Monotone bicompact scheme for quasilinear hyperbolic equations

Abstract

Hyperbolic equations and systems of equationscomprise a considerable part of mathematical modelsused in various applications [1]. Among them, someproblems are related to finding weak solutions, whichare described by piecewise smooth functions. Highorder accurate shockcapturing schemes are widelyused for the numerical solution of nonlinear problems.Important properties to be possessed by such schemesare monotonicity [1] and conservation [2]. For thisreason, the design of monotone and conservative difference schemes has been addressed in numerous publications [1, 2].For hyperbolic quasilinear equations and systemsof equations, bicompact difference schemes on a twopoint stencil were proposed in [3]. They have fourthorder accuracy in space and are absolutely stable, conservative, and monotone for local Courant numbersκ≥ [4]. Moreover, they are efficient and can besolved using by the running calculation method.In this paper, a new highorder accurate monotonebicompact scheme is proposed for quasilinear hyperbolic equations. Like the schemes in [3], it is absolutely stable, conservative, and efficient but, in contrast to the former, has the following additional advantages. First, the new scheme remains monotone forsmaller Courant numbers than that in [3]. Second,when applied to stiff problems, the scheme is morerobust than that in [3]. An