Filtering

Table of Contents

• Filter base type
• PDE filter
• Elementwise filter

Filter base type

filter_t is a base type for filters on field_t by calling subroutine filter_tapply(F_out, F_in), where F_out and F_in are the output and the input fields, respectively.

The derived type of filter_t includes PDE_filter_t, elementwise_filter_t, ...

PDE filter

The PDE filter is defined in the derived type PDE_filter_t. The filter is `based on the work of B. S. Lazarov, O. Sigmund that solves a Helmholtz-type differential equation to provide smoothing. The equation has the form

\[ -r^2 \nabla^2 X_\text{out} + X_\text{out} = X_\text{in}, \]

subject to Neumann boundary conditions. The filter has the following input parameters:

Name	Description	Admissible values	Default value
r	\(r\) is the above equation.	Real	-
tol	The desired tolerance used when solving the system.	Real	0.0000000001
max_iter	Maximum number of iterations when solving the system.	Integer	200
solver	Numerical solver used to solve the system.	cg,gmres, gmres	cg
preconditioner	Pre-conditioner used to solve the system.	ident, hsmg, jacobi	jacobi

Elementwise filter

The elementwise filter is defined in the derived type elementwise_filter_t. Basing on the spectral element discretization, it is performed by mapping the field into hierarchical polynomials element by element (currently only Legendre-like polynomials are supported) and then attenuating the kernel for different orders. One could directly set it up through the json file if the filter is needed in any section.

The user can also modify elementwise_filter_type and transfer_function according to their need.

elementwise_filter_type is an optional string to specify the type of usage of the polynomial for the filter.

Name	Polynomials
nonBoyd	i-th order polynomials L_i(x)
Boyd	L_i(x) for i<2, L_i(x) - L_{i-2}(x) for i>=2

The kernel is defined through an optional array transfer_function whose size is the number of polynomials. The indices of transfer_function corresponds to the polynomial order in ascending order. transfer_function is initialized to be all 1 such that the filter has no effect by default unless further defined.

One could set up the elementwise_filter in the following way for polynomial order 7.

"filter": {
    "type": "elementwise",
    "elementwise_filter_type": "Boyd",
    "transfer_function": [1,1,1,0.7,0.3,0,0,0]
}

One could also refer to dynamic_smagorinsky.f90 for an example of using elementwise_filter_t in the neko code.