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High order entropy stable schemes for the quasi-one-dimensional shallow water and compressible Euler equations

Jesse Chan, Khemraj Shukla, Xinhui Wu, Ruofeng Liu, Prani Nalluri

TL;DR

This work addresses instability of high-order schemes for nonlinear conservation laws with shocks by developing entropy-stable flux-difference schemes for quasi-1D shallow water and compressible Euler equations in channels and nozzles with varying width. It introduces new non-symmetric entropy-conservative fluxes and a generalized entropy conservation condition to accommodate width-dependent terms, embeds them in a summation-by-parts framework, and proves semi-discrete entropy stability, conservation properties, and, for SW, well-balancedness. The authors validate the approach with extensive numerical experiments, confirming entropy conservation/stability, high-order convergence (often $O(h^{N+1})$ for $N>1$), and lake-at-rest well-balancedness in continuous and, to an extent, discontinuous bathymetries and widths; nozzle flows further demonstrate applicability to subsonic and transonic regimes. The methods enable robust, high-accuracy simulations of quasi-1D flows in varying geometries, with potential impact on engineering applications in channels and nozzles where geometry varies spatially.

Abstract

High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This work extends high order entropy stable schemes to the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. We introduce new non-symmetric entropy conservative finite volume fluxes for both sets of quasi-1D equations, as well as a generalization of the entropy conservation condition to non-symmetric fluxes. When combined with an entropy stable interface flux, the resulting schemes are high order accurate, conservative, and semi-discretely entropy stable. For the quasi-1D shallow water equations, the resulting schemes are also well-balanced.

High order entropy stable schemes for the quasi-one-dimensional shallow water and compressible Euler equations

TL;DR

This work addresses instability of high-order schemes for nonlinear conservation laws with shocks by developing entropy-stable flux-difference schemes for quasi-1D shallow water and compressible Euler equations in channels and nozzles with varying width. It introduces new non-symmetric entropy-conservative fluxes and a generalized entropy conservation condition to accommodate width-dependent terms, embeds them in a summation-by-parts framework, and proves semi-discrete entropy stability, conservation properties, and, for SW, well-balancedness. The authors validate the approach with extensive numerical experiments, confirming entropy conservation/stability, high-order convergence (often for ), and lake-at-rest well-balancedness in continuous and, to an extent, discontinuous bathymetries and widths; nozzle flows further demonstrate applicability to subsonic and transonic regimes. The methods enable robust, high-accuracy simulations of quasi-1D flows in varying geometries, with potential impact on engineering applications in channels and nozzles where geometry varies spatially.

Abstract

High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This work extends high order entropy stable schemes to the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. We introduce new non-symmetric entropy conservative finite volume fluxes for both sets of quasi-1D equations, as well as a generalization of the entropy conservation condition to non-symmetric fluxes. When combined with an entropy stable interface flux, the resulting schemes are high order accurate, conservative, and semi-discretely entropy stable. For the quasi-1D shallow water equations, the resulting schemes are also well-balanced.
Paper Structure (29 sections, 4 theorems, 89 equations, 5 figures, 6 tables)

This paper contains 29 sections, 4 theorems, 89 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Entropy analysis for quasi-1D systems
  3. The quasi-1D shallow water equations
  4. The quasi-1D compressible Euler equations
  5. Continuous entropy analysis for quasi-1D systems
  6. Entropy conservative numerical fluxes for quasi-1D systems
  7. The quasi-1D shallow water equations
  8. The quasi-1D compressible Euler equations
  9. Entropy stable flux differencing schemes for quasi-1D systems
  10. Notation
  11. Algebraic formulation in terms of SBP operators
  12. Semi-discrete entropy analysis
  13. Wall boundary conditions
  14. Conservation
  15. High order accuracy
  16. ...and 14 more sections

Key Result

Lemma 1

Let $\eta(\bm{q})$ denote a differentiable convex entropy with respect to $\bm{q}$. Let $\bm{u} = a\bm{q}$, where $a(x) > 0$ is some scalar function, and let $S(\bm{u}) = a\eta(\bm{u} / a)$. Then, $S(\bm{u})$ is a differentiable convex function with respect to $\bm{u}$ and $\frac{\partial S}{\partia

Figures (5)

  • Figure 1: An example of a quasi-1D domain with spatially varying channel width $a(x)$ from carlberg2018conservative.
  • Figure 2: Water height and Froude number for the converging-diverging channel problem.
  • Figure 3: Entropy residual for an entropy conservative simulation of the quasi-1D compressible Euler equations.
  • Figure 4: Mach number and pressure for subsonic flow through a nozzle.
  • Figure 5: Mach number and pressure for transonic flow through a nozzle.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • proof
  • Remark 3