Splitting methods for Vlasov-Maxwell equations in plasma simulations

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Abstract

Since the splitting scheme with second order accuracy in time step for efficient computationof numerical solutions of the Vlasov-Poisson equations was published by Cheng and Knorr in1976, Vlasov simulations have made tremendous progress in the study of nonlinear plasmakinetic physics. The splitting scheme has since been extended to include electromagneticeffects and external magnetic field with complex magnetic field geometry. The splittingscheme splits the Vlasov equation into a series of lower-dimensional hyperbolic partialdifferential equations (i.e., the free-streaming and accelerating equations) in the spatial andvelocity space separately. These lower-dimensional equations have simple constant advectionspeed in the spatial dimensions and analytically tractable acceleration in the velocity space,and thus exact analytical solutions of these lower-dimensional equations can be obtained. Thelower-dimensional equations can be solved numerically by either performing interpolation orfinite difference or finite element or finite volume methods. Because the lower-dimensionalequations can be solved exactly by following the particle characteristics, the most naturalnumerical scheme is to perform interpolation. The important features of the splitting schemeare that not only the computation time is quite low, but also excellent results can be obtainedwith a few number of grid points. The advantage of the Vlasov simulations employing thesplitting scheme over the Particle-In-Cell methods has also been demonstrated on theconsideration of numerical noise, selective mode resolution, etc. Because the splitting methodcan be easily adapted to the massively parallel processor (MPP) technology, Vlasovsimulations using the splitting scheme should play a greater role in the study of nonlinearplasma kinetic physics. Here, we review the splitting scheme and its applications in theinteraction of nonlinear Langmuir solitons with plasma, and electron cyclotron waves.

Original languageEnglish
Title of host publicationEulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas
PublisherNova Science Publishers, Inc.
Pages1-21
Number of pages21
ISBN (Print)9781616684136
Publication statusPublished - 2010

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

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  • Cite this

    Cheng, C. Z. (2010). Splitting methods for Vlasov-Maxwell equations in plasma simulations. In Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas (pp. 1-21). Nova Science Publishers, Inc..