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Convergence analysis for a finite volume evolution Galerkin method for multidimensional hyperbolic systems

Mária Lukáčová-Medvidová, Zhuyan Tang, Yuhuan Yuan

TL;DR

This work develops a convergence theory for a genuinely multidimensional finite volume evolution Galerkin (FVEG) method for hyperbolic conservation laws, focusing on the linear wave equation and the nonlinear Euler equations. The method relies on the bicharacteristics-based evolution operator EG2 to predict edge states and construct fluxes, with entropy stability enforced via an entropy-conservative flux and diffusion-informed corrections. By combining stability, a consistency framework, and a generalized Lax equivalence principle for dissipative solutions, the authors prove convergence to a dissipative weak solution for Euler systems and to a weak solution for the wave system, with strong convergence guaranteed when a strong solution exists. Numerical results for both linear and nonlinear cases illustrate first-order convergence and robust performance on vortex, spiral, and 2D Riemann problems, underscoring the practical relevance of the theoretical results for multidimensional hyperbolic dynamics.

Abstract

We study the convergence of a finite volume method based on the method of bicharacteristics for multidimensional hyperbolic conservation laws. In particular, we concentrate on the linear wave equation system and nonlinear Euler equations of gas dynamics. We show the stability and the consistency of the numerical approximations. By means of the generalized Lax equivalence principle we prove the convergence of numerical solutions to the strong solution on the lifespan.

Convergence analysis for a finite volume evolution Galerkin method for multidimensional hyperbolic systems

TL;DR

This work develops a convergence theory for a genuinely multidimensional finite volume evolution Galerkin (FVEG) method for hyperbolic conservation laws, focusing on the linear wave equation and the nonlinear Euler equations. The method relies on the bicharacteristics-based evolution operator EG2 to predict edge states and construct fluxes, with entropy stability enforced via an entropy-conservative flux and diffusion-informed corrections. By combining stability, a consistency framework, and a generalized Lax equivalence principle for dissipative solutions, the authors prove convergence to a dissipative weak solution for Euler systems and to a weak solution for the wave system, with strong convergence guaranteed when a strong solution exists. Numerical results for both linear and nonlinear cases illustrate first-order convergence and robust performance on vortex, spiral, and 2D Riemann problems, underscoring the practical relevance of the theoretical results for multidimensional hyperbolic dynamics.

Abstract

We study the convergence of a finite volume method based on the method of bicharacteristics for multidimensional hyperbolic conservation laws. In particular, we concentrate on the linear wave equation system and nonlinear Euler equations of gas dynamics. We show the stability and the consistency of the numerical approximations. By means of the generalized Lax equivalence principle we prove the convergence of numerical solutions to the strong solution on the lifespan.

Paper Structure

This paper contains 20 sections, 7 theorems, 104 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Finite volume evolution Galerkin method
  3. Notations
  4. Finite volume evolution Galerkin method
  5. FVEG2 for the wave equation system
  6. Case $P=S$:
  7. Case $P=A$:
  8. FVEG2 for the Euler equations
  9. Case $P=S$:
  10. Case $P=A$:
  11. Stability
  12. Entropy stability of the FVEG flux
  13. Consistency
  14. Difference between $U^{EG}_{X}$ and $U_L, U_R$
  15. Consistency formulation
  16. ...and 5 more sections

Key Result

Lemma 3.1

Suppose that where $\{\bar{{\bm r}}^k_+\}^N_{k=1}$ is the right orthonormal eigenvectors of $\bar{\bm{A}}^{j+\frac{1}{2}}:=\bm{A}({\bm U}^j_{\sigma}(0))$, with the corresponding eigenvalues $\{\bar{\lambda}^k_+\}^N_{k=1}$. Then it holds where and $\tilde{c}>\frac{(c_1)^2c_2}{36} + \frac{1}{16} c_2c_1^2$ with $c_1=\max|\frac{\mathrm{d}{\bm r}^j}{\mathrm{d}{\bm U}}|$, $c_2=\max|\rho({\bm{A}}({\b

Figures (11)

  • Figure 1: Illustration of the update point values on a Cartesian mesh.
  • Figure 2: Illustration of possible update of point values for the linearized Euler equations.
  • Figure 3: Example \ref{['e1']}: first-order convergence rates.
  • Figure 4: Solutions of Example \ref{['e1']} at $T=0.1$ and $1/h=80$: $\phi$ (left), $u$ (middle), $v$ (right).
  • Figure 5: Solutions of Gresho problem computed at $T=1$ on a grid with $512\times512$: $\rho$ (left), $p$ (middle), $|{\bm u}|$ (right).
  • ...and 6 more figures

Theorems & Definitions (22)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4: Weak BV estimate
  • Remark 3.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 12 more