This paper proposes a second-order accurate numerical scheme for the Patlak-Keller-Segel system with various mobilities for the description of chemotaxis. Formulated in a variational structure, the entropy part is novelly discretized by a modified Crank-Nicolson approach so that the solution to the proposed nonlinear scheme corresponds to a minimizer of a convex functional. A careful theoretical analysis reveals that the unique solvability and positivity-preserving property could be theoretically justified. More importantly, such a second order numerical scheme is able to preserve the dissipative property of the original energy functional, instead of a modified one. To the best of our knowledge, the proposed scheme is the first second-order accurate one in literature that could achieve both the numerical positivity and original energy dissipation. In addition, an optimal rate convergence estimate is provided for the proposed scheme, in which rough and refined error estimate techniques have to be included to accomplish such an analysis. Ample numerical results are presented to demonstrate robust performance of the proposed scheme in preserving positivity and original energy dissipation in blowup simulations.

翻译：本文针对描述趋化现象的Patlak-Keller-Segel系统，提出了一种具有不同迁移率的二阶精度数值格式。基于变分结构构造，熵部分采用修正的Crank-Nicolson方法进行创新性离散，使得所提非线性格式的解对应于凸泛函的极小化器。严谨的理论分析表明，该格式的解的存在唯一性与保正性可得到理论保证。更重要的是，该二阶数值格式能够保持原始能量泛函（而非修正能量泛函）的耗散性质。据我们所知，所提格式是文献中首个能同时实现数值保正性与原始能量耗散的二阶精度格式。此外，本文给出了该格式的最优收敛阶估计，其中必须引入粗糙与精细误差估计技术以完成该分析。大量数值结果展示了所提格式在爆破模拟中保持正性与原始能量耗散的稳健性能。