Numerical solution of nonlinear hyperbolic conservation laws using exponential splines

Abstract

Previous theoretical (McCartin 1989a) and computational (McCartin 1989b) results on exponential splines are herein applied to provide approximate solutions of high order accuracy to nonlinear hyperbolic conservation laws. The automatic selection of certain “tension” parameters associated with the exponential spline allows the sharp resolution of shocks and the suppression of any attendant oscillations. Specifically, spatial derivatives are replaced by nodal derivatives of interpolatory splines and temporal discretization is achieved via a Runge-Kutta time stepping procedure. The fourth order accuracy of this scheme in both space and time (for uniform mesh and tension) is established and a linearized stability analysis is provided. The Lax-Wendroff theorem on convergence to weak solutions (Lax and Wendroff 1960) is then extended to spline approximations in conservation form. An implicit artificial viscosity term (Anderson et al. 1984) is included via upwinding in conservation form in order to assure convergence to the physically relevant weak solution. The efficacy of this procedure is illustrated on the inviscid Burgers' equation where the accurate capture of a travelling shockwave is demonstrated.