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Volume 17,     Number 3,     Fall 2009

 

APPROXIMATION OF NONCONSERVATIVE
HYPERBOLIC SYSTEMS BASED ON
DIFFERENT SHOCK CURVE DEFINITIONS
N. CHALMERS AND E. LORIN

Abstract. The aim of this paper is to lay a theoretical framework for developing numerical schemes for approximating Nonconservative Hyperbolic Systems (NCHSs). We first recall some key points of the theory of NCHSs, beginning with the definition nonconservative products proposed by Dal Maso, LeFloch, and Murat [16]. Next, we briefly introduce the vanishing viscosity solutions and shock curves derived from Bianchini and Bressan's center manifold technique [8], and their partial generalization recently proposed by Alouges and Merlet [5]. Approximation of these shock curves also proposed by Alouges and Merlet are then introduced and discussed. We then investigate the numerical implementation of these analytical approaches using Godunov-like schemes, which either use the approximate Shock curves of Alouges and Merlet directly in a Riemann solver, or use the framework of Dal Maso, LeFloch, and Murat, in combination with these approximate shock curves. To our knowledge, this work is the first attempt to numerically implement shock curves derived from Bianchini and Bressan's center manifold approach.

 

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