Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

Abstract

We develop method of characteristics schemes based on explicit Runge-Kutta and pseudo-Runge-Kutta third-and fourth-order solvers along the characteristics. Schemes based on Runge-Kutta solvers are found to be strongly unstable for certain physics-motivated models. In contrast, schemes based on pseudo-Runge-Kutta solvers are shown to be only weakly unstable for periodic boundary conditions and essentially stable for the more physically relevant nonreflecting boundary conditions. Our implementation of nonreflecting boundary conditions does not rely on interpolation. KEYWORDS coupled-mode equations, higher-order methods, method of characteristics 1 Here y ± and f ± are vectors of respective lengths N ± and subscripts denote partial differentiation. Functions f ± are, in general, nonlinear. Moreover, they, in principle, may contain small diffusion-like terms y ± xx. We will briefly comment on the latter possibility in the concluding section of this work, but in the main part of it we will assume that system (1a) is nondissipative.