On Unconditional Stability of Upwind Generalized Summation‐by‐Parts Schemes for Kinetic Equations in the Diffusion Limit

ABSTRACT Kinetic models describe a wide range of physical processes relevant to natural and engineering sciences where a large number of particles are involved. Recently an efficient numerical scheme for specific kinetic models modeling radiative transfer has been proposed by implicitly discretizing those terms in the micro–macro decomposition of the kinetic model which are linked to diffusion in the asymptotic limit. Unconditional stability for low‐order discontinuous Galerkin (DG) schemes in space has been rigorously proven in this previous work, meaning that for small scaling parameters near the macroscopic limit of the kinetic model, arbitrarily large time steps may be chosen. Upwind summation‐by‐parts (SBP) schemes provide a generic framework to construct robust, structure‐preserving approximations of higher order. In this work, the theoretical unconditional stability result is extended to higher order spatial discretization based on upwind SBP operators combined with implicit‐explicit Runge–Kutta (RK) time integrators. Numerical computations are carried out for a linear prototype kinetic model and a kinetic model with macroscopic limit given by the viscous Burgers equation.