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A Novel and Simple Invariant-Domain-Preserving Framework for PAMPA Scheme: 1D Case

Rémi Abgrall, Miaosen Jiao, Yongle Liu, Kailiang Wu

TL;DR

The paper addresses the challenge of preserving invariant-domain bounds in high-order PAMPA schemes for 1D hyperbolic conservation laws. It introduces a three-step IDP framework: (i) a local scaling limiter to enforce midpoint values within the invariant domain, (ii) a provably IDP update of cell averages using IDP numerical fluxes, and (iii) an unconditionally limiter-free update of point values via a non-conservative reformulation that maps to a domain-wide invariant set. Additional oscillation-control mechanisms (OE and MP) are integrated to damp nonphysical oscillations while preserving accuracy. Extensive 1D tests on advection, Burgers, Euler, and MHD demonstrate improved stability, positivity preservation, and high-order accuracy, with plans to extend to multidimensional problems.

Abstract

The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method, proposed in [R. Abgrall, Commun. Appl. Math. Comput., 5: 370-402, 2023], combines conservative and non-conservative formulations of hyperbolic conservation laws to evolve cell averages and point values. Solutions to hyperbolic conservation laws typically have an invariant domain, and ensuring numerical solutions stay within this domain is essential yet nontrivial. This paper presents a novel framework for designing efficient Invariant-Domain-Preserving (IDP) PAMPA schemes. We first analyze the IDP property for updated cell averages in the original PAMPA scheme, revealing the role of cell average decomposition and midpoint values in maintaining the invariant domain. This analysis highlights the difficulty of relying on continuous fluxes alone to preserve the invariant domain. Building on these insights, we introduce a simple IDP limiter for cell midpoint values, and propose a provably IDP PAMPA scheme that guarantees the preservation of the invariant domain for updated cell averages without requiring post-processing limiters. This approach contrasts with existing bound-preserving PAMPA schemes, which often require additional convex limiting to blend high-order and low-order solutions. Most notably, inspired by the Softplus and Clipped ReLU functions from machine learning, we propose an automatic IDP reformulation of the governing equations, resulting in an unconditionally limiter-free IDP scheme for evolving point values. We also introduce techniques to suppress spurious oscillations, enabling the scheme to capture strong shocks effectively. Numerical experiments on 1D problems, including the linear convection equation, Burgers equation, the compressible Euler equations, and MHD equations, demonstrate the accuracy and robustness of the proposed IDP PAMPA scheme.

A Novel and Simple Invariant-Domain-Preserving Framework for PAMPA Scheme: 1D Case

TL;DR

The paper addresses the challenge of preserving invariant-domain bounds in high-order PAMPA schemes for 1D hyperbolic conservation laws. It introduces a three-step IDP framework: (i) a local scaling limiter to enforce midpoint values within the invariant domain, (ii) a provably IDP update of cell averages using IDP numerical fluxes, and (iii) an unconditionally limiter-free update of point values via a non-conservative reformulation that maps to a domain-wide invariant set. Additional oscillation-control mechanisms (OE and MP) are integrated to damp nonphysical oscillations while preserving accuracy. Extensive 1D tests on advection, Burgers, Euler, and MHD demonstrate improved stability, positivity preservation, and high-order accuracy, with plans to extend to multidimensional problems.

Abstract

The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method, proposed in [R. Abgrall, Commun. Appl. Math. Comput., 5: 370-402, 2023], combines conservative and non-conservative formulations of hyperbolic conservation laws to evolve cell averages and point values. Solutions to hyperbolic conservation laws typically have an invariant domain, and ensuring numerical solutions stay within this domain is essential yet nontrivial. This paper presents a novel framework for designing efficient Invariant-Domain-Preserving (IDP) PAMPA schemes. We first analyze the IDP property for updated cell averages in the original PAMPA scheme, revealing the role of cell average decomposition and midpoint values in maintaining the invariant domain. This analysis highlights the difficulty of relying on continuous fluxes alone to preserve the invariant domain. Building on these insights, we introduce a simple IDP limiter for cell midpoint values, and propose a provably IDP PAMPA scheme that guarantees the preservation of the invariant domain for updated cell averages without requiring post-processing limiters. This approach contrasts with existing bound-preserving PAMPA schemes, which often require additional convex limiting to blend high-order and low-order solutions. Most notably, inspired by the Softplus and Clipped ReLU functions from machine learning, we propose an automatic IDP reformulation of the governing equations, resulting in an unconditionally limiter-free IDP scheme for evolving point values. We also introduce techniques to suppress spurious oscillations, enabling the scheme to capture strong shocks effectively. Numerical experiments on 1D problems, including the linear convection equation, Burgers equation, the compressible Euler equations, and MHD equations, demonstrate the accuracy and robustness of the proposed IDP PAMPA scheme.

Paper Structure

This paper contains 22 sections, 5 theorems, 82 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Invariant Domains in Hyperbolic Conservation Laws
  3. Review of the PAMPA Scheme
  4. Invariant-Domain-Preserving Analysis
  5. A Sufficient Condition for Ensuring IDP Cell Averages in PAMPA
  6. Challenges for IDP Cell Averages Using Only Continuous Flux
  7. Novel Efficient Invariant-Domain-Preserving PAMPA Framework
  8. Step 1: Local Scaling IDP Limiter for Midpoint Values
  9. Scalar Conservation Laws
  10. Hyperbolic Systems
  11. Step 2: Provably IDP Scheme for Updating Cell Averages without Any Extra Limiter
  12. Step 3: Unconditionally Limiter-Free IDP Scheme for Updating Point Values
  13. Suppression of Spurious Oscillations
  14. New OE Procedure for PAMPA Scheme
  15. MP Limiter for PAMPA Scheme
  16. ...and 7 more sections

Key Result

Proposition 2.1

If $\mathbf{U}_L, \mathbf{U}_R \in G$, there exists a suitable wave speed estimate $\lambda_{\max} (\mathbf{U}_L, \mathbf{U}_R)$ such that

Figures (10)

  • Figure 1: Example 7.2: Numerical solutions computed by PAMPA schemes with and without IDP technique.
  • Figure 2: Numerical results of PAMPA scheme with or without our IDP technique.
  • Figure 3: Example 7.5: Numerical solutions of IDP PAMPA scheme with 200 cells.
  • Figure 4: Example 7.6: Numerical solutions of IDP PAMPA scheme with 800 cells.
  • Figure 5: Example 7.7: Numerical results of IDP PAMPA scheme with 400 cells.
  • ...and 5 more figures

Theorems & Definitions (26)

  • Example 2.1: Scalar Conservation Laws
  • Example 2.2: Compressible Euler Equations
  • Example 2.3: Ideal MHD Equations
  • Proposition 2.1: Generalized Lax--Friedrichs Splitting Property
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Proof 1
  • Remark 4.4
  • Remark 4.5
  • ...and 16 more