Project B4 (finished)

Project B4 - Analytical Properties of Generalized Godunov Schemes

Principal Investigator: Armin Iske

Background and Motivation

Generalized Godunov schemes are a class of high order accurate finite volume schemes for the numerical solution of nonlinear hyperbolic conservation laws. At each time step, they use two main components: First, a reconstruction procedure recovers a piecewise smooth function from cell-averages. Second, that piecewise smooth function is used as initial data and we compute a high order accurate approximation of the exact evolution of that data. Examples of this approach are van Leer's MUSCL scheme, the GRP scheme of Ben-Artzi and Falcovitz and, more recently, the ADER scheme of Toro and Titarev. The numerical performance of ADER schemes is very promising, but it seems that few rigorous analysis of their theoretical properties, such as convergence proofs, formal derivation of the order of accuracy, entropy conditions etc., has been carried out so far.

Aims and Objectives

Our research focuses on the analysis of different ENO-type reconstruction methods and fluxes based on the solution of generalized Riemann problems to close these gaps in our understanding of this highly successful method.

PhD student: Claus Goetz

Claus Goetz successfully defended his PhD on 30 October 2013.

Publications

1. C. R. Goetz and A. Iske: Approximate solutions of generalized Riemann problems: The Toro-Titarev solver and the LeFloch-Raviart expansion. Numerical Methods for Hyperbolic Equations: Theory and Applications, M. E. Vásques-Cedón and A. Hidalgo and P. García-Navarro and L. Cea, eds., Taylor-Francis Group, 2013, pp. 267 - 275

2. C. R. Goetz and A. Iske: Approximate Solutions of Generalized Riemann problems for Nonlinear Systems of Hyperbolic Conservation Laws. Hamburger Beiträge zur Angewandten Mathematik 2013-02 (preprint, submitted to SIAM Journal of Numerical Analysis)