Finite-Difference Methods for Nonlinear Hyperbolic Systems

is obtained where A (u) is the Jacobian matrix of the components of / with respect to the components of u. Equation (1.2) is said to be hyperbolic if the eigenvalues of the matrix pi + 6A are real for all real numbers m, 0. Several authors have proposed finite-difference schemes for the numerical integration of (1.1) (or (1.2)). In [6], Lax and Wendroff introduced an explicit scheme which is stable if the Courant-Friedrichs-Lewy condition [2] is satisfied. In [10], Richtmyer showed how the Lax-Wendroff scheme could be written as a two-step process. Strang [13], has also considered the Lax-Wendroff scheme and in addition has examined the application of Runge-Kutta type methods to the integration of (1.1). Implicit methods, which are more difficult to apply, appear only to have been considered by Gary [4] although Richtmyer [10] has hinted at their possible use. In Section 2 we will develop a general two-step process and in particular a new predictor-corrector scheme. In Section 3 an implicit scheme, similar in nature to Gary's scheme, will be considered.