A time-implicit algorithm for solving the Vlasov-Poisson equation

摘要

) are exactly conserved while the error in energy remains bounded. We demonstrate this algorithm for both the nonrelativistic and relativistic Vlasov equation. (Our interest in the relativistic system is motivated by our ultimate goal of applying this algorithm to the Maxwell-Vlasov system to study intense laser-plasma interactions.) A straightforward implementation of the implicit algorithm requires solving a large nonlinear system of equations at each time step. Operator splitting can be used to convert the nonlinear system to a collection of independent tri-diagonal linear systems that can be efficiently solved using the Thomas method. We present two versions of the algorithm, one based on the operator splitting method and one using a Newton-Krylov method to solve the nonlinear system. We consider a number of benchmark examples with both the full system as well as the linearized equations. We discuss the relative merits of the two implementations.