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A note on the central-upwind scheme for nonlocal conservation laws

Jan Friedrich, Samala Rathan, Sanjibanee Sudha

TL;DR

The paper develops and analyzes central-upwind (CU) and Kurganov–Tadmor (KT) schemes for one-dimensional nonlocal conservation laws with spatial convolution kernels. It derives a detailed fully-discrete KT scheme and a semi-discrete CU flux, providing a convergence proof for the first-order CU scheme to the unique entropy solution and establishing second-order convergence under suitable assumptions, with extensions to systems of nonlocal balance laws. Numerical experiments on scalar and multilane balance-law models compare CU, KT, and Godunov-type schemes, showing CU achieves comparable accuracy with lower computational cost and can act as a robust black-box solver for general nonlocal systems. The work broadens the applicability of high-resolution, low-diffusion schemes to nonlocal problems by refining speed estimates at cell interfaces and clarifying the relationship between CU, KT, and traditional Godunov methods. Overall, CU fluxes emerge as practical, efficient tools for simulating nonlocal conservation laws in physics- and engineering-driven applications.

Abstract

The central-upwind flux is a widely used numerical flux function for local conservation laws. It has been investigated by Kurganov and Polizzi (2009) for a specific nonlocal conservation law and can be derived from a fully-discrete second-order scheme. Here, we derive this fully-discrete scheme in detail with a particular focus on the occurring nonlocal terms. In addition, we derive the central-upwind flux for a class of nonlocal conservation laws and use an estimate on the nonlocal speed which fixes the nonlocality at the cell interfaces. We prove that the resulting first-order numerical scheme converges to the correct solution. Under additional assumptions on the analytical flux we present a similar result for a second-order central-upwind scheme. Numerical examples compare the central-upwind schemes to Godunov-type schemes and the fully-discrete scheme.

A note on the central-upwind scheme for nonlocal conservation laws

TL;DR

The paper develops and analyzes central-upwind (CU) and Kurganov–Tadmor (KT) schemes for one-dimensional nonlocal conservation laws with spatial convolution kernels. It derives a detailed fully-discrete KT scheme and a semi-discrete CU flux, providing a convergence proof for the first-order CU scheme to the unique entropy solution and establishing second-order convergence under suitable assumptions, with extensions to systems of nonlocal balance laws. Numerical experiments on scalar and multilane balance-law models compare CU, KT, and Godunov-type schemes, showing CU achieves comparable accuracy with lower computational cost and can act as a robust black-box solver for general nonlocal systems. The work broadens the applicability of high-resolution, low-diffusion schemes to nonlocal problems by refining speed estimates at cell interfaces and clarifying the relationship between CU, KT, and traditional Godunov methods. Overall, CU fluxes emerge as practical, efficient tools for simulating nonlocal conservation laws in physics- and engineering-driven applications.

Abstract

The central-upwind flux is a widely used numerical flux function for local conservation laws. It has been investigated by Kurganov and Polizzi (2009) for a specific nonlocal conservation law and can be derived from a fully-discrete second-order scheme. Here, we derive this fully-discrete scheme in detail with a particular focus on the occurring nonlocal terms. In addition, we derive the central-upwind flux for a class of nonlocal conservation laws and use an estimate on the nonlocal speed which fixes the nonlocality at the cell interfaces. We prove that the resulting first-order numerical scheme converges to the correct solution. Under additional assumptions on the analytical flux we present a similar result for a second-order central-upwind scheme. Numerical examples compare the central-upwind schemes to Godunov-type schemes and the fully-discrete scheme.

Paper Structure

This paper contains 10 sections, 2 theorems, 49 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Kurganov-Tadmor and central-upwind scheme for nonlocal conservation laws
  3. The Kurganov-Tadmor scheme
  4. The central-upwind flux
  5. Convergence of the central-upwind scheme
  6. Extension to system of nonlocal balance laws
  7. Numerical examples
  8. Scalar conservation law
  9. Systems of balance laws
  10. Conclusion

Key Result

Theorem 2.2

Using an explicit Euler scheme for the semi-discrete scheme eq:semidiscrete with the central-upwind flux as in eq:upwindflux, the resulting numerical scheme converges, under the CFL condition for $(\Delta t,\Delta x)\to 0$ to the unique weak entropy solution of eq:conslaw--eq:Fsimple in the sense of friedrich2023numerical and satisfies the maximum principle $\rho_j^n\in I$ for $j\in \mathbb{Z}$ a

Figures (3)

  • Figure 1: Grid and reconstructions used during the fully-discrete scheme.
  • Figure 2: Approximate solutions of the CU (first- and second-order), Godunov and KT scheme for a discontinuous initial data at $T=1$ with $\Delta x=0.02$. The right panel shows a zoom into the solution.
  • Figure 3: Approximate solutions of the CU, second-order CU and KT scheme for a discontinuous initial data at $T=0.25$ with $\Delta x=0.01$.

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Remark 2.5