Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations

TL;DR

This paper introduces high-order explicit projective integration schemes for nonlinear kinetic equations like BGK and Boltzmann, enabling efficient simulations independent of stiffness through a telescopic hierarchy of integration levels.

Contribution

The authors develop a recursive telescopic projective integration method that reduces computational cost for stiff kinetic equations by decoupling the time step restriction from stiffness.

Findings

Method's cost is independent of stiffness with proper parameters.

Numerical results demonstrate effectiveness in 1D and 2D cases.

Hierarchical levels depend logarithmically on stiffness.

Abstract

We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.