Time discretisation

From CFD-Wiki

Introduction

In seeking a numerical solution for a partial differential equation, discretization has to be carried out in both space and time. Although, mathematically, the time dependent terms, i.e. transient terms, are just derivatives with respect to an independent variable (time), these terms require special treatment when looked upon from a physical point of view.

Without loss of generality, in the context of conservation laws, transient terms describe the accumulation in time, of a certain variable inside an infinitesimal control volume. Discretization of the transient terms is usually called temporal discretization or discretization in time. It is always desirable to seek a time dependent solution especially that the discretization of the transient terms is directly associated with the stability of a numerical solution. If the flow at hand is inherently steady, it is generally advisable to compute a time dependent solution and reach the steady state solution hereafter.

The following sections discuss several temporal discretization schemes.