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

doi:10.1111/sapm.12561

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

Mathematics

Research output: Contribution to journal › Article › peer-review

40 Downloads (Pure)

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

Access to Document

10.1111/sapm.12561Licence: Creative Commons: Attribution (CC BY)

NeedhamD2023RiemannFinal published version, 1.62 MBLicence: Creative Commons: Attribution (CC BY)

Cite this

@article{5aaef0610c8248dbac91ccf3d34de65b,

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. ",

keywords = "generalized Burgers equation, large-time structure, Riemann problem, spatially decaying sound speed",

author = "David Needham and John Meyer and John Billingham and Catherine Drysdale",

year = "2023",

month = apr,

day = "27",

doi = "10.1111/sapm.12561",

language = "English",

volume = "150",

pages = "963--995",

journal = "Studies in Applied Mathematics",

issn = "1467-9590",

publisher = "Wiley",

number = "4",

}

TY - JOUR

T1 - The Riemann problem for a generalized Burgers equation with spatially decaying sound speed

T2 - I Large-time asymptotics

AU - Needham, David

AU - Meyer, John

AU - Billingham, John

AU - Drysdale, Catherine

PY - 2023/4/27

Y1 - 2023/4/27

N2 - 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.

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.

KW - generalized Burgers equation

KW - large-time structure

KW - Riemann problem

KW - spatially decaying sound speed

U2 - 10.1111/sapm.12561

DO - 10.1111/sapm.12561

M3 - Article

SN - 1467-9590

VL - 150

SP - 963

EP - 995

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

IS - 4

ER -