Asymptotic Preserving Implicit-Explicit Runge--Kutta Methods for Nonlinear Kinetic Equations

Abstract

We discuss implicit-explicit (IMEX) Runge-Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are also studied. In the case of the Boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral. This reformulation of the collision operator permits us to construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator. Finally, we show some numerical results which confirm the theoretical analysis. Introduction.The numerical solution of Boltzmann-type equations close to fluid regimes represents a real challenge for numerical methods. In these regimes, in fact, the intermolecular collision rate grows exponentially and the collisional time becomes very small. On the other hand, the actual time scale for evolution is the fluid dynamic time scale, which can be much larger than the collisional time. A nondimensional measure of the importance of collisions is given by the Knudsen number which is large in the rarefied regions and small in the fluid ones. Standard computational approaches lose their efficiency due to the necessity of using very small time steps in deterministic schemes or, equivalently, a large number of collisions in probabilistic approaches. Unfortunately the use of implicit solvers originates a prohibitive computational cost due to the high dimensionality and the nonlinearity of the collision integral.Several authors have tackled the above problem for the Boltzmann equation in the recent past (see [2,18,15,20,19] and the references therein). In summary, the possible approaches that permit us to overcome such a difficulty can be subdivided into two main classes: domain decomposition strategies and asymptotic preserving (AP) schemes. The first class of methods permits us to avoid the problem of very small Knudsen number by identifying the regions where it is possible to use the reduced fluid model and the regions where the full kinetic model must be solved. The literature in this direction has a long history and we recall here references [6, 33] and some recent works by Degond and coauthors [11,12]. A closely related research approach

Multiscale Schemes for the BGK--Vlasov--Poisson System in the Quasi-Neutral and Fluid Limits. Stability Analysis and First Order Schemes

“…The development of AP and AA schemes for kinetic models in the fluid limit has been already successfully treated in [12,38,43] for the BGK equation and in [25,26] for the full Boltzmann operator.…”

“…Note that the implicit treatment of the relaxation source term is the basis of AP schemes in the fluid limit developed in [12,25,26,38,43]. Taking the velocity moments of (21a) leads to…”

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Steady State and Sign Preserving Semi-Implicit Runge--Kutta Methods for ODEs with Stiff Damping Term

“…The violation of such requirement may result in unphysical solutions. Examples of a positivity preserving IMEX-RK method can be found in [8,15], but the use of a more restrictive than usual time step may be required to maintain the sign preserving property of these schemes.…”

Asymptotic-Preserving Scheme Based on a Finite Volume/Particle-In-Cell Coupling for Boltzmann-BGK-Like Equations in the Diffusion Scaling

“…When the system stays near equilibrium, the problem can be reduced using a macroscopic description, only depending on t and x. Several strategies can be used to solved multiscale problems (see for example [9], [8], [11] or [2]), among them, the micro-macro decomposition introduced in [1] leads to a coupling of two equations: a macroscopic one for the mean part of f (in velocity) and a microscopic one for the remainder part (called perturbation).…”