Numerical Solution of the Vlasov–Ampère Equations

An implicit MacCormack-type scheme is constructed for a kinetic plasma model based on the Vlasov–Ampère equations. As compared with the explicit scheme, it has a weaker stability restriction, but preserves computational efficiency, i.e., it does not involve inner iterations. The error of the total energy corresponds to a second-order accurate algorithm, and the total charge (number of particles) is preserved at the grid level. The formation of plasma waves excited by a short intense laser pulse is modeled as an example.