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

David John Needham; John Christopher Meyer; John Billingham; Catherine Drysdale

doi:10.1111/sapm.12561

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

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

^*Corresponding author for this work

Mathematics

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Abstract

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

u_t + h_α(x) uu_x= u_xx,

with a spatially dependent, nonlinear sound speed, h_α(x) ≡ (1 + x²) ^-α 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 language	English
Pages (from-to)	963-995
Number of pages	33
Journal	Studies in Applied Mathematics
Volume	150
Issue number	4
Early online date	17 Jan 2023
DOIs	https://doi.org/10.1111/sapm.12561
Publication status	Published - 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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Cite this

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

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) -α 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. ",

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T2 - I Large-time asymptotics

AU - Needham, David John

AU - Meyer, John Christopher

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AU - Drysdale, Catherine

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AB - 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) -α 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.

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KW - large-time structure

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