High order semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics
High order semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics
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DOI：
10.1016/j.jcp.2020.110036
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发表时间：
2021-02-15
影响因子：
4.1
通讯作者：
Qiu, Jing-Mei
中科院分区：
文献类型：
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作者：
Cai, Xiaofeng;Boscarino, Sebastiano;Qiu, Jing-Mei
本文提出了一种求解非线性Vlasov动力学问题的半拉格朗日间断Galerkin方法-龙格-库塔指数积分器（SLDG-RKEI）。Celledoni等人（FGCS，2003）提出了无积分器的Runge-Kutta（RK）指数积分器（EI）。在非线性输运设置中，RKEI可以用于将非线性输运的演化分解为线性化动力学序列的组成。由此产生的线性化输运方程可以通过Cai等人（JSC，2017）提出的半拉格朗日（SL）间断Galerkin（DG）方法求解。该方法可以通过SLDG框架实现高阶空间精度，通过RK EI实现高阶时间精度。由于SL的性质，所提出的SLDG-RKEI方法不受CFL条件的约束，因此它们有可能使用比欧拉方法更大的时间步长。继承SLDG方法的优点，建议的SLDG-RKEI格式是质量守恒，正性保持，没有维度分裂误差，在解决复杂的解决方案结构表现良好，并可以与自适应的时间步长的演变。通过对非线性Vlasov-Poisson系统和指导中心Vlasov模型的经典测试问题，验证了SLDG-RKEI算法的性能.虽然这不是我们的重点，本文探讨的SLDG-RKEI格式的非线性双曲守恒律发展冲击，我们显示了一些初步的结果，格式的性能的Burgers方程。(C)2020爱思唯尔公司All rights reserved.
In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation. (C) 2020 Elsevier Inc. All rights reserved.