Dispersion

Let us return to the Lax method for hyperbolic equations. The solution of the differential equation has the form

$$u(x,t) = u_0 e^{\i(\omega t - k x)}$$ (2.42)

where $\omega = c k$. Let us substitute (2.42) into the Lax algorithm

$$v e^{\i(\omega t^{n+1} - k x_j)} = {\mathchoice{{\textstyl... ...\left(e^{+\i k\delta x} - e^{-\i k\delta x}\right) \right\}.$$ (2.43)

By cancelling the common factors we now obtain

$$e^{\i\omega\delta t} = \cos(k\delta x) - \i{c\delta t\over\delta x}\sin(k \delta x).$$ (2.44)

From this we can derive a dispersion relation, $\omega \mathrel{\rm vs}k$, for the discrete equation and compare the result with $\omega = c k$ for the original differential equation. Since, in general, $\omega$ could be complex we write it as $\omega = \Omega + \i\gamma$ and compare real and imaginary parts on both sides of the equation to obtain

$$\cos\Omega\delta t e^{-\gamma\delta t} = \cos k\delta x$$ (2.45)

$$\sin\Omega\delta t e^{-\gamma\delta t} = - {c\delta t\over \delta x}\sin k\delta x.$$ (2.46)

Taking the ratio of these or the sum of their squares respectively leads to the equations

$$\tan(\Omega\delta t) = {c\delta t\over\delta x}\tan(k\delta x)$$ (2.47)

$$e^{-2\gamma\delta t} = \cos^2(k\delta x) + \left(c\delta t\over\delta x\right)^2\sin^2(k\delta x).$$ (2.48)

The first of these equations tells us that in general the phase velocity is not $c$, although for long wavelengths, $k \delta x \ll 1$ and $\Omega\delta t \ll 1$, we recover the correct dispersion relationship. This is similar to the situation when we compare lattice vibrations with classical elastic waves: the long wavelength sound waves are OK but the shorter wavelengths deviate. The second equation (2.7b) describes the damping of the modes. Again for small $k\delta x$ and $\gamma\delta t$, $\gamma = 0$, but (e.g.) short wavelengths, $\lambda = 4\delta x$, are strongly damped. This may be a desirable property as short wavelength oscillations may be spurious. After all we should have chosen $\delta x$ to be small compared with any expected features. Nevertheless with this particular algorithm $\lambda = 2\delta x$ is not damped. Other algorithms, such as Lax-Wendroff, have been designed specifically to damp anything with a $k \delta x > {\mathchoice{{\textstyle{\frac14}}}{{\textstyle{\frac14}}}{{\scriptstyle{1/4}}}{{\scriptscriptstyle{1/4}}}}$.