一类连续性方程的全离散能量耗散且守恒的离散梯度粒子方法
Fully discrete energy-dissipative and conservative discrete gradient particle methods for a class of continuity equations
AI总结：
针对一类具有变分结构的非线性连续性方程，提出满足正则化能量相容性的粒子方法，结合离散梯度时间积分器实现全离散层面的能量耗散与多物理量守恒，并通过数值算例验证特性。
摘要：
保结构粒子方法近期已被提出用于一类非线性连续性方程，包括J. Carrillo、K. Craig、F. Patacchini发表于《Calc. Var.》2019年第58卷第53页的工作中的聚集扩散方程，以及J. Carrillo、J. Hu、L. Wang、J. Wu发表于《J. Comput. Phys. X》2020年第7卷第100066页的工作中的Landau方程。这些方程的一个共同特征是都具有某种变分形式，经适当正则化后，可得到在半离散层面同时耗散能量且守恒部分物理量的粒子近似。本文中，我们构建了带有与能量泛函变分导数相关的密度依赖双线性型的连续性方程，并证明合适的粒子方法满足与其正则化能量的相容性条件。这使得我们能够利用离散梯度时间积分器，并证明在全离散层面可同时实现能量耗散与质量守恒。针对Landau方程的情形，我们证明该方法在全离散层面还能守恒动量与动能。文中给出了若干数值算例，以验证所提方法的耗散与守恒特性。
英文摘要：
Structure-preserving particle methods have recently been proposed for a class of nonlinear continuity equations, including aggregation-diffusion equation in [J. Carrillo, K. Craig, F. Patacchini, Calc. Var., 58 (2019), pp. 53] and the Landau equation in [J. Carrillo, J. Hu., L. Wang, J. Wu, J. Comput. Phys. X, 7 (2020), pp. 100066]. One common feature to these equations is that they both admit some variational formulation, which upon proper regularization, leads to particle approximations dissipating the energy and conserving some quantities simultaneously at the semi-discrete level. In this paper, we formulate continuity equations with a density dependent bilinear form associated with the variational derivative of the energy functional and prove that appropriate particle methods satisfy a compatibility condition with its regularized energy. This enables us to utilize discrete gradient time integrators and show that the energy can be dissipated and the mass conserved simultaneously at the fully discrete level. In the case of the Landau equation, we prove that our approach also conserves the momentum and kinetic energy at the fully discrete level. Several numerical examples are presented to demonstrate the dissipative and conservative properties of our proposed method.
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