Temporal Discretization

Abstract

The methodologies for the temporal discretization of the governing equations can be divided into two main classes. One class comprises explicit time-stepping schemes and the other one consists of implicit schemes. We first concentrate on explicit methods and present Runge-Kutta and hybrid multistage schemes. In this connection, we discuss the treatment of source terms and the computation of the maximum time step. We then turn to implicit methodologies, where we present the various forms of the implicit operator and discuss popular solution schemes like ADI, LU-SGS, or Newton-Krylov. We also devote a section to implicit Runge-Kutta schemes. Finally, we present approaches for unsteady flows, in particular the dual time-stepping method.