Research Repository

# The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics

## Authors

David J. Needham

John C. Meyer

JOHN BILLINGHAM john.billingham@nottingham.ac.uk
Professor of Theoretical Mechanics

Catherine Drysdale

### Abstract

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, u t + h α ( x ) u u x = u x x , $$\begin{equation*} u_t + h_{\alpha }(x) u u_x = u_{xx}, \end{equation*}$$ with a spatially dependent, nonlinear sound speed, h α ( x ) ≡ ( 1 + x 2 ) − α $h_{\alpha }(x) \equiv (1+x^2)^{-\alpha }$ with α > 0 $\alpha >0$ , which decays algebraically with increasing distance from a fixed spatial origin. When α = 0 $\alpha =0$ , 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 → ∞ $t\rightarrow \infty$ , 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 = 1 2 $\alpha =\frac{1}{2}$ .

### Citation

Needham, D. J., Meyer, J. C., Billingham, J., & Drysdale, C. (2023). The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics. Studies in Applied Mathematics, https://doi.org/10.1111/sapm.12561

Journal Article Type Article Dec 29, 2022 Jan 17, 2023 Jan 17, 2023 Jan 19, 2023 Jan 19, 2023 Studies in Applied Mathematics 0022-2526 1467-9590 Wiley Peer Reviewed https://doi.org/10.1111/sapm.12561 Generalized Burgers equation, large‐time structure, Riemann problem, spatially decaying sound speed https://nottingham-repository.worktribe.com/output/16226443 https://onlinelibrary.wiley.com/doi/full/10.1111/sapm.12561