一种高阶多维特征追踪策略用于Vlasov-Poisson系统
A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov-Poisson System
AI总结：
本文提出一种高阶多维特征追踪策略，通过两阶段多导数预测-校正方法和VP系统的矩方程，实现高阶时间精度，避免时间步限制，但不保持质量守恒，通过数值实验验证其有效性。
摘要：
本文考虑了一种基于有限差分网格的半拉格朗日方法，用于求解Vlasov-Poisson (VP)系统。许多现有方法基于维度分离，将问题分解为求解线性输运问题。然而，这种分离会引入分离误差。若考虑多维问题而不进行分离，追踪高阶精度特征将面临困难。具体而言，特征的演化受电场支配，而电场由粒子密度分布通过泊松方程全局确定。本文提出一种新颖的策略，通过两阶段多导数预测-校正方法和VP系统的矩方程，实现高阶时间精度的特征追踪。通过准确定位特征点，使用加权本质非振荡(WENO)插值恢复网格点间的函数值，从而更新下一时间步的解。所提出算法不具有欧拉方法的时间步限制，具有高阶空间和时间精度。然而，此类有限差分算法不保持质量守恒；我们讨论了解决此问题的一种可能方法及其在数值稳定性方面的潜在挑战。通过经典测试问题如Landau阻尼和双流不稳定性，展示了所提出方案的性能。
英文摘要：
In this paper, we consider a finite difference grid-based semi-Lagrangian approach in solving the Vlasov-Poisson (VP) system. Many of existing methods are based on dimensional splitting, which decouples the problem into solving linear advection problems, see {\em Cheng and Knorr, Journal of Computational Physics, 22(1976)}. However, such splitting is subject to the splitting error. If we consider multi-dimensional problems without splitting, difficulty arises in tracing characteristics with high order accuracy. Specifically, the evolution of characteristics is subject to the electric field which is determined globally from the distribution of particle densities via the Poisson's equation. In this paper, we propose a novel strategy of tracing characteristics high order in time via a two-stage multi-derivative prediction-correction approach and by using moment equations of the VP system. With the foot of characteristics being accurately located, we proposed to use weighted essentially non-oscillatory (WENO) interpolation to recover function values between grid points, therefore to update solutions at the next time level. The proposed algorithm does not have time step restriction as Eulerian approach and enjoys high order spatial and temporal accuracy. However, such finite difference algorithm does not enjoy mass conservation; we discuss one possible way of resolving such issue and its potential challenge in numerical stability. The performance of the proposed schemes are numerically demonstrated via classical test problems such as Landau damping and two stream instabilities.
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