A MUSCL-Hancock scheme for non-local conservation laws

Abstract

In this article, we propose a MUSCL-Hancock-type second-order scheme for the discretization of a general class of non-local conservation laws and present its convergence analysis. The main difficulty in designing a MUSCL-Hancock-type scheme for non-local equations lies in the discretization of the convolution term, which we carefully formulate to ensure second-order accuracy and facilitate rigorous convergence analysis. We derive several essential estimates including $\mathrm{L}^\infty,$ bounded variation ($\mathrm{BV}$) and $\mathrm{L}^1$- Lipschitz continuity in time, which together with the Kolmogorov's compactness theorem yield the convergence of the approximate solutions to a weak solution. Further, by incorporating a mesh-dependent modification in the slope limiter, we establish convergence to the entropy solution. Numerical experiments are provided to validate the theoretical results and to demonstrate the improved accuracy of the proposed scheme over its first-order counterpart.