Chapter 6 - Temporal Discretisation

Abstract

This chapter considers the most popular explicit and implicit time-stepping methods in some detail. How the maximum allowable time step can be evaluated for a particular scheme is presented and the issues of the appropriate implementations on structured as well as on unstructured grids are discussed in the chapter. To provide a more complete overview, recently developed solution methods based on the Newton iteration as well as standard techniques like the explicit Runge–Kutta schemes are discussed in the chapter. The time-accurate solutions of unsteady flow problems are also discussed. Every explicit time-stepping scheme remains stable only up to a certain value of the time step. To be stable, a time-stepping scheme has to fulfill the so-called Courant–Friedrichs–Lewy (CFL) condition. The implicit operator constitutes a large, sparse, and non-symmetric block matrix with dimensions equal to the total number of cells (cell-centered scheme) or grid points (cell-vertex scheme). For the Navier–Stokes equations, the viscous fluxes in the implicit operator have to be accounted for. The implicit Lower-Upper Symmetric Gauss–Seidel (LU-SGS) scheme, which is also called the Lower–Upper Symmetric Successive Overrelaxation (LU-SSOR) scheme, became widely used because of its low numerical complexity and modest memory requirements, both of which are comparable to an explicit multistage scheme. The LU-SGS scheme can be implemented easily on vector and parallel computers. The scheme can also be used on structured as well as on unstructured grids. The simulation of unsteady flow phenomena is becoming increasingly important in many engineering disciplines.