The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large-time asymptotics |
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Authors: | David J Needham John C Meyer John Billingham Catherine Drysdale |
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Institution: | 1. School of Mathematics, University of Birmingham, Birmingham, UK;2. School of Mathematical Sciences, University of Nottingham, Nottingham, UK;3. ForBetterHealth, Clarksdale, Mississippi, USA |
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Abstract: | In this paper, we consider the classical Riemann problem for a generalized Burgers equation, with a spatially dependent, nonlinear sound speed, with , which decays algebraically with increasing distance from a fixed spatial origin. When , this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large-time structure of the associated Riemann problem, and obtain its detailed structure, as , via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at . |
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Keywords: | generalized Burgers equation large-time structure Riemann problem spatially decaying sound speed |
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