Study for Stability of Finite Difference Solution and its Modified Equation

The aim of this papar is to show the relation between the stability property of the finite difference equation for a linearized system of compressible flow equations and the property of its modified equation, that is the differential equation actually solved by a difference equation, as the eigenvalue problem of the coefficient matrix. Based on a finite difference equation by discretizing the linearized system of compressible flow equations by using the explicit scheme in time and 2 step Lax-Wendroff scheme or central differnce scheme in space, a modified equation of the finite difference equation is introduced. It is shown that the stability condition of the finite difference equation can be expressed by the eigenvalue of the lowest order diffusion term of the modified euation.