Nonoscillatory Interpolation Methods Applied to Vlasov-Based Models

Abstract

We demonstrate the ability of nonoscillatory interpolation strategies for solving efficiently the transport phase in kinetic systems with applications to charged particle transport in plasmas and semiconductors. Pointwise weighted essentially nonoscillatory (PWENO) interpolation is applied to obtain semi-Lagrangian and flux balance methods that together with splitting techniques form the building blocks of our numerical approach. These methods do not present the restrictive CFL condition typical of finite-difference methods with explicit time-solvers, and, moreover, they provide reliable results controlling parasite oscillations from classical polynomial interpolation while giving highly accurate approximations of smooth parts of the solutions. We perform and compare these methods in different benchmark problems for Vlasov or collisional models for charged particle transport.

Asymptotic-Preserving Particle-In-Cell method for the Vlasov–Poisson system near quasineutrality

“…We refer to the two celebrated books [3,27] for an overview of these methods. Recently, grid-based eulerian simulations have received a great deal attention [1,6,16,17,18,38,40] but particle methods are still the number-one method used for the numerical simulation of plasma kinetic models. The convergence of PIC methods has been mathematically investigated in [9,19].…”

Comparison of Vlasov solvers for spacecraft charging simulation

“…This idea is similar to the one used in WENO-type interpolations methods (see [3] and references therein) for reconstruction of smooth solutions.…”

“…We refer e.g. to [13,30] for details on this interpolation step, an alternative approach based on the WENO procedure has been proposed recently in [3]. However, most of the references we are aware of restrict to periodic boundary conditions and do not address the question of the interpolation rule to be adopted for boundary points.…”

Numerical Schemes of Diffusion Asymptotics and Moment Closures for Kinetic Equations

“…This update might make the relation between f and g consistent at the end of the second step but it leads to undesirable numerical divisions by the small parameter ε; but, for well-prepared initial data, this consistency can be imposed at the beginning. Similar arguments were already given in [30,31] to avoid this update of the fluctuations g. Before proceeding further with the analysis of this kinetic method, we show in Figure 1 a comparison between the results of the three discussed kinetic methods: a semi-lagrangian PWENO6,4-interpolation scheme [10] (SL-WENO), the asymptotic preserving method without update of g proposed above and the asymptotic preserving method with update of g. In all cases, we show the L 2 t,x,v -error between the kinetic results and the solutions of the heat equation, its ε → 0 asymptotic limit, in a log-plot depending on ε. The results show that the kinetic scheme proposed in this paper works perfectly in the ε → 0 regime while both the updated scheme and the SL-WENO scheme do not describe well the asymptotic limit.…”

“…One can repair this drawback by using interpolation procedures based on the WENO approach. We refer to [47,46] for the basis of the WENO method, and to [10] for a description of the adaptation to design an accurate interpolation method. Of course, for small ε's these computations become unbearably time consuming with large meshes and small time step due to the large velocities that are involved.…”