Using linear multistep methods for the time stepping in Vlasov–Poisson simulations

Abstract

The numerical simulation of hot and low density plasmas using the Vlasov–Poisson model is necessary for many practical applications such as the characterization of laboratory and astrophysical plasmas. The numerical treatment of the Vlasov equation is addressed using Eulerian methods when high precision and low noise are required. Among these methods, we highlight those based on finite-volumes without splitting, which have shown to be a good option for capturing small structures in phase space with low dissipation while preserving positivity. The problem is that kinetic simulations usually require the discretization of 3–6-dimensional phase-spaces which results in a huge number of ordinary differential equations (ODEs). This stands out the importance of using efficient schemes for the time integration. In this article, linear multistep methods are implemented for the time stepping of the resulting equations, and compared against traditional Runge–Kutta ones. Schemes with built-in error estimation are also implemented in an attempt to perform adaptive stepsize control. Their accuracy, stability and computational cost are compared through the simulation of classical benchmark problems for the two-dimensional Vlasov–Poisson system.