High Order Compact Generalized Finite Difference Methods for Solving Inviscid Compressible Flows

Abstract

This paper presents a novel generalized finite difference method that can achieve arbitrary order of accuracy on a compact stencil nodal set. Accurate reconstruction and flux evaluation are two key steps to achieve high order spatial accuracy. A newly developed variational reconstruction approach is utilized to obtain the piecewise higher order polynomial distribution of flow variables. The implementation of boundary conditions is of critical importance and a flexible variational extrapolation technique is proposed for the high order boundary treatment. The numerical flux derivatives are evaluated using a simple and efficient hybrid approach, in which the linear and high order terms of the flux function are treated differently. Several test cases are solved to verify the accuracy, efficiency, and shock capturing capability of the proposed numerical schemes for inviscid compressible flows.