Schemes

Modules
module	mod_finite_differences_scheme_fifth_o1
	Finite Difference Schemes for the third derivative at order 3.
module	mod_finite_differences_scheme_fifth_o2
	Finite Difference Schemes for the first derivative at order 5.
module	mod_finite_differences_scheme_first_o1
	Finite Difference Schemes for the first derivative at order 1.
module	mod_finite_differences_scheme_first_o2
	Finite Difference Schemes for the first derivative at order 2.
module	mod_finite_differences_scheme_first_o3
	Finite Difference Schemes for first derivative at order 3.
module	mod_finite_differences_scheme_first_o4
	Finite Difference Schemes for the first derivative at order 4.
module	mod_finite_differences_scheme_first_o5
	Finite Difference Schemes for the first derivative at order 5.
module	mod_finite_differences_scheme_first_o6
	Finite Difference Schemes for the first derivative at order 5.
module	mod_finite_differences_scheme_fourth_o1
	Finite Difference Schemes for fourth derivative at order 1.
module	mod_finite_differences_scheme_fourth_o2
	Finite Difference Schemes for the third derivative at order 3.
module	mod_finite_differences_scheme_fourth_o3
	Finite Difference Schemes for the first derivative at order 5.
module	mod_finite_differences_scheme_second_o1
	Finite Difference Schemes for second derivative at order 1.
module	mod_finite_differences_scheme_second_o2
	Finite Difference Schemes for second derivative at order 2.
module	mod_finite_differences_scheme_second_o3
	Finite Difference Schemes for second derivative at order 3.
module	mod_finite_differences_scheme_second_o4
	Finite Difference Schemes for second derivative at order 4.
module	mod_finite_differences_scheme_second_o5
	Finite Difference Schemes for the first derivative at order 5.
module	mod_finite_differences_scheme_sixth_o1
	Finite Difference Schemes for the first derivative at order 5.
module	mod_finite_differences_scheme_third_o1
	Finite Difference Schemes for second derivative at order 1.
module	mod_finite_differences_scheme_third_o2
	Finite Difference Schemes for third derivative at order 2.
module	mod_finite_differences_scheme_third_o3
	Finite Difference Schemes for the third derivative at order 3.
module	mod_finite_differences_scheme_third_o4
	Finite Difference Schemes for the first derivative at order 5.
module	mod_finite_differences_scheme_zero
	Finite Difference Schemes for the zero'th derivative (the value)
module	mod_finite_differences_weno_scheme_first_o2z
	Finite Difference Weno Scheme for first derivative at order 2Z.
module	mod_finite_differences_weno_scheme_first_o3
	Finite Difference Weno Scheme for first derivative at order 3.
module	mod_finite_differences_weno_scheme_first_o3z
	Finite Difference Weno Scheme for first derivative at order 3Z.
module	mod_finite_differences_weno_scheme_first_o5
	Finite Difference Weno Scheme for first derivative at order 5.
module	mod_finite_differences_weno_scheme_first_o5z
	Finite Difference Weno Scheme for first derivative at order 5Z.

Detailed Description

All the instances of schemes are (and has to be) created with the procedure fd_initialize()

Usage

Use an existing scheme

Schemes are used to directly or indirectly build FD schemes:

• the direct way consists in declaring a fd_scheme and assigning it with the result of a scheme builder
• the indirect way consists in supplying a scheme builder to the FD computer

Write your own scheme

Todo:

MCO: correct this depracated documentation If you have a particular numerical scheme that you want to implement, you should follow the main scheme_builder_non_uniform interface. To do so, simply write a function that simply:

1. define the fd_scheme's boundaries (thanks to the fd_scheme%declare_stencil member routine)
2. fill the fd_scheme%stencilweight array
3. returns the fd_scheme

Example

We propose here a special example that shows the capacity of FD schemes to touch a wide range of applications. We write the specific scheme for computing a Gaussian weighted mean depending on distances from the central point:

function fd_scheme_gaussian_mean( steps ) result(scheme)
  use mod_finite_differences
  double precision, dimension(:), intent(in) :: steps
  type(t_fd_scheme) :: scheme
  double precision :: sigma
  double precision :: dist
  integer :: sindex, eindex, i
 
  ! We assume the steps array to be even
  sindex = -size(steps)/2
  eindex =  size(steps)/2
  call scheme%declare_stencil( sindex, eindex )
  sigma = 0.25d0*sum(steps)
  do i=sindex,eindex
    ! Compute the distance from the point to the center element
    if( i==0 ) then
      dist = 0d0
    else
      dist = sum( steps(min(i,0)+eindex+1:max(i,0)+eindex) )
    end if
    scheme%stencil%weight(i) = exp(-0.5d0*(dist/sigma)**2)
  end do
  ! Normalize the weights
  scheme%stencil%weight(sindex:eindex) = scheme%stencil%weight(sindex:eindex) / sum(scheme%stencil%weight(sindex:eindex))
end function fd_scheme_gaussian_mean

Then, you can easily use your scheme - as long as it is defined in a Fortran module - with the FD computer

result = fd_compute( fd_scheme_gaussian_mean, dx(index-2:index+1), field(index-2:index+2) )

See unit_testing/discretization/node_level_schemes/finite_differences/fd_scheme/finite_differences.f90 for a detailed example.

List of available finite difference schemes

Note

For clarity, the definitions in the table below are simplified, please refer to the code for the actual value (especialy on non uniform grids).

Derivative	Order	Type	Name	Definition
First	1	Backward	t_fd_scheme_first_o1_backward	\( \frac{\phi_{i}-\phi_{i-1}}{\Delta x} \)
First	1	Forward	t_fd_scheme_first_o1_forward	\( \frac{\phi_{i-1}-\phi_{i}}{\Delta x} \)
First	2	Backward	t_fd_scheme_first_o2_backward	\( \frac{\phi_{i-2}-4\phi_{i-1}+3\phi_{i}}{2 \Delta x} \)
First	2	Centered	fd_scheme_first_o2_centered	\( \frac{\phi_{i+1}-\phi_{i-1}}{2 \Delta x} \)
First	2	Forward	t_fd_scheme_first_o2_forward	\( \frac{-\phi_{i+2}+4\phi_{i+1}-3\phi_{i}}{2 \Delta x} \)
Second	2	Centered	fd_scheme_second_o2_centered	\( \frac{\phi_{i+1}-2\phi_{i}+\phi_{i-1}}{\Delta x^2} \)