# Projective integration for nonlinear BGK kinetic equations

## TL;DR

This paper introduces a high-order, explicit projective integration scheme for nonlinear BGK kinetic equations that efficiently handles stiffness by combining small inner steps with an outer Runge-Kutta method, ensuring stability and accuracy.

## Contribution

The paper develops an asymptotic-preserving, fully explicit projective integration method for nonlinear BGK equations, allowing larger time steps independent of stiffness.

## Key findings

- Method achieves stability with outer time steps independent of stiffness
- Numerical results confirm high-order accuracy in 1D and 2D
- Inner steps effectively damp stiff components

## Abstract

We present a high-order, fully explicit, asymptotic-preserving projective integration scheme for the nonlinear BGK equation. The method first takes 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. Based on the spectrum of the linearized BGK operator, we deduce that, 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 BGK source term. We illustrate the method with numerical results in one and two spatial dimensions.