The construction of numerical scheme for hyperbolic equation and exact consideration of numerical diffusion and dispersion

Abstract

A new technique in the formulation of numerical scheme for hyperbolic equation is developed. It is different from the classical FD and FE methods. We begin with the algebraic equations with some undefined parameters, and get the difference equation through Taylor-series expansion. When the parameters in the partial difference equation are defined, the equation is what the scheme will simulate. The numerical example of the viscous Burgers equation shows the validity of the scheme. This method deals with the numerical viscosity and dispersion exactly, giving a preliminary explanation to some problems that the CFD face now.