A study of the role of weights in the spectral properties of the least square finite difference scheme and the least square upwind finite difference scheme

In this article, we analyze the performance of a family of nonlinear, shock-capturing mesh-free finite-difference schemes in wavenumber space. These finite difference methods are based on the use of a weighted least-square approximation procedure together with a Taylor series expansion of the unknown function. The influence of weights on the spectral properties of these schemes are studied using the approximate dispersion relation. The numerical study evidences that an appropriate choice of the weighting functions reduces dissipation and dispersion inherent to the schemes. The meshless schemes studied in this work, are then used to simulate 2D internal inviscid subsonic and transonic flows respectively in a channel with a bump. The results in 2D corroborate with the 1D analyses and reveal that when the supporting nodes closer to the reference node are assigned a greater weightage value in the computation of the 1st order spatial derivatives, better spectral properties are achieved at low wavenumber. However, the present numerical study shows that weighting function does not shift the frequency of the errors and thus in order to accelerate the convergence of these schemes, methods such as the classic multigrid can be used.