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.

中文翻译：

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聚合方程的二阶数值方法

受所谓的基于 TVD 限制器的双曲线守恒定律二阶方案的启发，我们开发了一种用于多维聚合方程的二阶精确数值方法。该方法允许在溶液的第一次爆破时间之后继续模拟。在原点可能存在 Lipschitz 奇点的对称 lambda-凸势势的情况下，我们证明该方法在 Monge--Kantorovich 距离中收敛于唯一的梯度流解。提出了几个数值实验来验证二阶收敛速度并探索该方案的性能。