! Time-dependent advection-reaction PDE in 1d. Demonstrates IMEX methods
!
!   u_t + a1*u_x = -k1*u + k2*v + s1
!   v_t + a2*v_x = k1*u - k2*v + s2
!   0 < x < 1
!   a1 = 1, k1 = 10^6, s1 = 0,
!   a2 = 0, k2 = 2*k1, s2 = 1
!
!   Initial conditions:
!   u(x,0) = 1 + s2*x
!   v(x,0) = k0/k1*u(x,0) + s1/k1
!
!   Upstream boundary conditions:
!   u(0,t) = 1-sin(12*t)^4
!

#include <petsc/finclude/petscts.h>
#include <petsc/finclude/petscdmda.h>
program main
  use petscts
  implicit none

!
!  Create an application context to contain data needed by the
!  application-provided call-back routines, FormJacobian() and
!  FormFunction(). We use a double precision array with six
!  entries, two for each problem parameter a, k, s.
!
  PetscReal user(6)
  integer user_a, user_k, user_s
  parameter(user_a=0, user_k=2, user_s=4)

  TS ts
  SNES snes
  SNESLineSearch linesearch
  Vec X
  Mat J
  PetscInt mx
  PetscErrorCode ierr
  DM da
  PetscReal ftime, dt
  PetscReal one, pone
  PetscInt im11, i2
  PetscBool flg

  im11 = 11
  i2 = 2
  one = 1.0
  pone = one/10

  PetscCallA(PetscInitialize(ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Create distributed array (DMDA) to manage parallel grid and vectors
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, im11, i2, i2, PETSC_NULL_INTEGER, da, ierr))
  PetscCallA(DMSetFromOptions(da, ierr))
  PetscCallA(DMSetUp(da, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!    Extract global vectors from DMDA
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(DMCreateGlobalVector(da, X, ierr))

! Initialize user application context
! Use zero-based indexing for command line parameters to match ex22.c
  user(user_a + 1) = 1.0
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-a0', user(user_a + 1), flg, ierr))
  user(user_a + 2) = 0.0
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-a1', user(user_a + 2), flg, ierr))
  user(user_k + 1) = 1000000.0
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-k0', user(user_k + 1), flg, ierr))
  user(user_k + 2) = 2*user(user_k + 1)
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-k1', user(user_k + 2), flg, ierr))
  user(user_s + 1) = 0.0
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-s0', user(user_s + 1), flg, ierr))
  user(user_s + 2) = 1.0
  PetscCallA(PetscOptionsGetReal(PETSC_NULL_OPTIONS, PETSC_NULL_CHARACTER, '-s1', user(user_s + 2), flg, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!    Create timestepping solver context
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(TSCreate(PETSC_COMM_WORLD, ts, ierr))
  PetscCallA(TSSetDM(ts, da, ierr))
  PetscCallA(TSSetType(ts, TSARKIMEX, ierr))
  PetscCallA(TSSetRHSFunction(ts, PETSC_NULL_VEC, FormRHSFunction, user, ierr))
  PetscCallA(TSSetIFunction(ts, PETSC_NULL_VEC, FormIFunction, user, ierr))
  PetscCallA(DMSetMatType(da, MATAIJ, ierr))
  PetscCallA(DMCreateMatrix(da, J, ierr))
  PetscCallA(TSSetIJacobian(ts, J, J, FormIJacobian, user, ierr))

  PetscCallA(TSGetSNES(ts, snes, ierr))
  PetscCallA(SNESGetLineSearch(snes, linesearch, ierr))
  PetscCallA(SNESLineSearchSetType(linesearch, SNESLINESEARCHBASIC, ierr))

  ftime = 1.0
  PetscCallA(TSSetMaxTime(ts, ftime, ierr))
  PetscCallA(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Set initial conditions
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(FormInitialSolution(ts, X, user, ierr))
  PetscCallA(TSSetSolution(ts, X, ierr))
  PetscCallA(VecGetSize(X, mx, ierr))
!  Advective CFL, I don't know why it needs so much safety factor.
  dt = pone*max(user(user_a + 1), user(user_a + 2))/mx
  PetscCallA(TSSetTimeStep(ts, dt, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!   Set runtime options
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(TSSetFromOptions(ts, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Solve nonlinear system
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(TSSolve(ts, X, ierr))

! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!  Free work space.
! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  PetscCallA(MatDestroy(J, ierr))
  PetscCallA(VecDestroy(X, ierr))
  PetscCallA(TSDestroy(ts, ierr))
  PetscCallA(DMDestroy(da, ierr))
  PetscCallA(PetscFinalize(ierr))
contains

! Small helper to extract the layout, result uses 1-based indexing.
  subroutine GetLayout(da, mx, xs, xe, gxs, gxe, ierr)
    use petscdm
    implicit none

