# 2.7 Eigenvalue Stability of Finite Difference Methods

## 2.7.3 Circulant Matrices

Measurable Outcome 2.10, Measurable Outcome 2.11

Though the eigenvalues of \(A\) typically require numerical techniques for the general problem, a special case of practical interest occurs when the matrix is ‘periodic'. That is, the column entries shift a column every row. Thus, the matrix has the form,

\[A = \left(\begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_{N } \\ a_ N & a_1 & a_2 & \ldots & a_{N-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_2 & a_3 & a_4 & \ldots & a_1 \end{array}\right)\] | (2.140)

This type of matrix is known as a circulant matrix. Circulant matrices have eigenvalues given by,

\[\lambda _ n = \sum _{j=1}^{N} a_ j e^{i\, 2\pi (j-1) \frac{n}{N}} \qquad \mbox{for} \quad n = 0, 1, \ldots , N-1 \label{equ:circulant_ eig}\] | (2.141)

As we saw in Section 2.7.2, when periodic boundary conditions are assumed, the central space discretization of one-dimensional convection gives purely imaginary eigenvalues, and when scaled by a timestep for which the CFL number is one, the eigenvalues stretch along the axis until \(\pm i\). Since for a convection problem with constant velocity and periodic boundary conditions gives a circulant matrix, we can use Equation (2.141) to determine the eigenvalues analytically. We begin by finding the coefficients, \(a_ j\). For a central space discretization, we find,

\[a_2 = -\frac{u}{2{\scriptstyle \Delta } x}, \quad a_ N = \frac{u}{2{\scriptstyle \Delta } x}, \mbox{ and for all other } j, \quad a_ j = 0.\] | (2.142)

Then, substituting these \(a_ j\) into Equation (2.141) gives,

| \(\displaystyle \lambda _ n\) | \(\displaystyle =\) | \(\displaystyle -\frac{u}{2{\scriptstyle \Delta } x} e^{i\, 2\pi \frac{n}{N}} + \frac{u}{2{\scriptstyle \Delta } x} e^{i\, 2\pi (N-1) \frac{n}{N}},\) |  | (2.143)
|  | \(\displaystyle =\) | \(\displaystyle -\frac{u}{2{\scriptstyle \Delta } x} e^{i\, 2\pi \frac{n}{N}} + \frac{u}{2{\scriptstyle \Delta } x} e^{i\, 2\pi n} e^{-i\, 2\pi \frac{n}{N}}.\) |  | (2.144)

Since \(e^{i\, 2\pi n} = 1\) (because \(n\) is an integer), then,

| \(\displaystyle \lambda _ n\) | \(\displaystyle =\) | \(\displaystyle -\frac{u}{2{\scriptstyle \Delta } x} e^{i\, 2\pi \frac{n}{N}} + \frac{u}{2{\scriptstyle \Delta } x} e^{-i\, 2\pi \frac{n}{N}},\) |  | (2.145)
|  | \(\displaystyle =\) | \(\displaystyle -\frac{u}{2{\scriptstyle \Delta } x}\left(e^{i\, 2\pi \frac{n}{N}} - e^{-i\, 2\pi \frac{n}{N}}\right),\) |  | (2.146)
|  | \(\displaystyle =\) | \(\displaystyle -i\frac{u}{{\scriptstyle \Delta } x}\sin \left(2\pi \frac{n}{N}\right).\) |  | (2.147)

Multiplying by the timestep,

\[\lambda _ n {\Delta t}= -i\frac{u{\Delta t}}{{\scriptstyle \Delta } x}\sin \left(2\pi \frac{n}{N}\right).\] | (2.148)

As observed in Section 2.7.2, the eigenvalues are purely imaginary and will extend to \(\pm i\) when \(\mathrm{CFL} = |u|{\Delta t}/{\scriptstyle \Delta } x= 1\).