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High-order WENO finite-difference methods for hyperbolic nonconservative systems of Partial Differential Equations

B. Ren, C. Parés

TL;DR

The paper develops a locally reconstructive, high-order WENO finite-difference framework for hyperbolic nonconservative PDEs by applying WENO to generalized fluxes derived from a path-based definition of nonconservative products. It introduces two strategies to incorporate source terms and achieve well-balanced schemes, including Roe-based linearizations when available, and extends the method from 1D to 2D. The approach is demonstrated on the 1D coupled Burgers system and both 1D and 2D two-layer shallow-water equations, showing high-order accuracy, robust shock-capturing, and preservation of water-at-rest steady states. The work highlights the importance of a symmetry property of the path in achieving high-order accuracy and discusses computational considerations, including the trade-offs between local versus global flux formulations and the cost of different path choices. Overall, the framework provides a unified, high-order mechanism to treat nonconservative terms consistently with conservation laws and to design well-balanced, high-fidelity simulations for multi-layer shallow-water flows and related systems.

Abstract

This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A new strategy is introduced here that allows non-conservative products to be written as the derivative of a generalized flux function that is defined locally on the basis of the selected family of paths. WENO reconstructions are then applied to this generalized flux. Moreover, if a Roe linearization is available, the generalized flux function can be evaluated through matrix vector operations instead of path-integrals. Two different known techniques are used to extend the methods to problems with source terms and the well-balanced properties of the resulting schemes are studied. These numerical schemes are applied to a coupled Burgers system and to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.

High-order WENO finite-difference methods for hyperbolic nonconservative systems of Partial Differential Equations

TL;DR

The paper develops a locally reconstructive, high-order WENO finite-difference framework for hyperbolic nonconservative PDEs by applying WENO to generalized fluxes derived from a path-based definition of nonconservative products. It introduces two strategies to incorporate source terms and achieve well-balanced schemes, including Roe-based linearizations when available, and extends the method from 1D to 2D. The approach is demonstrated on the 1D coupled Burgers system and both 1D and 2D two-layer shallow-water equations, showing high-order accuracy, robust shock-capturing, and preservation of water-at-rest steady states. The work highlights the importance of a symmetry property of the path in achieving high-order accuracy and discusses computational considerations, including the trade-offs between local versus global flux formulations and the cost of different path choices. Overall, the framework provides a unified, high-order mechanism to treat nonconservative terms consistently with conservation laws and to design well-balanced, high-fidelity simulations for multi-layer shallow-water flows and related systems.

Abstract

This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A new strategy is introduced here that allows non-conservative products to be written as the derivative of a generalized flux function that is defined locally on the basis of the selected family of paths. WENO reconstructions are then applied to this generalized flux. Moreover, if a Roe linearization is available, the generalized flux function can be evaluated through matrix vector operations instead of path-integrals. Two different known techniques are used to extend the methods to problems with source terms and the well-balanced properties of the resulting schemes are studied. These numerical schemes are applied to a coupled Burgers system and to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.
Paper Structure (33 sections, 3 theorems, 182 equations, 12 figures, 4 tables)

This paper contains 33 sections, 3 theorems, 182 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Path-Conservative WENO finite-difference reconstruction methods
  3. Conservative WENO finite difference schemes: a brief overview
  4. Extension to nonconservative systems
  5. Accuracy of the methods
  6. Problems with source terms and well-balanced property
  7. Well-balanced property
  8. Source terms
  9. Strategy 1
  10. Strategy 2
  11. Implementation
  12. Extension to 2D systems
  13. Homogeneous problems
  14. Problems with source terms
  15. Application to the two-layer shallow water model
  16. ...and 18 more sections

Key Result

Proposition 1

Let us consider a smooth solution $U(x,t)$ of eq:noncon and assume that $A(U)$ and $\Psi$ are smooth. We also assume that $\Psi$ satisfies for all $U, V \in \Omega$. Then we have where $p = 2k+1$ is the order of the reconstruction operator.

Figures (12)

  • Figure 1: Section \ref{['sec:coupled_rarefaction']}: smooth solution of the Coupled Burgers' system. Numerical solutions for $u$ (left) and $v$ (right) at $t=0.1$ obtained with the numerical methods consistent with the families of paths $\Psi_1$ and $\Psi_2$ using a mesh of 200 points.
  • Figure 2: Section \ref{['sec:coupled_shock']}: Riemann problem for the coupled Burgers' system. Numerical solutions for $u$ (left) and $v$ (right) at $t=0.1$ are obtained consistent with the families of paths $\Psi_1$ and $\Psi_2$ using a mesh of 200 points.
  • Figure 3: Section \ref{['subsec:small']}: small perturbation of a water-at-rest stationary solution with smooth (left) and discontinuous (right) bottom topographies. The numerical solution obtained with 200 points is compared with the reference solution obtained with 2000 points: water surface at $T = 0.15$.
  • Figure 4: Section \ref{['sec:Interface']}: Riemann problem 1. The numerical solution computed with 200 points is compared to a reference solution computed with 10000 points. Top left: water surface; top right: $h_1$ (zoom); bottom: $u_1$.
  • Figure 5: Section \ref{['sec:Interface']}: Riemann problem 2. The numerical solution obtained with $500$ points is compared to a reference solution computed with $5000$ points. Left: water surface; right: interface.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Remark 2