    DM da
    PetscInt mx, xs, xe, gxs, gxe
    PetscErrorCode ierr
    PetscInt xm, gxm
    PetscCall(DMDAGetInfo(da, PETSC_NULL_INTEGER, mx, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, PETSC_NULL_ENUM, PETSC_NULL_ENUM, PETSC_NULL_ENUM, PETSC_NULL_ENUM, ierr))
    PetscCall(DMDAGetCorners(da, xs, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, xm, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, ierr))
    PetscCall(DMDAGetGhostCorners(da, gxs, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, gxm, PETSC_NULL_INTEGER, PETSC_NULL_INTEGER, ierr))
    xs = xs + 1
    gxs = gxs + 1
    xe = xs + xm - 1
    gxe = gxs + gxm - 1
  end subroutine

  subroutine FormIFunctionLocal(mx, xs, xe, gxs, gxe, x, xdot, f, a, k, s, ierr)
    implicit none
    PetscInt mx, xs, xe, gxs, gxe
    PetscScalar x(2, xs:xe)
    PetscScalar xdot(2, xs:xe)
    PetscScalar f(2, xs:xe)
    PetscReal a(2), k(2), s(2)
    PetscErrorCode ierr
    PetscInt i
    do i = xs, xe
      f(1, i) = xdot(1, i) + k(1)*x(1, i) - k(2)*x(2, i) - s(1)
      f(2, i) = xdot(2, i) - k(1)*x(1, i) + k(2)*x(2, i) - s(2)
    end do
  end subroutine

  subroutine FormIFunction(ts, t, X, Xdot, F, user, ierr)
    use petscts
    implicit none

    TS ts
    PetscReal t
    Vec X, Xdot, F
    PetscReal user(6)
    PetscErrorCode ierr
    integer user_a, user_k, user_s
    parameter(user_a=1, user_k=3, user_s=5)

    DM da
    PetscInt mx, xs, xe, gxs, gxe
    PetscScalar, pointer :: xx(:), xxdot(:), ff(:)

    PetscCall(TSGetDM(ts, da, ierr))
    PetscCall(GetLayout(da, mx, xs, xe, gxs, gxe, ierr))

! Get access to vector data
    PetscCall(VecGetArrayRead(X, xx, ierr))
    PetscCall(VecGetArrayRead(Xdot, xxdot, ierr))
    PetscCall(VecGetArray(F, ff, ierr))

    PetscCall(FormIFunctionLocal(mx, xs, xe, gxs, gxe, xx, xxdot, ff, user(user_a), user(user_k), user(user_s), ierr))

    PetscCall(VecRestoreArrayRead(X, xx, ierr))
    PetscCall(VecRestoreArrayRead(Xdot, xxdot, ierr))
    PetscCall(VecRestoreArray(F, ff, ierr))
  end subroutine

  subroutine FormRHSFunctionLocal(mx, xs, xe, gxs, gxe, t, x, f, a, k, s, ierr)
    implicit none
    PetscInt mx, xs, xe, gxs, gxe
    PetscReal t
    PetscScalar x(2, gxs:gxe), f(2, xs:xe)
    PetscReal a(2), k(2), s(2)
    PetscErrorCode ierr
    PetscInt i, j
    PetscReal hx, u0t(2)
    PetscReal one, two, three, four, six, twelve
    PetscReal half, third, twothird, sixth
    PetscReal twelfth

    one = 1.0
    two = 2.0
    three = 3.0
    four = 4.0
    six = 6.0
    twelve = 12.0
    hx = one/mx
!     The Fortran standard only allows positive base for power functions; Nag compiler fails on this
    u0t(1) = one - abs(sin(twelve*t))**four
    u0t(2) = 0.0
    half = one/two
    third = one/three
    twothird = two/three
    sixth = one/six
    twelfth = one/twelve
    do i = xs, xe
      do j = 1, 2
        if (i == 1) then
          f(j, i) = a(j)/hx*(third*u0t(j) + half*x(j, i) - x(j, i + 1)  &
&              + sixth*x(j, i + 2))
        else if (i == 2) then
          f(j, i) = a(j)/hx*(-twelfth*u0t(j) + twothird*x(j, i - 1)    &
&              - twothird*x(j, i + 1) + twelfth*x(j, i + 2))
        else if (i == mx - 1) then
          f(j, i) = a(j)/hx*(-sixth*x(j, i - 2) + x(j, i - 1)             &
&         - half*x(j, i) - third*x(j, i + 1))
        else if (i == mx) then
          f(j, i) = a(j)/hx*(-x(j, i) + x(j, i - 1))
        else
          f(j, i) = a(j)/hx*(-twelfth*x(j, i - 2)                      &
&              + twothird*x(j, i - 1)                                 &
&              - twothird*x(j, i + 1) + twelfth*x(j, i + 2))
        end if
      end do
    end do
  end subroutine

  subroutine FormRHSFunction(ts, t, X, F, user, ierr)
    use petscts
    implicit none

