poisson_basic.f90

Example showing how to use multigrid and compare with an analytic solution. A standard 5-point Laplacian is used.

#include "../src/cpp_macros.h"
 
program poisson_basic
  use m_af_all
  use m_gaussians
 
  implicit none
 
  integer, parameter :: box_size          = 16
  integer, parameter :: n_iterations      = 10
  integer            :: domain_size(NDIM)
  real(dp)           :: domain_len(NDIM)
  integer            :: i_phi
  integer            :: i_sol
  integer            :: i_rhs
  integer            :: i_err
  integer            :: i_tmp
  integer            :: i_gradx
  integer            :: i_egradx
  type(af_t)         :: tree
  type(ref_info_t)   :: refine_info
  integer            :: mg_iter
  real(dp)           :: residu(2), anal_err(2)
  character(len=100) :: fname
  type(mg_t)         :: mg
  type(gauss_t)      :: gs
  integer            :: count_rate,t_start,t_end
 
  print *, "Running poisson_basic_" // dimname
  print *, "Number of threads", af_get_max_threads()
 
  ! The manufactured solution exists of Gaussians
  ! Amplitudes:  [1.0_dp, 1.0_dp]
  ! Sigmas    :  [0.04_dp, 0.04_dp]
  ! Locations :  x, y, z = 0.25 or x, y, z = 0.75
  call gauss_init(gs, [1.0_dp, 1.0_dp], [0.04_dp, 0.04_dp], &
       reshape([dtimes(0.1_dp), &
       dtimes(0.75_dp)], [ndim,2]))
 
  call af_add_cc_variable(tree, "phi", ix=i_phi)
  call af_add_cc_variable(tree, "sol", ix=i_sol)
  call af_add_cc_variable(tree, "rhs", ix=i_rhs)
  call af_add_cc_variable(tree, "err", ix=i_err)
  call af_add_cc_variable(tree, "tmp", ix=i_tmp)
  call af_add_cc_variable(tree, "Dx",  ix=i_gradx)
  call af_add_cc_variable(tree, "eDx", ix=i_egradx)
 
  domain_len(1)   = 3.0_dp
  domain_len(2:)  = 1.0_dp
  domain_size(1)  = 3 * box_size
  domain_size(2:) = box_size
 
  ! Initialize tree
  call af_init(tree, & ! Tree to initialize
       box_size, &     ! A box contains box_size**DIM cells
       domain_len, &
       domain_size)
 
  call system_clock(t_start, count_rate)
  do
     ! For each box, set the initial conditions
     call af_loop_box(tree, set_initial_condition)
 
     ! This updates the refinement of the tree, by at most one level per call.
     ! The second argument is a subroutine that is called for each box that can
     ! be refined or derefined, and it should set refinement flags. Information
     ! about the changes in refinement are returned in the third argument.
     call af_adjust_refinement(tree, refine_routine, refine_info)
 
     ! If no new boxes have been added, exit the loop
     if (refine_info%n_add == 0) exit
  end do
  call system_clock(t_end, count_rate)
 
  write(*,"(A,Es10.3,A)") " Wall-clock time generating AMR grid: ", &
       (t_end-t_start) / real(count_rate,dp), " seconds"
 
  call af_print_info(tree)
 
  ! Initialize the multigrid options. This performs some basics checks and sets
  ! default values where necessary.
  mg%i_phi        = i_phi       ! Solution variable
  mg%i_rhs        = i_rhs       ! Right-hand side variable
  mg%i_tmp        = i_tmp       ! Variable for temporary space
  mg%sides_bc     => sides_bc   ! Method for boundary conditions
 
  ! This routine does not initialize the multigrid fields boxes%i_phi,
  ! boxes%i_rhs and boxes%i_tmp. These fileds will be initialized at the
  ! first call of mg_fas_fmg
  call mg_init(tree, mg)
 
  print *, "Multigrid iteration | max residual | max error"
  call system_clock(t_start, count_rate)
 
  do mg_iter = 1, n_iterations
     ! Perform a FAS-FMG cycle (full approximation scheme, full multigrid). The
     ! third argument controls whether the residual is stored in i_tmp. The
     ! fourth argument controls whether to improve the current solution.
     call mg_fas_fmg(tree, mg, set_residual=.true., have_guess=(mg_iter>1))
 
     ! Compute the error compared to the analytic solution
     call af_loop_box(tree, set_error)
 
