Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems

Marco Artiano, Hendrik Ranocha

Abstract

Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes.

On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems

Abstract

Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes.
Paper Structure (41 sections, 28 theorems, 223 equations, 3 figures, 10 tables)

This paper contains 41 sections, 28 theorems, 223 equations, 3 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Hyperbolic conservation laws
  3. Tadmor condition for non-convex functionals
  4. Nonconservative hyperbolic systems
  5. An algebraic characterization of the fluctuation form
  6. Specialized semi-discretizations for nonconservative systems
  7. Algebraic characterization of the third form
  8. Algebraic characterization of the first form
  9. Algebraic characterization of the second form
  10. Algebraic characterization of the fourth form
  11. Entropy conservation/stability condition for nonconservative systems
  12. A generalized semi-discretization for nonconservative systems
  13. Constructing entropy-conservative/stable fluxes
  14. Variable-coefficient advection equation
  15. Coupled Burgers' equation
  16. ...and 26 more sections

Key Result

Lemma 2.2

Let $Y \subset \mathbb{R}^n$ be open, and $\boldsymbol{\omega}\colon Y \to \mathbb{R}^n$, $\boldsymbol{f}\colon Y \to \mathbb{R}^n$, $F\colon Y \to \mathbb{R}$, and $\boldsymbol{f}^{\mathrm{num}}\colon Y \times Y \to \mathbb{R}^n$ satisfying be given. Then, $\exists F^\mathrm{num}\colon Y \times Y \to \mathbb{R}$ satisfying $\forall \boldsymbol{u} \in Y\colon F^\mathrm{num}(\boldsymbol{u},\boldsy

Figures (3)

  • Figure 1: Warped mesh used for the well-balancedness test of the 2D hyperbolized Sainte-Marie system and for the convergence test of the 2D compressible Euler equations in nonconservative form RANOCHA2025113471.
  • Figure 2: Contour of the surface pressure at day 10 for the baroclinic instability test case, with polynomial degree 5.
  • Figure 3: Evolution of the entropy over 20 days for the baroclinic instability test case, with polynomial degree 5 and the entropy stable numerical scheme \ref{['eq:es_flux']}.

Theorems & Definitions (61)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Tadmor
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 51 more