The Riemann problem for a generalized Burgers equation with spatially decaying sound speed: I Large-time asymptotics

David Needham*, John Meyer, John Billingham, Catherine Drysdale

*Corresponding author for this work

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Abstract

In this paper, we consider the classical Riemann problem for a generalized Burgers equation,

ut + hα(x) uux = uxx,

with a spatially dependent, nonlinear sound speed, hα(x) ≡ (1 + x2)  with α > 0, which decays algebraically with increasing distance from a fixed spatial origin. When α = 0, 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 t → ∞, 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  α = ½. 

Original languageEnglish
Pages (from-to)963-995
Number of pages33
JournalStudies in Applied Mathematics
Volume150
Issue number4
Early online date17 Jan 2023
DOIs
Publication statusPublished - 27 Apr 2023

Keywords

  • generalized Burgers equation
  • large-time structure
  • Riemann problem
  • spatially decaying sound speed

ASJC Scopus subject areas

  • Applied Mathematics

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