High-order, low dispersive and low dissipative explicit schemes for multiple-scale and boundary problems

Explicit high-order numerical schemes are proposed for the accurate computation of multiple-scale problems and for the implementation of boundary conditions. Specific high-order node-centered finite differences and selective filters removing grid-to-grid oscillations are first designed for the discretization of the buffer region between a Δx-grid domain and 2Δx-grid domain. The coefficients of these matching schemes are chosen so that the maximum order of accuracy is reached. Non-centered finite differences and selective filters are then developed with the aim of accurately computing boundary conditions. They are constructed by minimizing the dispersion and the dissipation errors in the wave number space for waves down to four points per wavelength. The dispersion and dissipation properties of the matching and the boundary schemes are described in detail, and their accuracy limits are determined, to show that these schemes calculate accurately waves with at least five points per wavelength. Test problems, including linear convection, wall reflection and acoustic scattering around a cylinder, are finally solved to illustrate the accuracy of the schemes.