this transformation
!  when setting matrix entries:
!    (A) MatSetValuesLocal(), using the local ordering (including
!        ghost points!)
!        - Set matrix entries using the local ordering
!          by calling MatSetValuesLocal()
!    (B) MatSetValues(), using the global ordering

!        - Set matrix entries using the global ordering by calling
!          MatSetValues()
!  Option (A) seems cleaner/easier in many cases, and is the procedure
!  used in this example.
!
        subroutine FormJacobian(snes, X, jac, jac_prec, user, ierr)
          use ex5f90module
          implicit none

!  Input/output variables:
          SNES snes
          Vec X
          Mat jac, jac_prec
          type(userctx) user
          PetscErrorCode ierr
          DM da

!  Declarations for use with local arrays:
          PetscScalar, pointer :: lx_v(:)
          Vec localX

!  Scatter ghost points to local vector, using the 2-step process
!     DMGlobalToLocalBegin(), DMGlobalToLocalEnd()
!  Computations can be done while messages are in transition,
!  by placing code between these two statements.

          PetscCallA(SNESGetDM(snes, da, ierr))
          PetscCallA(DMGetLocalVector(da, localX, ierr))
          PetscCallA(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX, ierr))
          PetscCallA(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX, ierr))

!  Get a pointer to vector data
          PetscCallA(VecGetArray(localX, lx_v, ierr))

!  Compute entries for the locally owned part of the Jacobian preconditioner.
          PetscCallA(FormJacobianLocal(lx_v, jac_prec, user, ierr))

!  Assemble matrix, using the 2-step process:
!     MatAssemblyBegin(), MatAssemblyEnd()
!  Computations can be done while messages are in transition,
!  by placing code between these two statements.

          PetscCallA(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY, ierr))
          if (jac /= jac_prec) then
            PetscCallA(MatAssemblyBegin(jac_prec, MAT_FINAL_ASSEMBLY, ierr))
          end if
          PetscCallA(VecRestoreArray(localX, lx_v, ierr))
          PetscCallA(DMRestoreLocalVector(da, localX, ierr))
          PetscCallA(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY, ierr))
          if (jac /= jac_prec) then
            PetscCallA(MatAssemblyEnd(jac_prec, MAT_FINAL_ASSEMBLY, ierr))
          end if

!  Tell the matrix we will never add a new nonzero location to the
!  matrix. If we do it will generate an error.

          PetscCallA(MatSetOption(jac, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE, ierr))

        end

! ---------------------------------------------------------------------
!
!  FormJacobianLocal - Computes Jacobian matrix used to compute the preconditioner,
!  called by the higher level routine FormJacobian().
!
!  Input Parameters:
!  x        - local vector data
!
!  Output Parameters:
!  jac_prec - Jacobian matrix used to compute the preconditioner
!  ierr     - error code
!
!  Notes:
!  This routine uses standard Fortran-style computations over a 2-dim array.
!
!  Notes:
!  Due to grid point reordering with DMDAs, we must always work
!  with the local grid points, and then transform them to the new
!  global numbering with the "ltog" mapping
!  We cannot work directly with the global numbers for the original
!  uniprocessor grid!
!
!  Two methods are available for imposing this transformation
!  when setting matrix entries:
!    (A) MatSetValuesLocal(), using the local ordering (including
!        ghost points!)
!        - Set matrix entries using the local ordering
!          by calling MatSetValuesLocal()
!    (B) MatSetValues(), using the global ordering
!        - Then apply this map explicitly yourself
!        - Set matrix entries using the global ordering by calling
!          MatSetValues()
!  Option (A) seems cleaner/easier in many cases, and is the procedure
!  used in this example.
!
        subroutine FormJacobianLocal(x, jac_prec, user, ierr)
          use ex5f90module
          implicit none

!  Input/output variables:
          type(userctx) user
          PetscScalar x(user%gxs:user%gxe, user%gys:user%gye)
          Mat jac_prec
          PetscErrorCode ierr

!  Local variables:
          PetscInt row, col(5), i, j
          PetscInt ione, ifive
          PetscScalar two, one, hx, hy, hxdhy
          PetscScalar hydhx, sc, v(5)

!  Set parameters
          ione = 1
          ifive = 5
          one = 1.0
          two = 2.0
          hx = one/(user%mx - 1)
          hy = one/(user%my - 1)
          sc = hx*hy
          hxdhy = hx/hy
          hydhx = hy/hx

!  Compute entries for the locally owned part of the Jacobian.
!   - Currently, all PETSc parallel matrix formats are partitioned by
!     contiguous chunks of rows across the processors.
!   - Each processor needs to insert only elements that it owns
!     locally (but any non-local elements will be sent to the
!     appropriate processor during matrix assembly).
!   - Here, we set all entries for a particular row at once.
!   - We can set matrix entries either using either
!     MatSetValuesLocal() or MatSetValues(), as discussed above.
!   - Note that MatSetValues() uses 0-based row and column numbers
!     in Fortran as well as in C.

          do 20 j = user%ys, user%ye
            row = (j - user%gys)*user%gxm + user%xs - user%gxs - 1
            do 10 i = user%xs, user%xe
              row = row + 1
!           boundary points
              if (i == 1 .or. j == 1 .or. i == user%mx .or. j == user%my) then
                col(1) = row
                v(1) = one
                PetscCallA(MatSetValuesLocal(jac_prec, ione, [row], ione, col, v, INSERT_VALUES, ierr))
!           interior grid points
              else
                v(1) = -hxdhy
                v(2) = -hydhx
                v(3) = two*(hydhx + hxdhy) - sc*user%lambda*exp(x(i, j))
                v(4) = -hydhx
                v(5) = -hxdhy
                col(1) = row - user%gxm
                col(2) = row - 1
                col(3) = row
                col(4) = row + 1
                col(5) = row + user%gxm
                PetscCallA(MatSetValuesLocal(jac_prec, ione, [row], ifive, col, v, INSERT_VALUES, ierr))
              end if
10            continue
20            continue

            end

!
!/*TEST
!
!   test:
!      nsize: 4
!      args: -snes_mf -pc_type none -da_processors_x 4 -da_processors_y 1 -snes_monitor_short -ksp_gmres_cgs_refinement_type refine_always
!      requires: !single
!
!   test:
!      suffix: 2
!      nsize: 4
!      args: -da_processors_x 2 -da_processors_y 2 -snes_monitor_short -ksp_gmres_cgs_refinement_type refine_always
!      requires: !single
!
!   test:
!      suffix: 3
!      nsize: 3
!      args: -snes_fd -snes_monitor_short -ksp_gmres_cgs_refinement_type refine_always
!      requires: !single
!
!   test:
!      suffix: 4
!      nsize: 3
!      args: -snes_mf_operator -snes_monitor_short -ksp_gmres_cgs_refinement_type refine_always
!      requires: !single
!
!   test:
!      suffix: 5
!      requires: !single
!
!TEST*/