Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov–Maxwell system

The Time Splitting Scheme (TSS) has been examined within the context of the one-dimensional (1D) relativistic Vlasov–Maxwell model. In the strongly relativistic regime of the laser–plasma interaction, the TSS cannot be applied to solve the Vlasov equation. We propose a new semi-Lagrangian scheme based on a full 2D advection and study its advantages over the classical Splitting procedure. Details of the underlying integration of the Vlasov equation appear to be important in achieving accurate plasma simulations. Examples are given which are related to the relativistic modulational instability and the self-induced transparency of an ultra-intense electromagnetic pulse in the relativistic regime.