Convergence analysis of an explicit splitting method for laser plasma interaction simulations

Abstract

Abstract. The convergence of a triple splitting method originally proposed by Tückmantel, Pukhov, Liljo, and Hochbruck for the solution of a simple model that describes laser plasma interactions with overdense plasmas is analyzed. For classical explicit integrators it is the large density parameter that imposes a restriction on the time step size to make the integration stable. The triple splitting method contains an exponential integrator in its central component and was specifically designed for systems that describe laser plasma interactions and overcomes this restriction. We rigorously analyze a slightly generalized version of the original method. This analysis enables us to identify modifications of the original scheme such that a second-order convergent scheme is obtained.Key words. exponential integrators, highly oscillatory problems, trigonometric integrators, splitting methods AMS subject classifications. 65P101. Introduction. We consider the numerical solution of a system of equations describing laser plasma interactions with an overdense plasma that is a simplified version of the models considered in [14,16]. Essentially as in [17], the laser is described by Maxwell's equations, and the plasma is modeled as a fluid only, in contrast to [14,16]. After discretizing in space with a fixed spatial grid size h, a system of ordinary differential equations is obtained. This highly oscillatory system of ordinary differential equations is discretized in time by a triple splitting method with filter functions. The introduction of filter functions is a widely-used method to avoid resonance effects in splitting methods applied to oscillatory differential equations; see, e.g., [4, 6, 7, 11] and [8].The situation to consider now is slightly unusual as it is the localized overdense plasma and not the spatial discretization that gives rise to fast oscillations in the solution. The plasma frequency is several orders of magnitude larger than the laser frequency, which has to be resolved by the spatial grid. Hence, it is the plasma density ρ that imposes a step size restriction in explicit Runge-Kutta or multistep methods. To overcome the restriction on the time step size due to the plasma density, a triple splitting method with filter functions was introduced in [13,17] for this model problem. An astute choice of filter functions results in a method that shows excellent behavior in numerical experiments. The numerical experiments in [17] indicate convergence of second order in the time step size τ independent of ρ. A more detailed experiment, which is reported in Section 8.1, reveals that the method from [17] is not of second order in τ independent of the plasma density but is merely stable.By our convergence analysis of the triple splitting we are able to formulate conditions on the filter functions to obtain second-order convergence in τ independent of the plasma density ρ. These conditions can be fulfilled by slightly modifying the choice of the filter functions originally proposed in [13,17]. The modification comes ...