Conservative stabilized Runge-Kutta methods for the Vlasov-Fokker-Planck equation

Ibrahim Almuslimani; Nicolas Crouseilles

doi:10.1016/j.jcp.2023.112241

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arXiv · 2212.13943

Conservative stabilized Runge-Kutta methods for the Vlasov-Fokker-Planck equation

Ibrahim Almuslimani·Nicolas Crouseilles

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Abstract

In this work, we aim at constructing numerical schemes, that are as efficient as possible in terms of cost and conservation of invariants, for the Vlasov--Fokker--Planck system coupled with Poisson or Ampère equation. Splitting methods are used where the linear terms in space are treated by spectral or semi-Lagrangian methods and the nonlinear diffusion in velocity in the collision operator is treated using a stabilized Runge--Kutta--Chebyshev (RKC) integrator, a powerful alternative of implicit schemes. The new schemes are shown to exactly preserve mass and momentum. The conservation of total energy is obtained using a suitable approximation of the electric field. An H-theorem is proved in the semi-discrete case, while the entropy decay is illustrated numerically for the fully discretized problem. Numerical experiments that include investigation of Landau damping phenomenon and bump-on-tail instability are performed to illustrate the efficiency of the new schemes.

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Ibrahim Almuslimani, Nicolas Crouseilles. 2023-06-10. Conservative stabilized Runge-Kutta methods for the Vlasov-Fokker-Planck equation. https://doi.org/10.1016/j.jcp.2023.112241

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