Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson

A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov-Poisson equations is provided. It is shown that under suitable assumptions the error is of order O(τ 2 + h q + h q /τ ), where τ is the size of a time step, h is the cell size, and q the order of the discontinuous Galerkin approximation. In order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed.

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“…To conclude this section, let us note that alternative strategies have been introduced in the literature (see, e.g., [23,10]). …”

Section: Without Loss Of Generality Let Us Consider a Translation Of mentioning

confidence: 99%

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

Einkemmer¹,

Ostermann

2014

SIAM J. Numer. Anal.

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“…Our code employs a semi-Lagrangian approach based on a discontinuous Galerkin formulation. The algorithm has been proposed independently by [7,29,27]. A more detailed treatise on the particular implementation we employ can be found in [9].…”

Section: Description Of the Algorithm And Codementioning

confidence: 99%

Semi-Lagrangian Vlasov simulation on GPUs

Einkemmer

2020

Computer Physics Communications

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In this paper, our goal is to efficiently solve the Vlasov equation on GPUs. A semi-Lagrangian discontinuous Galerkin scheme is used for the discretization. Such kinetic computations are extremely expensive due to the high-dimensional phase space. The SLDG code abstracts the number of dimensions and uses a shared code base for both GPU and CPU based simulations. We investigate the performance of the implementation on a range of both Tesla (V100, Titan V, K80) and consumer (GTX 1080 Ti) GPUs. Our implementation is typically able to achieve a performance of approximately 470 GB/s on a single GPU and 1600 GB/s on four V100 GPUs connected via NVLink. This results in a speed up of about a factor of ten (comparing a single GPU with a dual socket Intel Xeon Gold node) and approximately a factor of 35 (comparing a single node with and without GPUs). In addition, we investigate the effect of single precision computation on the performance of the SLDG code and demonstrate that a template based dimension independent implementation can achieve good performance regardless of the dimensionality of the problem.

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“…This problem has been numerically investigated by several authors, e.g. see [41,14,49,57,29]. First, we perform the comparison for Q k SLDG-split and P k SLDG-(QC)-timek + 1, k = 1, 2.…”

Section: Vlasov-poisson Systemmentioning

confidence: 99%

A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations

2017

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Abstract. In this paper, we develop a class of high order conservative semi-Lagrangian (SL) discontinuous Galerkin (DG) methods for solving multi-dimensional linear transport equations. The methods rely on a characteristic Galerkin weak formulation, leading to L 2 stable discretizations for linear problems. Unlike many existing SL methods, the high order accuracy and mass conservation of the proposed methods are realized in a non-splitting manner. Thus, the detrimental splitting error, which is known to significantly contaminate long term transport simulations, will be not incurred. One key ingredient in the scheme formulation, borrowed from CSLAM [Lauritzen, Nair & Ullrich, 2010], is the use of Green's theorem which allows us to convert volume integrals into a set of line integrals. The resulting line integrals are much easier to approximate with high order accuracy, hence facilitating the implementation. Another novel ingredient is the construction of quadratic curves in approximating sides of upstream cell, leading to quadratic-curved quadrilateral upstream cells. Formal third order accuracy is obtained by such a construction. The desired positivity-preserving property is further attained by incorporating a high order bound-preserving filter. To assess the performance of the proposed methods, we test and compare the numerical schemes with a variety of configurations for solving several benchmark transport problems with both smooth and nonsmooth solutions. The efficiency and efficacy are numerically verified.

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Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson

Cited by 38 publications

References 26 publications

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

Semi-Lagrangian Vlasov simulation on GPUs

A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations

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