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High Order Numerical Methods Preserving Invariant Domain for Hyperbolic and Related Systems

Kailiang Wu, Xiangxiong Zhang, Chi-Wang Shu

TL;DR

This paper surveys invariant-domain-preserving (IDP) numerical schemes for hyperbolic and related PDEs, emphasizing convex invariant domains to guarantee physically meaningful solutions and stability. It contrasts two high-order IDP strategies: Zhang–Shu’s intrinsic weak IDP with polynomial limiters and flux-corrected transport (FCT)–style flux limiters applicable to FV, DG, FD, and CFEM discretizations, with particular attention to challenging systems like MHD and RMHD. The work covers a wide array of models (gas dynamics, shallow water, ten-moment closures, ideal and relativistic MHD) and discusses both first-order IDP proofs and advanced high-order constructions, including geometric quasi-linearization (GQL) to handle nonlinear, implicit constraints. Through extensive theory and numerical experiments, the paper highlights the practical robustness and accuracy of IDP schemes, outlining pathways for designing reliable, high-order solvers for complex applications.

Abstract

Admissible states in hyperbolic systems and related equations often form a convex invariant domain. Numerical violations of this domain can lead to loss of hyperbolicity, resulting in illposedness and severe numerical instabilities. It is therefore crucial for numerical schemes to preserve the invariant domain to ensure both physically meaningful solutions and robust computations. For complex systems, constructing invariant-domain-preserving (IDP) schemes is highly nontrivial and particularly challenging for high-order accurate methods. This paper presents a comprehensive survey of IDP schemes for hyperbolic and related systems, with a focus on the most popular approaches for constructing provable IDP schemes. We first give a systematic review of the fundamental approaches for establishing the IDP property in first-order accurate schemes, covering finite difference, finite volume, finite element, and residual distribution methods. Then we focus on two widely used and actively developed classes of high order IDP schemes as well as their recent developments, most of which have emerged in the past decade. The first class of methods seeks an intrinsic weak IDP property in high-order schemes and then designs polynomial limiters to enforce a strong IDP property at the points of interest. This generic approach applies to high-order finite volume and discontinuousGalerkin schemes. The second class is based on the flux limiting approaches, which originated from the flux-corrected transport method and can be adapted to a broader range of spatial discretizations, including finite difference and continuous finite element methods. In this survey, we elucidate the main ideas in the construction of IDP schemes, provide some new perspectives and insights, with extensive examples, and numerical experiments in gas dynamics and magnetohydrodynamics.

High Order Numerical Methods Preserving Invariant Domain for Hyperbolic and Related Systems

TL;DR

This paper surveys invariant-domain-preserving (IDP) numerical schemes for hyperbolic and related PDEs, emphasizing convex invariant domains to guarantee physically meaningful solutions and stability. It contrasts two high-order IDP strategies: Zhang–Shu’s intrinsic weak IDP with polynomial limiters and flux-corrected transport (FCT)–style flux limiters applicable to FV, DG, FD, and CFEM discretizations, with particular attention to challenging systems like MHD and RMHD. The work covers a wide array of models (gas dynamics, shallow water, ten-moment closures, ideal and relativistic MHD) and discusses both first-order IDP proofs and advanced high-order constructions, including geometric quasi-linearization (GQL) to handle nonlinear, implicit constraints. Through extensive theory and numerical experiments, the paper highlights the practical robustness and accuracy of IDP schemes, outlining pathways for designing reliable, high-order solvers for complex applications.

