ABSTRACT
In this chapter we begin the study of finite difference methods for solving hyperbolic systems of equations. Some excellent general references for this material are books by Hirsch [179], Leveque [180], and Roache [24]. If we denote by U the vector of unknowns, then a conservative system of equations can be written (in one dimension) either as
∂U ∂t
+ ∂F ∂x
= 0, (9.1)
or, in the non-conservative form
∂U ∂t
+A · ∂U ∂x
= 0. (9.2)
The matrix A = ∂F/∂U has real eigenvalues for a pure hyperbolic system. Since the stability analysis we perform is linear, the stability properties of the same method as applied to either Eq. (9.1) or Eq. (9.2) will be identical. Systems of linear equations will be of the form Eq. (9.2) from the outset. We begin by examining the numerical stability of several finite difference methods applied to the system Eq. (9.2).
