Question 1: Consider the wave equation
$$
u_t + c(x) u_x = 0 ,
$$
where $x\in \Omega \subset \Bbb R$ and $c(x)$ is a function of $x$.(a) Show that the Lax-Wendroff scheme for this PDE is given by
$$
u_j^{n+1} = u_j^n – c_j \Delta t \frac{D_x u_j^n}{2 \Delta x} + \frac{c_j^2 \Delta t^2}{2} \frac{\delta_x^2 u_j^n}{\Delta x^2} + \frac{c_j \Delta t^2}{8 \Delta x ^2} (D_x c_j)(D_x u_j^n) ,
$$
where $D_x$ is the first central difference operator, $\delta_x^2$ is the second central difference operator, and $\Delta t$ and $\Delta x$ are the mesh-spacing in $t$ and $x$, respectively. The $j$ and $n$ are space and time indices, respectively, and $u_j^n$ is the grid function such that $u_j^n\approx u(x_j,t_n)$ and $c_j \approx c(x_j)$.
I would also like to answer the following question
Assuming $c(x)$ is a constant, and given that the initial value problem
is well posed, carry out a von Neumann Stability analysis and hence
show that the scheme is convergent provided $$\bigg \lvert \frac{c\Delta t}{
\Delta x} \bigg \rvert \leq1$$
I have tried to do the stability analysis but I am not getting anywhere near what the question requires. Could someone please check my working and provide some advice.
The scheme can be written as
$$u_{j}^{n+1} = u_j^n – \frac{p}{2}(u_{j+1}-u_{j-1}) + \frac{p^2}{2}(u_{j+1}^n-2u_j^n+u_{j-1}^n) + \frac{p^2}{8}(u_{j+1}-u_{j-1}), \quad \text{where $p = \frac{c\Delta t}{5 \Delta x}$}$$
I think this is how the scheme should be written. However, I am not sure about the $D_{x}c_{j}$ term, since it is a constant, should I just ignore the $D_x$ operator?
For the stability analysis, using a trial solution of the form $u_{j}^{n} = A \xi^{n}e^{i \omega j}$ gives
$$A \xi^{n+1} e^{i \omega j} = A \xi^{n} e^{i \omega j} – \frac{p}{2}\left[ A \xi^{n} e^{i \omega (j+1)} – A \xi^{n} e^{i \omega (j-1)} \right] + \frac{p^{2}}{2} \left[ A \xi^{n} e^{i \omega (j+1)} – 2 A \xi^{n} e^{i \omega j} + A \xi^{n} e^{i \omega (j-1)} \right] + \frac{p^2}{8}\left[ A \xi^{n} e^{i \omega (j+1)} + A \xi^{n} e^{i \omega (j-1)} \right]$$
Dividing through and by $u_j^n = A \xi^{n} e^{i \omega j}$ I get
$$\xi = 1 + \frac{p}{2}\left[ e^{i\omega} – e^{-i\omega} \right] – \frac{p^{2}}{2} \left[ e^{i\omega} – 2 + e^{-i\omega} \right] + \frac{p^{2}}{8} \left[ e^{i \omega} + e^{-i \omega} \right] = 0$$
Than, using Eulers identity
$$\implies \xi = 1 + \frac{p^{2}}{2}\cdot 2 \cos \omega – \frac{p^{2}}{2} (2 \cos \omega – 2) – \frac{p^{2}}{8}\cdot 2 \cos \omega$$
Can anyone help me from here?
Best Answer
Recall how the Lax-Wendroff method is obtained in the constant-speed case [1]:
Thus, the following scheme is obtained: $$ u_j^{n+1} = u_j^n - c\Delta t \frac{D_x u_j^n}{2 \Delta x} + \frac{c^2\Delta t^2}{2} \frac{\delta_x^2 u_j^n}{\Delta x^2} \, . $$ This method can be adapted to the variable velocity case.
Let us analyze the stability of the Lax-Wendroff scheme above (constant-speed case). Assuming a perturbation of the form $u_j^{n} = \xi^n \text{e}^{\text i k x_j}$, one has \begin{aligned} \xi &= 1 - c \Delta t \frac{\text{e}^{\text i k \Delta x} - \text{e}^{-\text i k \Delta x}}{2 \Delta x} + \frac{c^2 \Delta t^2}{2} \frac{\text{e}^{\text i k \Delta x} - 2 + \text{e}^{-\text i k \Delta x}}{\Delta x^2} \\ &= 1 - \text{i} \kappa \sin (k\Delta x) + \kappa^2 \left(\cos (k\Delta x) - 1\right) . \end{aligned} where $\kappa = c\frac{\Delta t}{\Delta x}$ is the Courant number. Thus, the squared modulus of the amplification factor is \begin{aligned} |\xi|^2 &= (1 - \kappa^2 (1-\cos(k\Delta x)))^2 + \kappa^2 (1-\cos^2(k\Delta x))\\ &= 1 - 2\kappa^2 (1-\cos(k\Delta x)) + \kappa^2 (1-\cos^2(k\Delta x)) + \kappa^4 (1-\cos(k\Delta x))^2 \\ &= 1 - \kappa^2 (1-\kappa^2) (1-\cos(k\Delta x))^2 \\ &= 1 - 4\kappa^2 (1-\kappa^2) \sin^4\left(\tfrac{1}{2}k\Delta x\right) . \end{aligned} Finally, the Lax-Wendroff scheme is Neumann-stable provided that $|\xi|^2 \leq 1$, which implies that the CFL condition $|\kappa| \leq 1$ is satisfied.
[1] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.