Abstract

Admissible states in hyperbolic systems and related equations often form a convex invariant domain. Numerical violations of this domain can lead to loss of hyperbolicity, resulting in illposedness and severe numerical instabilities. It is therefore crucial for numerical schemes to preserve the invariant domain to ensure both physically meaningful solutions and robust computations. For complex systems, constructing invariant-domain-preserving (IDP) schemes is highly nontrivial and particularly challenging for high-order accurate methods. This paper presents a comprehensive survey of IDP schemes for hyperbolic and related systems, with a focus on the most popular approaches for constructing provable IDP schemes. We first give a systematic review of the fundamental approaches for establishing the IDP property in first-order accurate schemes, covering finite difference, finite volume, finite element, and residual distribution methods. Then we focus on two widely used and actively developed classes of high order IDP schemes as well as their recent developments, most of which have emerged in the past decade. The first class of methods seeks an intrinsic weak IDP property in high-order schemes and then designs polynomial limiters to enforce a strong IDP property at the points of interest. This generic approach applies to high-order finite volume and discontinuousGalerkin schemes. The second class is based on the flux limiting approaches, which originated from the flux-corrected transport method and can be adapted to a broader range of spatial discretizations, including finite difference and continuous finite element methods. In this survey, we elucidate the main ideas in the construction of IDP schemes, provide some new perspectives and insights, with extensive examples, and numerical experiments in gas dynamics and magnetohydrodynamics.

Paper Structure

This paper contains 87 sections, 18 theorems, 248 equations, 19 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Scalar conservation laws and order barriers
  4. First order schemes for systems
  5. High order schemes for systems
  6. Scope and organization of this paper
  7. Examples of invariant domains
  8. Gas dynamics equations
  9. Single layer shallow water equations
  10. Two-layer shallow water equations
  11. Ten-moment Gaussian closure model
  12. Ideal MHD equations
  13. Relativistic MHD equations
  14. First order schemes preserving invariant domains
  15. Monotone schemes for 1D scalar conservation laws
  16. ...and 72 more sections

Key Result

Lemma 2

\newlabellf-fact-NS0 Consider any ${\bf u}=(\rho, {\bm m}, E)^\top$, and where $p$, $\boldsymbol{\tau}$, and $\mathbf{q}$ are not necessarily dependent on ${\bf u}$. Let $e=\rho^{-1} E-\frac{1}{2} \rho^{-2}|{\bm m}|^2$. For any unit vector $\mathbf{n}$, let $v=\rho^{-1}{\bm m} \cdot\mathbf{n}$, $q=\mathbf{q}\cdot\mathbf{n}$ and $\tau=\mathbf{n}\cdot\boldsymbol{\tau}$.

Figures (19)

  • Figure 1: Notation for continuous finite element method. Left: $\mathcal{N}_i=\{i, j_1, j_2, j_3, j_4, j_5\}$.
  • Figure 1: One example of the special quadrature for quadratic polynomials. Left: 1D cell. Middle: 2D rectangle. Right: 2D triangle. For 1D, it is simply 3-point Gauss-Lobatto quadrature. The points in cyan color are the Gauss quadrature points for computing numerical flux integrals in high order schemes, and red points and cyan points together form a special quadrature that is exact for quadratic polynomials with positive weight.
  • Figure 1: An illustration of the parameters for enforcing positivity of both density and pressure: the first step is to find a box region $S_\rho$ (cyan color rectangle bounded by dashed lines) with four vertices $(0,0), A^1, A^2, A^3$, the second step is to find $B^i=rA^i$ with $r\in[0,1]$ such that $p(B^i)\geq\epsilon_p$ for each $i$. The convex hull of vertices $(0,0), B^1, B^2, B^3$ is the green polygon $S_p\subset S_{\rho, p}$, and the largest rectangle inside $S_p$ with two sides along axes is the red rectangle $R_{\rho,p}\subset S_{\rho, p}$.
  • Figure 1: (Example \ref{['Ex:ShockCloud']}) The density logarithm (top), thermal pressure (middle), and magnitude of magnetic field (bottom). Left: third order IDP DG scheme. Right: fifth order IDP finite volume scheme.
  • Figure 2: An illustration of notations for computing ${\bf c}_{ij}^T$ and ${\bf c}_{ij}$.
  • ...and 14 more figures

Theorems & Definitions (80)

  • Example 2.1: Positivity density and pressure for ideal gas
  • Example 2.2: Invariant domain with minimum entropy principle
  • Example 2.3: Invariant domain for any EOS
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 1: IDP numerical flux
  • Example 3.1: Godunov scheme
  • Example 3.2: Global Lax--Friedrichs flux
  • Example 3.3: Local Lax--Friedrichs flux
  • ...and 70 more