    TS ts
    PetscReal t
    Vec X, F
    PetscReal user(6)
    PetscErrorCode ierr
    integer user_a, user_k, user_s
    parameter(user_a=1, user_k=3, user_s=5)
    DM da
    Vec Xloc
    PetscInt mx, xs, xe, gxs, gxe
    PetscScalar, pointer :: xx(:), ff(:)

    PetscCall(TSGetDM(ts, da, ierr))
    PetscCall(GetLayout(da, mx, xs, xe, gxs, gxe, ierr))

!     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(DMGetLocalVector(da, Xloc, ierr))
    PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, Xloc, ierr))
    PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, Xloc, ierr))

! Get access to vector data
    PetscCall(VecGetArrayRead(Xloc, xx, ierr))
    PetscCall(VecGetArray(F, ff, ierr))

    PetscCall(FormRHSFunctionLocal(mx, xs, xe, gxs, gxe, t, xx, ff, user(user_a), user(user_k), user(user_s), ierr))

    PetscCall(VecRestoreArrayRead(Xloc, xx, ierr))
    PetscCall(VecRestoreArray(F, ff, ierr))
    PetscCall(DMRestoreLocalVector(da, Xloc, ierr))
  end subroutine

! ---------------------------------------------------------------------
!
!  IJacobian - Compute IJacobian = dF/dU + shift*dF/dUdot
!
  subroutine FormIJacobian(ts, t, X, Xdot, shift, J, Jpre, user, ierr)
    use petscts
    implicit none

    TS ts
    PetscReal t, shift
    Vec X, Xdot
    Mat J, Jpre
    PetscReal user(6)
    PetscErrorCode ierr
    integer user_a, user_k, user_s
    parameter(user_a=0, user_k=2, user_s=4)

    DM da
    PetscInt mx, xs, xe, gxs, gxe
    PetscInt i, i1, row, col
    PetscReal k1, k2
    PetscScalar val(4)

    PetscCall(TSGetDM(ts, da, ierr))
    PetscCall(GetLayout(da, mx, xs, xe, gxs, gxe, ierr))

    i1 = 1
    k1 = user(user_k + 1)
    k2 = user(user_k + 2)
    do i = xs, xe
      row = i - gxs
      col = i - gxs
      val(1) = shift + k1
      val(2) = -k2
      val(3) = -k1
      val(4) = shift + k2
      PetscCall(MatSetValuesBlockedLocal(Jpre, i1, [row], i1, [col], val, INSERT_VALUES, ierr))
    end do
    PetscCall(MatAssemblyBegin(Jpre, MAT_FINAL_ASSEMBLY, ierr))
    PetscCall(MatAssemblyEnd(Jpre, MAT_FINAL_ASSEMBLY, ierr))
    if (J /= Jpre) then
      PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY, ierr))
      PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY, ierr))
    end if
  end subroutine

  subroutine FormInitialSolutionLocal(mx, xs, xe, gxs, gxe, x, a, k, s, ierr)
    implicit none
    PetscInt mx, xs, xe, gxs, gxe
    PetscScalar x(2, xs:xe)
    PetscReal a(2), k(2), s(2)
    PetscErrorCode ierr

    PetscInt i
    PetscReal one, hx, r, ik
    one = 1.0
    hx = one/mx
    do i = xs, xe
      r = i*hx
      if (k(2) /= 0.0) then
        ik = one/k(2)
      else
        ik = one
      end if
      x(1, i) = one + s(2)*r
      x(2, i) = k(1)*ik*x(1, i) + s(2)*ik
    end do
  end subroutine

  subroutine FormInitialSolution(ts, X, user, ierr)
    use petscts
    implicit none

    TS ts
    Vec X
    PetscReal user(6)
    PetscErrorCode ierr
    integer user_a, user_k, user_s
    parameter(user_a=1, user_k=3, user_s=5)

    DM da
    PetscInt mx, xs, xe, gxs, gxe
    PetscScalar, pointer :: xx(:)

    PetscCall(TSGetDM(ts, da, ierr))
    PetscCall(GetLayout(da, mx, xs, xe, gxs, gxe, ierr))

! Get access to vector data
    PetscCall(VecGetArray(X, xx, ierr))

    PetscCall(FormInitialSolutionLocal(mx, xs, xe, gxs, gxe, xx, user(user_a), user(user_k), user(user_s), ierr))

    PetscCall(VecRestoreArray(X, xx, ierr))
  end subroutine
end program
!/*TEST
!
!    test:
!      args: -da_grid_x 200 -ts_arkimex_type 4
!      requires: !single
!      output_file: output/empty.out
!
!TEST*/