The spatial resolution properties of composite compact finite differencing

Improving the boundary efficiency of a compact finite difference scheme through optimising its composite template

“…This was first suggested by Jordan [12] whose composite template approach consists of taking the Fourier transform of each row of the matrix system in Eq.(1). This results in a pseudo-wavenumber curve for each grid point, with…”

“…In past procedures the numerical stability is often treated as an afterthought, and is only considered after the wavenumber optimisation of the finite difference coefficients is completed ( [1,[11][12][13]). …”

“…Jordan [12] introduced an alternative approach for analysing spectral resolution properties through composite templates. Unlike the more traditional decoupled Fourier approach where the resolution of each differencing stencil is studied separately, this consists of Fourier analysis of the whole matrix system of equations, consisting of both the interior and boundary stencils.…”

“…This paper aims to extend the composite template strategy of Jordan [12] by redefining how the composite template modified wavenumber is evaluated. Unlike the original approach, Fourier analysis will not be conducted until the matrix system of equations has already been rearranged for the derivative.…”

New optimized fourth-order compact finite difference schemes for wave propagation phenomena

“…Comparisons between this algorithm with the standard fourth-order four-stage RK formulation (RK44) are shown. The accuracy, stability, and resolution, the so-called 'tri-testing' by Jordan [21], for the fully discrete schemes, are investigated. Finally, using benchmark propagation problems involving the simple convection equation, and the nonlinear shallow water equations (NSWEs) in one and two dimensions, the performance of the proposed schemes, has been verified.…”