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High-order In-cell Discontinuous Reconstruction path-conservative methods for nonconservative hyperbolic systems -- DR.MOOD generalization

Ernesto Pimentel-García, Manuel J. Castro, Christophe Chalons, Carlos Parés

TL;DR

The paper develops DR.MOOD, a general framework for nonconservative hyperbolic systems that combines unlimited high-order path-conservative methods, an in-cell discontinuous reconstruction, and MOOD limiting to robustly capture discontinuities. By extending MOOD to nonconservative contexts through Taylor-based reconstructions and a robust first-order correction, the approach achieves exact capture of isolated shocks while preserving high-order accuracy in smooth regions. The authors validate DR.MOOD on Modified Shallow Water and Two-Layer Shallow Water systems, demonstrating correct weak-solutions convergence, reduced time stepping in marked regions, and notable efficiency gains compared with full high-order in-cell reconstructions. The results indicate that the method effectively controls numerical viscosity near discontinuities and offers a viable path for extending high-order, shock-capturing schemes to broader 1D nonconservative problems, with future work including well-balanced analysis and multidimensional extensions.

Abstract

In this work we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in-cell discontinuous reconstruction operator are the key points to develop a new family of high-order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two-Layer Shallow Water system.

High-order In-cell Discontinuous Reconstruction path-conservative methods for nonconservative hyperbolic systems -- DR.MOOD generalization

TL;DR

The paper develops DR.MOOD, a general framework for nonconservative hyperbolic systems that combines unlimited high-order path-conservative methods, an in-cell discontinuous reconstruction, and MOOD limiting to robustly capture discontinuities. By extending MOOD to nonconservative contexts through Taylor-based reconstructions and a robust first-order correction, the approach achieves exact capture of isolated shocks while preserving high-order accuracy in smooth regions. The authors validate DR.MOOD on Modified Shallow Water and Two-Layer Shallow Water systems, demonstrating correct weak-solutions convergence, reduced time stepping in marked regions, and notable efficiency gains compared with full high-order in-cell reconstructions. The results indicate that the method effectively controls numerical viscosity near discontinuities and offers a viable path for extending high-order, shock-capturing schemes to broader 1D nonconservative problems, with future work including well-balanced analysis and multidimensional extensions.

Abstract

In this work we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in-cell discontinuous reconstruction operator are the key points to develop a new family of high-order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two-Layer Shallow Water system.
Paper Structure (24 sections, 1 theorem, 95 equations, 10 figures, 2 tables)

This paper contains 24 sections, 1 theorem, 95 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. MOOD methods for nonconservative systems.
  3. First-order path-conservative methods
  4. Unlimited high-order path-conservative methods
  5. Second-order method
  6. Third-order method
  7. MOOD procedure
  8. DR.MOOD methods
  9. First-order in-cell discontinuous reconstruction path-conservative methods
  10. Cell marking
  11. Time step
  12. Shock-capturing property
  13. DR.MOOD procedure
  14. Numerical tests
  15. Modified Shallow Water system
  16. ...and 9 more sections

Key Result

Theorem 3.2

Assume that ${\bf u}_l$ and ${\bf u}_r$ can be joined by an entropy shock of speed $\sigma$. Then, the numerical method eq:discrete1 provides an exact numerical solution of the Riemann problem with initial conditions in the sense that where ${\bf u}(x,t)$ is the exact solution.

Figures (10)

  • Figure 3: Cell classification.
  • Figure 4: Modified Shallow Water system. Test 1: Numerical solutions obtained using the MOOD approach with the first- , second- and third-order methods with and without in-cell discontinuous reconstruction based on the Roe matrix at time $t=0.15$ with 1000 cells. Top: variable $h$ (left), zoom middle state (right). Down: variable $q$ (left), zoom middle state (right).
  • Figure 5: Modified Shallow Water system. Test 2: Numerical solutions obtained using the MOOD approach with the first- , second- and third-order methods with and without in-cell discontinuous reconstruction based on the Roe matrix at time $t=0.15$ with 1000 cells. Top: variable $h$ (left), zoom middle state (right). Down: variable $q$ (left), zoom middle state (right).
  • Figure 6: Modified Shallow Water system. Test 2: Numerical solutions obtained using 1000- and 5000-cell meshes with the first- , second- and third-order DR.MOOD methods at time $t=0.15$. Top: variable $h$ (left), zoom middle state (right). Down: variable $q$ (left), zoom middle state (right).
  • Figure 7: Modified Shallow Water system. Test 2: Numerical solutions obtained using the first- and second-order in-cell discontinuous reconstruction methods with and without the MOOD approach at time $t=0.15$ with 1000 cells. Top: variable $h$ (left), zoom middle state (right). Down: variable $q$ (left), zoom middle state (right).
  • ...and 5 more figures

Theorems & Definitions (3)

  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3