A second-order asymptotic-preserving and positivity-preserving exponential Runge-Kutta method for a class of stiff kinetic equations
This work provides a robust numerical method for simulating kinetic equations across multiple regimes, benefiting computational physicists and engineers modeling rarefied gas dynamics.
The authors developed a second-order time discretization method for stiff kinetic equations that is both asymptotic-preserving (captures the Euler limit without resolving small Knudsen numbers) and positivity-preserving (maintains non-negativity of the solution). The method applies to BGK, Fokker-Planck, and Boltzmann equations, and numerical tests confirm its theoretical properties.
We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) -- can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving -- can preserve the non-negativity of the solution which is a probability density function for arbitrary Knudsen numbers. The method is based on a new formulation of the exponential Runge-Kutta method and can be applied to a large class of stiff kinetic equations including the BGK equation (relaxation type), the Fokker-Planck equation (diffusion type), and even the full Boltzmann equation (nonlinear integral type). Furthermore, we show that when coupled with suitable spatial discretizations the fully discrete scheme satisfies an entropy-decay property. Various numerical results are provided to demonstrate the theoretical properties of the method.