ABSTRACT
As an example of the von Neumann method of stability analysis, let's perfonn a stability analysis of the FTCS approximation of the diffusion equation, Eq. (10.25):
/;//+1 = /;// + d(/;~I - 2/;// +/;"-1) (10.42) The required Fourier components are given by Eq. (10.40). Substituting Eq. (10.40) into Eq. (10.42) gives
/;//+1 = /;// + d(f/l/O _ 2/;// +/;//e-1o) which can be written as
(1O.43a)
(10.43b)/;//+1 =/;//[1 + d(e1o + e-1o _ 2)] = /;// [1 + 2d(e'O ~ e-10 - 1)] Introducing the relationship between the cosine and exponential functions, Eq. (10.41), yields
/;//+1 = /;//[1 + 2d(cos e - I)] Thus, the amplification factor G is defined as
I G = I + 2d(cos e - 1) I
(10.44)
(10.45)
The amplification factor G is the single step exact solution of the finite difference equation for the general Fourier component, which must be less than unity in magnitude to ensure a bounded solution. For a specific wave number km and grid spacing &, Eq. (10.45) can be analysed to detennine the range of values of the diffusion number d for which IGI .:s 1. In the infinite Fourier series representation of the property distribution, km ranges from -00 to +00. The grid spacing & can range from zero to any finite value up to L, where L is the length of the physical space. Consequently, the product (km&) = eranges continuously from -00 to +00. To ensure that the FDE is stable for an arbitrary property distribution and arbitrary &, Eq. (10.45) must be analysed to detennine the range of values of d for which IGI .:s 1 as e ranges continuously from -00 to +00.
