SSD - Helzel Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

Prof. Dr.Christiane Helzel - A Third Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Differential Equations

Institute of Mathematics, Heinrich-Heine-Universität Düsseldorf

Abstract

The wave propagation algorithm of LeVeque and its implementation in the software package Clawpack are widely used for the approximation of hyperbolic problems. The method belongs to the class of truly multidimensional, high-resolution finite volume methods. Furthermore, it can be characterised as a one-step Lax-Wendroff type method, i.e. the PDE is solved simultaneously in space and time. Approximations obtained with this method are second order accurate for smooth solutions and avoid unphysical oscillations near discontinuities or steep gradients. 
 
Second order accurate methods are often a good choice in terms of balance between computational cost and desired resolution, especially for solutions dominated by shock waves or contact discontinuities and relatively simple structures between these discontinuities. However, for problems containing complicated smooth solution structures, where the accurate resolution of small scales is require, schemes with a higher order of accuracy are more efficient and computationally affordable.
 
I will present my recent work towards the construction of a third order accurate wave propagation algorithm for hyperbolic pdes. The resulting method shares main properties with the original method, i.e. it is based on a wave decomposition at grid cell interfaces, it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of
a wave limiter. Furthermore, I will compare this new method with other recently proposed third order accurate finite volume methods.