iextm_time_scheme.f90

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! Implemented as shown in:
! Brandon E. Merrill, Yulia T. Peet, Paul F. Fischer, & James W. Lottes (2016).
! A spectrally accurate method for overlapping grid solution of incompressible
! Navier–Stokes equations. Journal of Computational Physics, 307, 60-93.
module iextm_time_scheme
  use num_types, only : rp
  use time_scheme, only : time_scheme_t
  use math, only : rzero
  use utils, only : neko_error
  implicit none
  private
 
  type, public, extends(time_scheme_t) :: iextm_time_scheme_t
   contains
     procedure, nopass :: compute_coeffs => iextm_time_scheme_compute_coeffs
  end type iextm_time_scheme_t
 
contains
 
  subroutine iextm_time_scheme_compute_coeffs(coeffs, dt, order)
    implicit none
    real(kind=rp), intent(out) :: coeffs(4)
    real(kind=rp), intent(in) :: dt(10)
    integer, intent(in) :: order
    real(kind=rp) :: x(0:3), xtest
    integer :: i, j
    real(kind=rp) :: basis
 
    call rzero(coeffs, 4)
 
    ! This is the direct way, working for constant tstep
    !select case (order)
    !case (1)
    !   coeffs(1) = 1.0_rp
    !case (2)
    !   coeffs(2) = -1.0_rp
    !   coeffs(1) = 2.0_rp
    !case (3)
    !   coeffs(3) = 1.0_rp
    !   coeffs(2) = -3.0_rp
    !   coeffs(1) = 3.0_rp
    !case default
    !   call neko_error("The order of the IEXTm time scheme must be 1 to 3.")
    !end select
 
    if (order .lt. 1 .or. order .gt. 3) then
       call neko_error("The order of the IEXTm time scheme must be 1 to 3.")
    end if
 
    ! Create the nodes to build the interpolation basis
    ! This simply places old points as nodes in a 1d grid
    x(0) = dt(1) ! Current timestep
    do i = 1, order
       x(i) = x(i-1) - dt(i)
    end do
    ! Define the test point
    xtest = x(0)
 
    do i = 1, order
       basis = 1.0_rp
       do j = 1, order
          if (j .ne. i) then
             basis = basis * (xtest - x(j)) / (x(i) - x(j))
          end if
       end do
       coeffs(i) = basis
    end do
 
  end subroutine iextm_time_scheme_compute_coeffs
 
end module iextm_time_scheme