Overview of Elementary Temporal Discretizations

Abstract

A major element of any CFD procedure is the solution of the convection and diffusion transport equations. These govern such things as momentum and energy conservation and also the transport of passive scalars and certain turbulence parameters. To a lesser extent, CFD predictions can involve the solution of Lagrangian particle transport equations. For unsteady flows (and also the particle transport equations) the temporal discretization process needs to incorporate assumptions of how variables will change with time and also how this time variation varies over each cell. Generally, the process involves two key initial stages. First, just the spatial terms are discretized (again profile assumptions are involved and these are discussed in Chapter 3). These terms are generally represented here by the global symbol A. The temporal derivative remains, giving a semi-discrete Ordinary Differential Equation (\( \partial \phi /\partial t = A \)). This process of reducing the full governing equations to semi-discrete Ordinary Differential Equations (ODEs) is called the Method of Lines. The second stage involves discretization of the ODEs. There are numerous discretization assumptions and methods originating from general solution procedures for ODEs. The most common procedures, which have found application in CFD, will be described in this chapter. Also, such things as their stability, accuracy, computational economy and compatibility with different computer architectures will be discussed.