Exponential Runge-Kutta schemes for inhomogeneous Boltzmann equations with high order of accuracy

We consider the development of exponential methods for the robust time discretization of space inhomogeneous Boltzmann equations in stiff regimes. Compared to the space homogeneous case, or more in general to the case of splitting based methods, studied in Dimarco Pareschi (SIAM J. Num. Anal. 2011) a major difficulty is that the local Maxwellian equilibrium state is not constant in a time step and thus needs a proper numerical treatment. We show how to derive asymptotic preserving (AP) schemes of arbitrary order and in particular using the Shu-Osher representation of Runge-Kutta methods we explore the monotonicity properties of such schemes, like strong stability preserving (SSP) and positivity preserving. Several numerical results confirm our analysis.

💡 Research Summary

The paper addresses the challenging problem of time integration for the space‑inhomogeneous Boltzmann equation in regimes where the collision term is stiff (i.e., the Knudsen number ε is very small). In such regimes the distribution function rapidly relaxes toward a local Maxwellian equilibrium that varies in both space and time. Traditional splitting‑based or IMEX schemes either treat the equilibrium as frozen over a time step or require costly implicit solves, which limits their accuracy and efficiency for non‑homogeneous problems.

To overcome these limitations the authors develop a family of Exponential Runge‑Kutta (ERK) methods. The key idea is to separate the linear stiff part, which can be integrated exactly via the exponential operator e^{-(t‑tⁿ)/ε}, from the nonlinear collision operator Q(f,f). By writing the exact Duhamel formula for the solution over a step