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The Lax-Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws

Janina Bender, Thomas Izgin, Philipp Öffner, Davide Torlo

TL;DR

This work addresses the gap in convergence theory for hyperbolic conservation laws when using nonlinear, positivity-preserving Patankar-type time integrators. By introducing a total time variation ($\mathrm{TTV}$) bound and embedding Patankar schemes within a difference-scheme framework, the authors extend the classical Lax--Wendroff theorem to nonlinear time integration. They establish conditions under which MP difference and Runge--Kutta (NSARK) methods converge to the weak (entropy) solution, and validate the theory with extensive numerical tests on Burgers' and Buckley--Leverett equations, as well as shallow-water models, including high-order in time and space with WENO reconstructions. The results provide a rigorous foundation for MP methods in hyperbolic PDEs, enabling unconditionally positivity-preserving, high-order schemes that converge to physically relevant solutions while guiding future work on removing the $\mathrm{TTV}$ assumption and designing entropy-preserving MP schemes.

Abstract

For hyperbolic conservation laws, the famous Lax-Wendroff theorem delivers sufficient conditions for the limit of a convergent numerical method to be a weak (entropy) solution. This theorem is a fundamental result, and many investigations have been done to verify its validity for finite difference, finite volume, and finite element schemes, using either explicit or implicit linear time-integration methods. Recently, the use of modified Patankar (MP) schemes as time-integration methods for the discretization of hyperbolic conservation laws has gained increasing interest. These schemes are unconditionally conservative and positivity-preserving and only require the solution of a linear system. However, MP schemes are by construction nonlinear, which is why the theoretical investigation of these schemes is more involved. We prove an extension of the Lax-Wendroff theorem for the class of MP methods. This is the first extension of the Lax--Wendroff theorem to nonlinear time integration methods with just an additional hypothesis on the total time variation boundedness of the numerical solutions. We provide some numerical simulations that validate the theoretical observations.

The Lax-Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws

TL;DR

This work addresses the gap in convergence theory for hyperbolic conservation laws when using nonlinear, positivity-preserving Patankar-type time integrators. By introducing a total time variation () bound and embedding Patankar schemes within a difference-scheme framework, the authors extend the classical Lax--Wendroff theorem to nonlinear time integration. They establish conditions under which MP difference and Runge--Kutta (NSARK) methods converge to the weak (entropy) solution, and validate the theory with extensive numerical tests on Burgers' and Buckley--Leverett equations, as well as shallow-water models, including high-order in time and space with WENO reconstructions. The results provide a rigorous foundation for MP methods in hyperbolic PDEs, enabling unconditionally positivity-preserving, high-order schemes that converge to physically relevant solutions while guiding future work on removing the assumption and designing entropy-preserving MP schemes.

Abstract

For hyperbolic conservation laws, the famous Lax-Wendroff theorem delivers sufficient conditions for the limit of a convergent numerical method to be a weak (entropy) solution. This theorem is a fundamental result, and many investigations have been done to verify its validity for finite difference, finite volume, and finite element schemes, using either explicit or implicit linear time-integration methods. Recently, the use of modified Patankar (MP) schemes as time-integration methods for the discretization of hyperbolic conservation laws has gained increasing interest. These schemes are unconditionally conservative and positivity-preserving and only require the solution of a linear system. However, MP schemes are by construction nonlinear, which is why the theoretical investigation of these schemes is more involved. We prove an extension of the Lax-Wendroff theorem for the class of MP methods. This is the first extension of the Lax--Wendroff theorem to nonlinear time integration methods with just an additional hypothesis on the total time variation boundedness of the numerical solutions. We provide some numerical simulations that validate the theoretical observations.
Paper Structure (17 sections, 6 theorems, 59 equations, 6 figures)

This paper contains 17 sections, 6 theorems, 59 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Consistency and convergence for difference schemes for conservation laws
  3. Patankar-type schemes
  4. Modified Patankar for Production-Destruction Systems
  5. Modified Patankar for difference schemes
  6. Lax--Wendroff theorem for MP difference schemes
  7. Extension to Patankar-type RK
  8. Numerical results
  9. First order tests
  10. Burgers' equation
  11. Buckley--Leverett equation
  12. High order tests
  13. High order in time
  14. High order in space and time with WENO-FV
  15. Conclusions and perspectives
  16. ...and 2 more sections

Key Result

Theorem 1

Consider a sequence of step sizes $\Delta x_{\{l\}} > 0$ with $\lim_{l{ \rightarrow \infty}} \Delta x_{\{l\}} = 0$. Let $\left( U^{\{l\}} \right)_l$ be a sequence of piece-wise constant functions constructed from numerical solutions obtained by a difference scheme consistent with eq:InitProb with ti

Figures (6)

  • Figure 1: Simulations (top) and numerical shock location errors (bottom) for Burgers' equation \ref{['eq:Burgers']} with upwind numerical flux and MPE. CFL=1 on the left and CFL=2.1 on the right
  • Figure 2: Simulations for explicit Euler (first line), MPE (third line) and numerical shock location errors for explicit Euler (second line) and MPE (last line) for Buckley--Leverett equation \ref{['eq:Buckley']} with upwind numerical flux and MPE. $\text{CFL}=0.99$ (left), $\text{CFL}=1.20$ (center) and $\text{CFL}=1.99$ (right)
  • Figure 3: Burgers' equation \ref{['eq:Burgers']} simulations and error of the shock location with MPDeC of order 2 (left), 3 (center) and 4 (right) and upwind numerical flux with different CFLs
  • Figure 4: Total time variation for various methods and different CFL number for Burgers' equation \ref{['eq:Burgers']} with IC \ref{['eq:RP']} with $u_1=2$ and $u_2=10^{-13}$. Observe that for different methods different scales are used.
  • Figure 5: Burgers' (top) and SW dam break (bottom), solutions and shock location error refining the mesh: MPDeC2 (left), MPSSPRK3 (center) and MPDeC5 (right)
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 1: Lax--Wendroff leveque_numerical_1992
  • Example 1: Modified Patankar Euler
  • Lemma 2: MPE weights convergence
  • proof
  • Theorem 3: Lax--Wendroff for MPE difference schemes
  • proof
  • Lemma 4: Extended Patankar weights convergence
  • proof
  • Theorem 5: Lax--Wendroff for MPRK schemes
  • proof
  • ...and 3 more