Asymptotic-Preserving Schemes for Multiscale Hyperbolic and Kinetic Equations

Abstract

Abstract Hyperbolic and kinetic equations often possess small spatial and temporal scales that lead to various asymptotic limits. Numerical approximation of these equations is challenging due to the presence of stiff source, collision, forcing terms, or when different scales coexist. Asymptotic-preserving (AP) schemes are numerical methods that are efficient in these asymptotic regimes. They are designed to capture the asymptotic limit at the discrete level without resolving small scales. This chapter aims to review the current status of AP schemes for a large class of hyperbolic and kinetic equations. We will first use simple models to illustrate the basic design principles, and then describe several generic AP strategies for handling general equations. Various aspects of the AP schemes for different asymptotic regimes, including some recent development, will be discussed as well.