Hopping numerical approximations of the hyperbolic equation

Abstract

Polynomial functions can be used to derive numerical schemes for an approximate solution of hyperbolic equations. A conventional derivation technique requires a polynomial to pass through every node values of a continuous computational stencil, leading to severe manifestation of the Gibbs phenomenon and strict time-step limitation. To overcome the problem, this paper introduces polynomials that skip regularly ('hop' over) one or more nodes from the computational grid. Polynomials hopping over odd and even nodes yield a series of explicit numerical schemes of a required accuracy, with Lax-Friedrichs method being a particular simplest case. The schemes have two times wider stability interval compared to conventional continuous-stencil explicit methods. Convex combinations of odd- and even-node-based updates improve further accuracy and stability of the method. Out of considered combinations (up to third-order accuracy), derived odd-order methods are stable for the Courant number ranging from 0 to 3, and even-order ones from 0 to 5. A 2-D extension of the hopping polynomial method exhibits similar properties. Copyright © 2007 John Wiley & Sons, Ltd.