A convergent MUSCL–Hancock scheme for non-local conservation laws

A MUSCL–Hancock (MH) second-order scheme for the discretization of a general class of non-local conservation laws is developed, and its convergence analysis is presented. The MH scheme is a single-stage fully discrete second-order method, which is well-known for its simplicity and computational efficiency in the context of conservation laws. The core difficulty in designing such a scheme for non-local problems originates from the convolution operator embedded in the flux functions. This is handled through a novel strategy that ensures second-order accuracy and facilitates a rigorous theoretical analysis. Several essential estimates including L∞, bounded variation (BV) and L1-Lipschitz continuity in time are derived. Applying these results within the framework of Kolmogorov’s compactness theorem, we prove the convergence of a subsequence of approximate solutions to a weak solution. In general, establishing a discrete entropy inequality for second-order schemes remains elusive, making the proof of entropy convergence difficult. To overcome this, we resort to an approach in which a mesh-dependent modification to the classical slope limiter is utilized to 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.