2.7 Eigenvalue Stability of Finite Difference Methods

2.7.1 Fourier Analysis of PDEs

Measurable Outcome 2.2, Measurable Outcome 2.11

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