4 - Consistency and Stability Analysis

Summary

In this chapter, we will introduce the concept of consistency, stability, and convergence of finite difference schemes. Towards this end, various finite difference schemes are introduced and applied to well-known partial differential equations (PDEs) of the parabolic, hyperbolic and elliptic type.

Consistency and Stability Analysis

Analytical or closed form solutions of PDEs provide closed-form solutions/expressions that depict the variation of the dependant variables in the domain. The method of finite differences provide the numerical solutions with the values of the dependent variables at discrete grid points in the domain. Consider Figure 3.3, which shows a domain of interest in the xy plane in which the spacing of the grid points in the x-direction is uniform, and is given by Δx. Similarly, the spacing of the grid points in the y-direction is also uniform, and is given by Δy. In reality, it is not mandatory that Δx or Δy be constant (uniform). However, generally, PDEs are numerically solved on a grid that has uniform spacing in each (x and y) direction, as this simplifies the programming, and often yields higher accuracy solutions. In the present chapter, we assume uniform spacing in each coordinate direction. However, in general Δx ≠ Δy.

Basic Aspects of Finite Differences

Consider the following one-dimensional unsteady state linear heat conduction equation. Here, u (temperature) is a function of x and t (time), whereas α is a constant known as coefficient of thermal diffusivity:

Letting the time derivative in Equation (4.1) be replaced by a forward difference approximation, and the spatial derivative by a central difference approximation (this finite difference scheme is called FTCS, forward in time central in space scheme), one gets

Equation (4.2) is an explicit finite difference scheme as the unknown is explicitly expressed in terms of known terms and. Considering the truncation error of the FTCS scheme, one obtains

The terms in the square brackets in Equation (4.3) are the truncation error terms of the FTCS scheme.Observing Equation (4.3), it can be inferred that as Δx→0 and Δt→0, the truncation error approaches zero.