An Analysis of the Influence of Data Extrema on Some First and Second Order Central Approximations of Hyperbolic Conservation Laws

We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov’s method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov’s method. It turns out, that schemes with a large viscosity coefficient are prone to oscillations at data extrema. For all LFt schemes except for the classical Lax-Friedrichs method, occurring oscillations are damped in the course of a computation. This damping effect also holds for Rusanov’s method. Concerning the NT schemes, the non-staggered version may yield oscillatory results, while it can be shown rigorously that the staggered NT scheme does not produce oscillations when using the classical minmod-limiter under a restriction on the time step size. Note that this restriction is not the same as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar problems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper. Mathematics Subject Classification. 35L65, 65M06, 65M12. Received: November 4, 2004. Revised: March 25, 2005. Introduction The phenomenon we discuss within this paper is concerned with oscillations appearing at local data extrema in the numerical simulation of hyperbolic conservation laws when using central methods. The aim of this paper is on the one hand the detailed investigation of the influence of the coefficients of numerical viscosity defining especially the Lax-Friedrichs type (LFt) central schemes on the occurrence of this phenomenon. The properties found for these methods also shed light on the corresponding properties of Rusanov’s method. On the other hand, we clarify the role of the formulation of methods using staggered or non-staggered grids, respectively, with respect to the subject of our investigation. This is especially important for the construction of higher-order central schemes for which the Nessyahu-Tadmor (NT) schemes we investigate are the classical prototype.