Title: A Class of Residual Schemes for Conservation Laws: Design Criteria, Analysis and Applications Speaker: Mario Ricchiuto, INRIA Futurs project Scalapplix and IBM Universite’ de Bordeaux 1 Date/Time: Thursday, March 29, 2007, 10:30am – 11:30am Location: CSRI Building/Room 90 (Sandia NM) Brief Abstract: The objective of this talk is to review the basics of a class of compact discretizations for conservation laws known as Residual Distribution (RD) or Fluctuation Splitting (FS) schemes. Based on ideas pioneered by P.L. Roe in the beginning of the 80’s [1,2], the interest in these schemes has been growing ever since, due to the original combination of properties borrowed from both the Finite Elements and Finite Volumes approaches. Residual Distribution is by construction a form of weighted residual discretizations whose accuracy is determined by the functional (viz. polynomial) space chosen to represent the discrete unknown [3]. In this respect, FS is very close to standard Finite Element methodologies. However, a structural property of RD discretizations is a strong conservative character, which makes them in a way similar to Finite Volume schemes [4,5]. Another common point between these two approaches is the use of upwinding as a means of stabilizing the discrete equations [3,6]. In this talk we give an overview of the properties of RD schemes, present the general design criteria used in the construction of discretizations of this type, as well as the basic tools available for their analysis. In particular, the presentation will be articulated in three parts. In part I, we illustrate the very basic idea of Fluctuation Splitting, and the heuristics that have led to the first successful FS schemes, which motivated the further development of the approach. Part II, is devoted to the analysis of RD discretizations. Some theoretical results (e.g. Lax-Wendroff theorem, truncation error estimates, positivity and discrete max principles) are briefly reviewed, with the objective of deducing general design criteria. Finally, in part III we describe some of the most recent developments, illustrating how the criteria presented in part II are used to construct very high order non-oscillatory discretizations. Applications to the Euler equations and to other systems of conservation laws will be shown. [1] P.L. Roe. Fluctuations and Signals - A Framework for Numerical Evolution Problems. Num. Meth. for Fluids Dynamics, K.W. Morton and M.J. Baines Ed.s, Academic Press, 1982. CSRI POC: Ed Love, (505) 845-3600 |