A Survey of Time-Differencing Schemes for the Oscillation and Decay Equations

In atmospheric dynamics, the governing equations are usually non-linear partial differential equations. Some knowledge of finite-difference approximations to ordinary differential equations (especially first order) is needed, however. In fact, if we linearize a governing partial differential equation and assume a wave form for the solution, the equation simply reduces to an ordinary differential equation. An example of this was given in Chapter 2. The stability of a finite difference approximation to such an ordinary differential equation can be examined using von Neumann's method, as explained in Chapter 2. In this Chapter, we deliberately side-step the complexities of space differencing and consider the problem of time differencing in isolation.