poisson_helmholtz.f90

Example showing how to use multigrid for Helmholtz equation and compare with an analytic solution. A standard 5 or 7 point stencil is used.

#include "../src/cpp_macros.h"
!> \example poisson_helmholtz.f90
!>
!> Example showing how to use multigrid for Helmholtz equation and compare with
!> an analytic solution. A standard 5 or 7 point stencil is used.
program poisson_helmholtz
  use m_af_all
  use m_gaussians
 
  implicit none
 
  integer, parameter  :: box_size     = 8
  integer, parameter  :: n_iterations = 10
  integer             :: i_phi
  integer             :: i_rhs
  integer             :: i_err
  integer             :: i_tmp
  real(dp), parameter :: lambda       = 1.0e3_dp
 
  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_helmholtz_" // dimname // ""
  print *, "Number of threads", af_get_max_threads()
 
  call gauss_init(gs, [1.0_dp, 1.0_dp], [0.04_dp, 0.04_dp], &
       reshape([dtimes(0.25_dp), &
       dtimes(0.75_dp)], [ndim,2]))
 
  call af_add_cc_variable(tree, "phi", ix=i_phi)
  call af_add_cc_variable(tree, "rhs", ix=i_rhs)
  call af_add_cc_variable(tree, "tmp", ix=i_tmp)
  call af_add_cc_variable(tree, "err", ix=i_err)
 
  ! Initialize tree
  call af_init(tree, & ! Tree to initialize
       box_size, &     ! A box contains box_size**DIM cells
       [dtimes(1.0_dp)], &
       [dtimes(box_size)])
 
  do
     call af_loop_box(tree, set_initial_condition)
     call af_adjust_refinement(tree, refine_routine, refine_info)
     if (refine_info%n_add == 0) exit
  end do
 
  call af_print_info(tree)
 
  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
  mg%helmholtz_lambda = lambda
 
  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
     call mg_fas_fmg(tree, mg, set_residual=.true., have_guess=(mg_iter>1))
     call af_loop_box(tree, set_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))
 
     write(fname, "(A,I0)") "output/poisson_helmholtz_" // dimname // "_", mg_iter
     call af_write_vtk(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"
 
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 < 5) 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)
 
    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) - &
            lambda * 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)
 
    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)
    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_helmholtz