2.7 Eigenvalue Stability of Finite Difference Methods

2.7.3 Circulant Matrices

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 $$\mathrm{CFL} = |u|{\Delta t}/{\scriptstyle \Delta } x= 1$$.