标题： 保守的多维半拉格朗日有限差分格式：稳定性及其在动理学和流体模拟中的应用 显示英文标题

标题： Conservative Multi-Dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations

摘要： 本文中，我们提出了一种适用于多维问题且无需维度分裂的质量守恒半拉格朗日有限差分格式。 半拉格朗日格式通过从网格点向后追踪特征线来实现，不一定守恒总质量。 为了确保质量守恒，我们提出了一种基于通量差形式的保守修正过程。 该过程保证了局部质量守恒，同时引入了稳定性的时间步长约束。 我们从常微分方程的角度理论上研究了这些稳定性约束，假设空间微分算子被精确计算，并从傅里叶分析的角度研究了线性偏微分方程。 该格式通过经典的二维线性被动传输问题进行了测试，例如线性对流、旋转和旋涡变形。 该格式被用于求解非线性Vlasov-Poisson系统，使用了[Qiu和Russo, 2016]中提出的高阶追踪机制。 这种高阶特征追踪方案被推广到非线性引导中心Vlasov模型和不可压缩欧拉系统。 通过我们广泛的数值测试，所提出的保守半拉格朗日格式的有效性得到了数值证明。 显示英文摘要

摘要： In this paper, we propose a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems without dimensional splitting. The semi-Lagrangian scheme, based on tracing characteristics backward in time from grid points, does not necessarily conserve the total mass. To ensure mass conservation, we propose a conservative correction procedure based on a flux difference form. Such procedure guarantees local mass conservation, while introducing time step constraints for stability. We theoretically investigate such stability constraints from an ODE point of view by assuming exact evaluation of spatial differential operators and from the Fourier analysis for linear PDEs. The scheme is tested by classical two dimensional linear passive-transport problems, such as linear advection, rotation and swirling deformation. The scheme is applied to solve the nonlinear Vlasov-Poisson system using a a high order tracing mechanism proposed in [Qiu and Russo, 2016]. Such high order characteristics tracing scheme is generalized to the nonlinear guiding center Vlasov model and incompressible Euler system. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by our extensive numerical tests.