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Monotone-based Numerical Schemes for Two-Dimensional Systems of Nonlocal Conservation Laws

Anika Beckers, Jan Friedrich

TL;DR

This work addresses two-dimensional systems of nonlocal conservation laws in which fluxes depend on local densities and nonlocal convolutions. It develops a class of monotone finite-volume schemes that handle nonlocal terms via interface flux approximations and proves convergence to the unique weak entropy solution under broad regularity assumptions, establishing an $O(\sqrt{Δt})$ error bound through Kuznetsov-type analysis. The framework accommodates general nonlinear fluxes and includes a numerical encryption-decryption scenario to validate practical robustness; existence/uniqueness are established, and numerical results illustrate convergence behavior for both non-smooth and smooth initial data. Overall, the paper provides a rigorous, applicable methodology for simulating 2D nonlocal balance laws with guaranteed convergence and quantifiable rates.

Abstract

We present a class of numerical schemes for two-dimensional systems of nonlocal conservation laws, which are based on utilizing well-known monotone numerical flux functions after suitably approximating the nonlocal terms. The considered systems are weakly coupled by the nonlocal terms and the underlying flux function is rather general to guarantee that our results are applicable to a wide range of common nonlocal models. We state sufficient conditions to ensure the convergence of the monotone-based numerical schemes to the unique weak entropy solution. Moreover, we provide an error estimate that yields the convergence rate of $\mathcal{O}(\sqrt{Δt})$ for the numerical approximations of the solution. Our results include an existence and uniqueness proof of the nonlocal system, too. Numerical results illustrate our theoretical findings.

Monotone-based Numerical Schemes for Two-Dimensional Systems of Nonlocal Conservation Laws

TL;DR

This work addresses two-dimensional systems of nonlocal conservation laws in which fluxes depend on local densities and nonlocal convolutions. It develops a class of monotone finite-volume schemes that handle nonlocal terms via interface flux approximations and proves convergence to the unique weak entropy solution under broad regularity assumptions, establishing an error bound through Kuznetsov-type analysis. The framework accommodates general nonlinear fluxes and includes a numerical encryption-decryption scenario to validate practical robustness; existence/uniqueness are established, and numerical results illustrate convergence behavior for both non-smooth and smooth initial data. Overall, the paper provides a rigorous, applicable methodology for simulating 2D nonlocal balance laws with guaranteed convergence and quantifiable rates.

Abstract

We present a class of numerical schemes for two-dimensional systems of nonlocal conservation laws, which are based on utilizing well-known monotone numerical flux functions after suitably approximating the nonlocal terms. The considered systems are weakly coupled by the nonlocal terms and the underlying flux function is rather general to guarantee that our results are applicable to a wide range of common nonlocal models. We state sufficient conditions to ensure the convergence of the monotone-based numerical schemes to the unique weak entropy solution. Moreover, we provide an error estimate that yields the convergence rate of for the numerical approximations of the solution. Our results include an existence and uniqueness proof of the nonlocal system, too. Numerical results illustrate our theoretical findings.
Paper Structure (11 sections, 12 theorems, 125 equations, 3 figures)

This paper contains 11 sections, 12 theorems, 125 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and assumptions
  3. A class of monotone-based first order schemes
  4. Convergence proof
  5. Error estimate
  6. Numerical examples
  7. Encryption-decryption
  8. Non-smooth initial data
  9. Smooth initial data
  10. Conclusion
  11. Appendix

Key Result

Theorem 3.4

Let the Asm. asm hold and $\boldsymbol{\rho}_0 \in \textnormal{BV}(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$. Then a monotone-based numerical scheme eq:1stscheme with a flux function from Def. def:flux converges for $\Delta x_1, \Delta x_2, \Delta t \to 0$ in $\left(L_{loc}^{1}\mleft( \mathbb{R}^2

Figures (3)

  • Figure 1: Discontinuous initial data (left) being encrypted until $t=0.75$ (middle) and decrypted (right) each plotted in the domain $[-4,4]^2$. Here an Upwind-type scheme was used with $N=6400$ cells in each direction.
  • Figure 2: Convergence analysis for discontinuous initial data \ref{['eq:nonsmoothinitial']}. The left side displays the decrease of the error employing Upwind-type flux on different grids with logarithmic axes. On the right, a table summarizes the errors and convergence rates for different grid sizes and using both the Lax-Friedrichs-type ACG15 and Upwind-type fluxes.
  • Figure 3: Encypted density at $t=0.75$ of the smooth inital function \ref{['eq:smoothinitial']} using an Upwind-type flux for $N=6400$ cells in each direction (left) and a table with errors and convergence rates for different grid sizes using the Lax-Friedrichs and Upwind-type numerical flux function.

Theorems & Definitions (28)

  • Definition 2.1: Weak entropy solution
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1: Decoupling of the system
  • Definition 3.2: Montone-based numerical flux function
  • Remark 3.3: Multiplicative flux functions
  • Theorem 3.4
  • Remark 3.5: Dimensional splitting
  • Theorem 3.6: Maximum principle
  • proof
  • ...and 18 more