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

TL;DR

This paper develops efficient, conservative numerical schemes for the Vlasov-Fokker-Planck system using stabilized Runge-Kutta methods, ensuring mass, momentum, and energy conservation, with proven entropy decay and successful numerical tests.

Contribution

Introduction of stabilized Runge-Kutta-Chebyshev schemes that conserve key invariants for the Vlasov-Fokker-Planck equation, improving efficiency over implicit methods.

Findings

Exact mass and momentum conservation achieved.

Energy conservation via electric field approximation.

Numerical validation with Landau damping and bump-on-tail instability.

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 Amp\`ere 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.