Conservative Multi-dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations
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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 and guiding center Vlasov model using high order tracing schemes. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by extensive numerical tests.
KeywordsSemi-Lagrangian Conservative High order WENO Linear stability analysis Fourier analysis Vlasov–Poisson system
The final draft of this paper is completed, while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Collaboration@ICERM program, supported by the National Science Foundation under Grant No. DMS-1439786.
- 28.Xiong, T., Qiu, J.-M., Russo, G.: High order multi-dimensional characteristic tracing for the incompressible Euler equation and the guiding-center Vlasov equation. J. Sci. Comput. (2018). https://doi.org/10.1007/s10915-018-0705-y)