A second-order numerical method for the aggregation equations

TL;DR

This paper introduces a second-order numerical scheme for multi-dimensional aggregation equations, capable of handling solution blow-up and proven to converge to the unique gradient flow solution under certain conditions.

Contribution

It develops a TVD limiter-based second-order method for aggregation equations, extending simulation capabilities beyond blow-up times and providing convergence proof.

Findings

The scheme achieves second-order convergence rate.

Numerical experiments validate the method's accuracy.

The method effectively handles solution blow-up.

Abstract

Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, lambda-convex potentials with a possible Lipschitz singularity at the origin we prove that the method converges in the Monge--Kantorovich distance towards the unique gradient flow solution. Several numerical experiments are presented to validate the second-order convergence rate and to explore the performance of the scheme.