About non linear stabilization for scalar hyperbolic problems

Abstract

This paper deals with the numerical approximation of linear and non linear hyperbolic problems. We are mostly interested in the development of parameter free methods that satisfy a local maximum principle. We focus on the scalar case, but extensions to systems are relatively straightforward when these techniques are combined with the ideas contained in Abgrall (J. Comput. Phys., 214(2):773–808, 2006). In a first step, we precise the context, give conditions that guaranty that, under standard stability assumptions, the scheme will converge to weak solutions. In a second step, we provide conditions that guaranty an arbitrary order of accuracy. Then we provide several examples of such schemes and discuss in some details two versions. Numerical results support correctly our initial requirements: the schemes are accurate and satisfy a local maximum principle, even in the case of non smooth solutions. © Springer Science+Business Media LLC 2017.