     ! Determine the minimum and maximum residual and error
     call af_tree_min_cc(tree, i_tmp, residu(1))
     call af_tree_max_cc(tree, i_tmp, residu(2))
     call af_tree_min_cc(tree, i_err, anal_err(1))
     call af_tree_max_cc(tree, i_err, anal_err(2))
     write(*,"(I8,13x,2(Es14.5))") mg_iter, maxval(abs(residu)), &
          maxval(abs(anal_err))
 
     ! This writes a Silo output file containing the cell-centered values of the
     ! leaves of the tree (the boxes not covered by refinement).
     write(fname, "(A,I0)") "output/poisson_basic_" // dimname // "_", mg_iter
     call af_write_silo(tree, trim(fname))
  end do
  call system_clock(t_end, count_rate)
 
  write(*, "(A,I0,A,E10.3,A)") &
       " Wall-clock time after ", n_iterations, &
       " iterations: ", (t_end-t_start) / real(count_rate, dp), &
       " seconds"
 
  ! This call is not really necessary here, but cleaning up the data in a tree
  ! is important if your program continues with other tasks.
  call af_destroy(tree)
 
contains
 
  ! Return the refinement flags for box
  subroutine refine_routine(box, cell_flags)
    type(box_t), intent(in) :: box
    integer, intent(out)     :: cell_flags(DTIMES(box%n_cell))
    integer                  :: IJK, nc
    real(dp)                 :: rr(NDIM), dr2, drhs
 
    nc = box%n_cell
    dr2 = maxval(box%dr)**2
 
    do kji_do(1,nc)
       rr = af_r_cc(box, [ijk])
 
       ! This is an estimate of the truncation error in the right-hand side,
       ! which is related to the fourth derivative of the solution.
       drhs = dr2 * box%cc(ijk, i_rhs)
 
       if (abs(drhs) > 1e-3_dp .and. box%lvl < 7 - ndim) then
          cell_flags(ijk) = af_do_ref
       else
          cell_flags(ijk) = af_keep_ref
       end if
    end do; close_do
  end subroutine refine_routine
 
  ! This routine sets the initial conditions for each box
  subroutine set_initial_condition(box)
    type(box_t), intent(inout) :: box
    integer                     :: IJK, nc
    real(dp)                    :: rr(NDIM), grad(NDIM)
 
    nc = box%n_cell
 
    do kji_do(1,nc)
       ! Get the coordinate of the cell center at IJK
       rr = af_r_cc(box, [ijk])
 
       ! And set the rhs values
       box%cc(ijk, i_rhs) = gauss_laplacian(gs, rr)
       call gauss_gradient(gs, rr, grad)
       box%cc(ijk, i_gradx) = grad(1)
       box%cc(ijk, i_sol) = gauss_value(gs, rr)
    end do; close_do
  end subroutine set_initial_condition
 
  ! Set the error compared to the analytic solution
  subroutine set_error(box)
    type(box_t), intent(inout) :: box
    integer                     :: IJK, nc
    real(dp)                    :: rr(NDIM), gradx
 
    nc = box%n_cell
 
    do kji_do(1,nc)
       rr = af_r_cc(box, [ijk])
       box%cc(ijk, i_err) = box%cc(ijk, i_phi) - gauss_value(gs, rr)
#if NDIM == 1
       gradx = 0.5_dp * (box%cc(i+1, i_phi) - &
            box%cc(i-1, i_phi)) / box%dr(1)
#elif NDIM == 2
       gradx = 0.5_dp * (box%cc(i+1, j, i_phi) - &
            box%cc(i-1, j, i_phi)) / box%dr(1)
#elif NDIM == 3
       gradx = 0.5_dp * (box%cc(i+1, j, k, i_phi) - &
            box%cc(i-1, j, k, i_phi)) / box%dr(1)
#endif
       box%cc(ijk, i_egradx) = gradx - box%cc(ijk, i_gradx)
 
    end do; close_do
  end subroutine set_error
 
  ! This routine sets boundary conditions for a box
  subroutine sides_bc(box, nb, iv, coords, bc_val, bc_type)
    type(box_t), intent(in) :: box
    integer, intent(in)     :: nb
    integer, intent(in)     :: iv
    real(dp), intent(in)    :: coords(NDIM, box%n_cell**(NDIM-1))
    real(dp), intent(out)   :: bc_val(box%n_cell**(NDIM-1))
    integer, intent(out)    :: bc_type
    integer                 :: n
 
    ! We use dirichlet boundary conditions
    bc_type = af_bc_dirichlet
 
    ! Below the solution is specified in the approriate ghost cells
    do n = 1, box%n_cell**(ndim-1)
       bc_val(n) = gauss_value(gs, coords(:, n))
    end do
  end subroutine sides_bc
 
end program poisson_basic