# Numerical schemes for linear advection: stability, dissipation, dispersion

A generalized numerical scheme for the linear hyperbolic equation $$u_t + au_x = 0$$ has the following form $$\frac{u_j^{n+1} - u_j^n}{\Delta t} + a\frac{u_{j+1}^{n} - u_{j-1}^n}{2\Delta x} - \chi \frac{u_{j+1}^{n} - 2u_{j}^n + u_{j-1}^n}{\Delta x^2} = 0 .$$ a) Find the values of $$\chi$$ so that the above scheme is the central-explicit, forward, backward, Lax-Friedrich and Lax-Wendroff scheme.

b) What is the order of truncation for each scheme?

c) What is the stability condition for each of these schemes?

d) In what range of $$\chi$$ the scheme is unconditionally unstable?

e) Classify the schemes in (a) as (i) dissipative, and (ii) dispersive.

## ATTEMPT

(a) Notice that if $$\chi = 0$$, then we obtain the central explicit scheme. If we write the numerical discretization as

$$u_j^{n+1} = u_j^n - \frac{ a \Delta t }{2 \Delta x} (u_{j+1}^n - u_{j-1}^n) + \frac{ \chi \Delta t }{\Delta x^2} (u_{j+1}^n - 2 u_j^n + u_{j-1}^n )$$

And one can see that if $$\chi = a \Delta t/2$$, then we have the LW scheme. If $$\chi = a \Delta x/2$$, then we have

$$u_j^{n+1} = u_j^n + \frac{ a \Delta t }{\Delta x} (u_{j+1}^n - u_j^n)$$

which is forward explicit and if $$\chi = - a \Delta x/2$$, we obtain backward explicit. Now if $$\chi = \frac{ \Delta x^2}{2 \Delta t }$$, then we obtain Lax-Friedrichs.

We go to part (c) as (b) is trivial. Perhaps we can combine (c) and (d) in a single problem if we apply the discrete fourier transform:

$$\hat{u}^{n+1} = \hat{u}^n - r( e^{ij \xi} -e^{-ij \xi } ) \hat{u}^n + \frac{ 2 \chi r }{a \Delta x} (e^{ij \xi} -e^{-ij \xi } -2 ) \hat{u}^n$$

Now, am I on the right track here? Is this the correct approach? As for e), I need some suggestions, how would we approach it?

The proposed values of $$\chi$$ in a) are correct. One notes that the method is a straightforward finite-difference approximation of the advection-diffusion method $$u_t + au_x = \chi u_{xx}$$. Introducing the Courant number $$r = |a| {\Delta t}/{\Delta x}$$, the diffusion constants from a) rewrite as $$\chi \in |a|\frac{\Delta x}{2} \times \left\lbrace \frac{1}{r},\, 1,\, r,\, 0 \right\rbrace ,$$ which correspond to the numerical viscosities of the Lax-Friedrichs, upwind, Lax-Wendroff, and central explicit method, respectively. One notes that if $$0, then those values of $$\chi$$ are sorted in decreasing order.
As done in this post, let us assume a perturbation of the form $$u_j^n = \xi^n e^{-\text i k j \Delta x}$$, where $$\xi = \text e^{\text i \omega \Delta t}$$. Note that $$\omega$$ is a complex number, with real part $$\omega_R$$ and imaginary part $$\omega_I$$. Injecting this Ansatz in the time-stepping formula, Euler's formulas lead to $$\xi = 1 + \text i r \sin( k \Delta x) + 2\lambda \left( \cos(k \Delta x) -1 \right) ,$$ where $$r$$ is the Courant number and $$\lambda = {\chi}\frac{\Delta t}{\Delta x^2} \in \left\lbrace \frac{1}{2},\, \frac{r}{2},\, \frac{r^2}{2},\, 0 \right\rbrace$$ denotes the Fourier number. From this equation, one equates the real and imaginary parts: \begin{aligned} \text{Re}\, \xi &= e^{-\omega_I \Delta t}\cos(\omega_R \Delta t) = 1 + 2\lambda \left( \cos(k \Delta x) -1 \right) ,\\ \text{Im}\, \xi &= e^{-\omega_I \Delta t}\sin(\omega_R \Delta t) = r \sin( k \Delta x) \, . \end{aligned} The stability can be analyzed in terms of the squared amplification factor $$|\xi|^2$$ which has to be smaller than one. Hence, we must have $$|\xi|^2 - 1 = 4\lambda^2 - 4 \lambda + r^2 + 4 \lambda (1-2\lambda) Z + (4\lambda^2 - r^2) Z^2 \leq 0$$ for all $$Z = \cos( k \Delta x)$$ in $$[-1, 1]$$. The dispersion is analyzed in terms of the phase velocity $${\omega_R}/{k}$$ with $${\omega_R}\Delta t = \arctan\left[{\text{Im}\, \xi}/{\text{Re}\, \xi}\right]$$. The attenuation is analyzed in terms of $${\omega_I}\Delta t = -\ln |\xi|$$ which is non-negative if $$|\xi| \leq 1$$ (cf. e.g. (1) p. 182).