A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

doi:10.1007/s42967-021-00156-z

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (1): 170-198.doi: 10.1007/s42967-021-00156-z

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A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

Linjin Li¹, Jingmei Qiu¹, Giovanni Russo²

1 Department of Mathematical Sciences, University of Delaware, Newark, DE 19717, USA;
2 Department of Mathematics and Computer Science, University of Catania, Catania, Italy

收稿日期:2021-02-11 修回日期:2021-06-30 出版日期:2023-03-20 发布日期:2023-03-08
通讯作者: Jingmei Qiu,E-mail:jingqiu@udel.edu;Linjin Li,E-mail:llj@udel.edu;Giovanni Russo,E-mail:grusso@dmi.unict.it E-mail:jingqiu@udel.edu;llj@udel.edu;grusso@dmi.unict.it

A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

Linjin Li¹, Jingmei Qiu¹, Giovanni Russo²

1 Department of Mathematical Sciences, University of Delaware, Newark, DE 19717, USA;
2 Department of Mathematics and Computer Science, University of Catania, Catania, Italy

Received:2021-02-11 Revised:2021-06-30 Online:2023-03-20 Published:2023-03-08
Contact: Jingmei Qiu,E-mail:jingqiu@udel.edu;Linjin Li,E-mail:llj@udel.edu;Giovanni Russo,E-mail:grusso@dmi.unict.it E-mail:jingqiu@udel.edu;llj@udel.edu;grusso@dmi.unict.it

摘要/Abstract

摘要： In this paper, we propose a new conservative high-order semi-Lagrangian finite difference (SLFD) method to solve linear advection equation and the nonlinear Vlasov and BGK models. The finite difference scheme has better computational flexibility by working with point values, especially when working with high-dimensional problems in an operator splitting setting. The reconstruction procedure in the proposed SLFD scheme is motivated from the SL finite volume scheme. In particular, we define a new sliding average function, whose cell averages agree with point values of the underlying function. By developing the SL finite volume scheme for the sliding average function, we derive the proposed SLFD scheme, which is high-order accurate, mass conservative and unconditionally stable for linear problems. The performance of the scheme is showcased by linear transport applications, as well as the nonlinear Vlasov-Poisson and BGK models. Furthermore, we apply the Fourier stability analysis to a fully discrete SLFD scheme coupled with diagonally implicit Runge-Kutta (DIRK) method when applied to a stiff two-velocity hyperbolic relaxation system. Numerical stability and asymptotic accuracy properties of DIRK methods are discussed in theoretical and computational aspects.

关键词: Semi-Lagrangian, WENO, Finite difference, Vlasov-Poisson, BGK equation, Linear stability

Abstract: In this paper, we propose a new conservative high-order semi-Lagrangian finite difference (SLFD) method to solve linear advection equation and the nonlinear Vlasov and BGK models. The finite difference scheme has better computational flexibility by working with point values, especially when working with high-dimensional problems in an operator splitting setting. The reconstruction procedure in the proposed SLFD scheme is motivated from the SL finite volume scheme. In particular, we define a new sliding average function, whose cell averages agree with point values of the underlying function. By developing the SL finite volume scheme for the sliding average function, we derive the proposed SLFD scheme, which is high-order accurate, mass conservative and unconditionally stable for linear problems. The performance of the scheme is showcased by linear transport applications, as well as the nonlinear Vlasov-Poisson and BGK models. Furthermore, we apply the Fourier stability analysis to a fully discrete SLFD scheme coupled with diagonally implicit Runge-Kutta (DIRK) method when applied to a stiff two-velocity hyperbolic relaxation system. Numerical stability and asymptotic accuracy properties of DIRK methods are discussed in theoretical and computational aspects.

Key words: Semi-Lagrangian, WENO, Finite difference, Vlasov-Poisson, BGK equation, Linear stability

中图分类号:

65M25

Linjin Li, Jingmei Qiu, Giovanni Russo. A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 170-198.

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A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models

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