# 2.7 Eigenvalue Stability of Finite Difference Methods

## 2.7.1 Fourier Analysis of PDEs

We will develop Fourier analysis in one dimension. The basic ideas extend easily to multiple dimensions. We will consider the convection-diffusion equation,

 $\frac{\partial U}{\partial t} + u\frac{\partial U}{\partial x} = \mu \frac{\partial ^2 U}{\partial x^2}.$ (2.124)

We will assume that the velocity, $$u$$, and the viscosity, $$\mu$$ are constant.

The solution is assumed to be periodic over a length $$L$$. Thus,

 $U(x+mL,t) = U(x,t)$ (2.125)

where $$m$$ is any integer.

A Fourier series with periodicity over length $$L$$ is given by,

 $U(x,t) = \sum _{m=-\infty }^{+\infty } \hat{U}_ m(t)e^{i k_ m x} \qquad \mbox{where} \qquad k_ m = \frac{2\pi m}{L}. \label{equ:Fourier}$ (2.126)

$$k_ m$$ is generally called the wavenumber, though $$m$$ is the number of waves occurring over the length $$L$$. We note that $$\hat{U}_ m(t)$$ is the amplitude of the $$m$$-th wavenumber and it is generally complex (since we have used complex exponentials). Substituting the Fourier series into the convection-diffusion equation gives,

 $\frac{\partial }{\partial t}\left[\sum _{m=-\infty }^{+\infty } \hat{U}_ m(t)e^{i k_ m x}\right] + u \frac{\partial }{\partial x}\left[\sum _{m=-\infty }^{+\infty } \hat{U}_ m(t)e^{i k_ m x}\right] = \mu \frac{\partial ^2}{\partial x^2}\left[\sum _{m=-\infty }^{+\infty } \hat{U}_ m(t)e^{i k_ m x}\right].$ (2.127)
 $\sum _{m=-\infty }^{+\infty } \frac{{\rm d}\hat{U}_ m}{{\rm d}t} e^{i k_ m x} + u \sum _{m=-\infty }^{+\infty } i k_ m \hat{U}_ m e^{i k_ m x} = \mu \sum _{m=-\infty }^{+\infty } (i k_ m)^2 \hat{U}_ m e^{i k_ m x}.$ (2.128)

Noting that $$i^2 = -1$$ and collecting terms gives,

 $\sum _{m=-\infty }^{+\infty } \left[\frac{{\rm d}\hat{U}_ m}{{\rm d}t} + \left(i u k_ m + \mu k_ m^2\right) \hat{U}_ m\right] e^{i k_ m x} = 0. \label{equ:Fourier_ preorth}$ (2.129)

The next step is to utilize the orthogonality of the different Fourier modes over the length $$L$$, specifically,

 $\int _0^ L e^{-i k_ n x} e^{i k_ m x} dx = \left\{ \begin{array}{cc} 0 & \mbox{if } m \neq n \\ L & \mbox{if } m = n \end{array}\right. \label{equ:Fourier_ orth}$ (2.130)

By multiplying Equation (2.129) by $$e^{-i k_ n x}$$ and integrating from $$0$$ to $$L$$, the orthogonality condition gives,

 $\frac{{\rm d}\hat{U}_ n}{{\rm d}t} + \left(i u k_ n + \mu k_ n^2\right) \hat{U}_ n = 0, \qquad \mbox{for any integer value of } n. \label{equ:Fourier_ condif1d}$ (2.131)

Thus, the evolution of the amplitude for an individual wavenumber is independent of the other wavenumbers. The solution to Equation (2.131),

 $\hat{U}_ n(t) = \hat{U}_ n(0) e^{-i u k_ n t} e^{-\mu k_ n^2 t}.$ (2.132)

The convection term, which results in the complex time dependent behavior, $$e^{-i u k_ n t}$$, only oscillates and does not change the magnitude of $$\hat{U}_ n$$. The diffusion term causes the magnitude to decrease as long as $$\mu > 0$$. But, if the diffusion coefficient were negative, then the magnitude would increase unbounded with time. Thus, in the case of the convection-diffusion PDE, as long as $$\mu \geq 0$$, this solution is stable.

## Exercise

For the convection-diffusion equation analyzed above, how does the rate of damping compare for a wavelength $$L$$ mode to a wavelength $$L/10$$ mode?

Exercise 1

Answer: The $$L_1$$ mode has a damping rate $$\mu k_ m^2 = \mu (2\pi /L)^2$$ the $$L/10$$ mode has a damping rate of $$\mu k_ m^2 = \mu (20\pi /L)^2$$