Francis Filbet

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.

Numerical Simulation of the Smoluchowski Coagulation Equation

In this paper, we develop a numerical scheme for the Smoluchowski coagulation equation, which relies on a conservative formulation and a finite volume approach. Several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model. In particular the gelation phenomenon and the long time behavior of the solution are numerically studied.

High order numerical methods for the space non-homogeneous Boltzmann equation

In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) positive and flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrators in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods.

An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation

In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme.

Approximation of Hyperbolic Models for Chemosensitive Movement

Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first- and second-order well-balanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On the other hand, a high-order finite difference weighted essentially nonoscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme to exactly preserve steady states and to retain high-order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments.

A FINITE VOLUME SCHEME FOR NONLINEAR DEGENERATE PARABOLIC EQUATIONS

We propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional. For some of these models (porous media equation, drift-diffusion system for semiconductors) it has been proved that the transient solution converges to a steady state when time goes to infinity. The present scheme preserves steady states and provides a satisfying long time behavior. Moreover, it remains valid and second order accurate in space even in the degenerate case. After describing the numerical scheme, we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic.

High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

The main purpose of the paper is to show how to use implicit–explicit Runge–Kutta methods in a much more general context than usually found in the literature, obtaining very effective schemes for a large class of problems. This approach gives a great flexibility, and allows, in many cases the construction of simple linearly implicit schemes without any Newton’s iteration. This is obtained by identifying the (possibly linear) dependence on the unknown of the system which generates the stiffness. Only the stiff dependence is treated implicitly, then making the whole method much simpler than fully implicit ones. The resulting schemes are denoted as semi-implicit R–K. We adopt several semi-implicit R–K methods up to order three. We illustrate the effectiveness of the new approach with many applications to reaction–diffusion, convection diffusion and nonlinear diffusion system of equations.

Vlasov simulations of beams with a moving grid

Vlasov simulations can for some situations be a valuable alternative to PIC simulations for the study of intense beam propagation. However, as they rely on a phase-space grid which is fixed for the whole simulation, important computing effort can be wasted in zones where no particles are present at some given time. In order to overcome this drawback, we introduce here a new method which makes use of a phase-space grid which is uniform at any given time, but moves in time according to the evolution of the envelope of the beam.

Numerical methods for the Vlasov equation

In this paper, we give a fairly exhaustive review of the literature on numerical simulations of the Vlasov equation.We first recall the range of applications of the Vlasov equation and present the different approaches for the discretization.We briefly describe Lagrangian and Eulerian schemes and give a few numerical results comparing these methods.

An adaptive numerical method for the Vlasov equation based on a multiresolution analysis

Simulation of some problems in plasma physics or for high intensity beams requires the numerical resolution of the Vlasov equation on a mesh of phase space which doubles the number of dimensions. In order to optimize the number of mesh points where the distribution is computed, we developed a Vlasov solver using a multi resolution analysis where the distribution function is expanded on a wavelet basis spanning several scales. The idea of the method is to use a semi-Lagrangian type algorithm, where the characteristics are followed backwards, with time splitting between position and velocity advance. The interpolation points are chosen adaptively so as to put the computational effort where necessary. Our initial results using this method are presented.