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Finite difference alternative WENO schemes with Riemann invariant-based local characteristic decompositions for compressible Euler equations

Yue Wu, Chi-Wang Shu

TL;DR

This work develops a finite-difference A-WENO scheme for the compressible Euler equations that uses a Riemann invariant–based local characteristic decomposition to reduce LCD cost. By selecting transform variables that sparsify the eigenstructure, the method maintains high-order accuracy and the E-property while enabling substantial LCD-cost reductions and robust stability via positivity-preserving limiters. Across 1D and 2D benchmarks, the RI-based approach achieves up to about 88% LCD-time savings and ~23% overall per-step savings at ninth order, with accuracy and non-oscillatory behavior preserved in shocks and smooth flows. The findings suggest significant practical impact for high-order simulations and potential generalization to other EOS and hyperbolic systems, including MHD and multi-component Euler equations.

Abstract

The weighted essentially non-oscillatory (WENO) schemes are widely used for hyperbolic conservation laws due to the ability to resolve discontinuities and maintain high-order accuracy in smooth regions at the same time. For hyperbolic systems, the WENO procedure is usually performed on local characteristic variables that are obtained by local characteristic decompositions to avoid oscillation near shocks. However, such decompositions are often computationally expensive. In this paper, we study a Riemann invariant-based local characteristic decomposition for the compressible Euler equations that reduces the cost. We apply the WENO procedure to the local characteristic fields of the Riemann invariants, where the eigenmatrix is sparse and thus the computational cost can be reduced. It is difficult to obtain the cell averages of Riemann invariants from those of the conserved variables due to the nonlinear relation between them, so we only focus on the finite difference alternative WENO versions. The efficiency and non-oscillatory property of the proposed schemes are well demonstrated by our numerical results.

Finite difference alternative WENO schemes with Riemann invariant-based local characteristic decompositions for compressible Euler equations

TL;DR

This work develops a finite-difference A-WENO scheme for the compressible Euler equations that uses a Riemann invariant–based local characteristic decomposition to reduce LCD cost. By selecting transform variables that sparsify the eigenstructure, the method maintains high-order accuracy and the E-property while enabling substantial LCD-cost reductions and robust stability via positivity-preserving limiters. Across 1D and 2D benchmarks, the RI-based approach achieves up to about 88% LCD-time savings and ~23% overall per-step savings at ninth order, with accuracy and non-oscillatory behavior preserved in shocks and smooth flows. The findings suggest significant practical impact for high-order simulations and potential generalization to other EOS and hyperbolic systems, including MHD and multi-component Euler equations.

Abstract

The weighted essentially non-oscillatory (WENO) schemes are widely used for hyperbolic conservation laws due to the ability to resolve discontinuities and maintain high-order accuracy in smooth regions at the same time. For hyperbolic systems, the WENO procedure is usually performed on local characteristic variables that are obtained by local characteristic decompositions to avoid oscillation near shocks. However, such decompositions are often computationally expensive. In this paper, we study a Riemann invariant-based local characteristic decomposition for the compressible Euler equations that reduces the cost. We apply the WENO procedure to the local characteristic fields of the Riemann invariants, where the eigenmatrix is sparse and thus the computational cost can be reduced. It is difficult to obtain the cell averages of Riemann invariants from those of the conserved variables due to the nonlinear relation between them, so we only focus on the finite difference alternative WENO versions. The efficiency and non-oscillatory property of the proposed schemes are well demonstrated by our numerical results.

Paper Structure

This paper contains 30 sections, 7 theorems, 36 equations, 12 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemann invariants
  4. The compressible Euler equations
  5. Numerical methods
  6. The A-WENO scheme with transform variables
  7. Reducing the cost of LCD
  8. Properties of the new scheme
  9. Positivity-preserving limiters
  10. Numerical results
  11. Efficiency test
  12. 1D tests
  13. Accuracy test 1
  14. Accuracy test 2
  15. The Sod shock tube problem
  16. ...and 15 more sections

Key Result

Corollary 2.1

If $w(\bm u)$ is an $i$-Riemann invariant, then for any $C^1$ function $g: \mathbb{R} \mapsto \mathbb{R}$, $g(w(\bm u))$ is also an $i$-Riemann invariant. Moreover, a differentiable function of several $i$-Riemann invariants is also an $i$-Riemann invariant.

Figures (12)

  • Figure 4.1: The Sod shock tube problem at $T = 2$ with $N=200$ grids.
  • Figure 4.2: The Lax shock tube problem at $T = 1.3$ with $N=200$ grids.
  • Figure 4.3: The LeBlanc shock tube problem at $T = 5 \times 10^{-5}$ with $N=2000$ grids.
  • Figure 4.4: The critical double rarefaction problem at $T = 3.3$ with $N=200$ grids.
  • Figure 4.5: The blast wave interaction problem at $T = 0.038$.
  • ...and 7 more figures

Theorems & Definitions (16)

  • Definition 2.1: hyperbolicity in 1D
  • Definition 2.2: hyperbolicity in multi-D
  • Remark 2.1
  • Definition 2.3: Riemann invariant
  • Corollary 2.1: closedness under composition of differentiable functions
  • Theorem 2.1: jump of Riemann invariants 10.1007/978-1-4612-0873-0
  • Proposition 2.1: change of variables
  • Corollary 2.2: sparsity of the eigenmatrix
  • Remark 2.2: coordinate system of Riemann invariants 10.1137/22M1536479
  • Proposition 2.2: Riemann invariants of the 2D compressible Euler equations
  • ...and 6 more