High-order compact filters with variable cut-off wavenumber and stable boundary treatment

Abstract

This paper presents a new system of compact discrete filters based on a seven-point stencil and pentadiagonal matrix formulation which can be readjusted with ease for many different high-order finite difference schemes. The filter coefficients are determined by imposing a required cut-off wavenumber which is a free parameter for the system. The filters incorporate a high-order boundary treatment which enables the closure of the pentadiagonal matrix system and ensures its stable implementation. The overall accuracy of the proposed filter system is proved to be sixth order following grid convergence tests. It is found that a locally non-uniform allocation of cut-off wavenumbers at the boundary nodes (boundary weighting) can enhance the numerical stability of nonlinear solutions especially where a complex geometry is concerned. An optimal value of the boundary weighting factor can be obtained through eigenvalue analysis. The eigenvalue analysis procedure shown in this paper will set an exemplar for the others to customise and optimise the filters for their own use. The new filters are presented in a differential form which has computational benefits in implementation. The accuracy and performance of the proposed filters are demonstrated through a variety of benchmark test cases.