Convergence of Lax-Wendroff method

Question

What is the convergence condition of the Lax-Wendroff scheme to solve $U_t+AU_x=0$, where $$A=\begin{pmatrix}2&1&0 \\ 1&1&2\\0&2&-1\end{pmatrix}$$

I don't know what I should do here.

Thanks in advance.

Answer 1

By construction, the Lax-Wendroff method $$ U_i^{n+1} = U_i^{n} - \frac{\Delta t}{2\Delta x} A (U_{i+1}^{n}-U_{i-1}^{n}) + \frac{{\Delta t}^2}{2{\Delta x}^2} A^2 (U_{i+1}^{n}-2U_{i}^{n}+U_{i-1}^{n}) $$ is consistent with the linear system of conservation laws $\partial_t U + A \partial_x U = 0$ (cf. this post and related ones), where $U_i^{n} \simeq U(i\Delta x, n\Delta t)$. It remains to prove stability to have a convergent method. The Lax-Wendroff method is not a monotone scheme, therefore the technique consisting in proving directly $\|U^{n+1}\| \leq \|U^{n}\|$ will not work. However, for linear problems, the stability condition can be obtained using Fourier analysis (von Neumann approach). To do so, the matrix $A$ is diagonalized as $A = R\Lambda R^{-1}$, where $$ \Lambda = \text{diag}(\lambda_1,\lambda_2,\lambda_3) = \begin{pmatrix}3&0&0 \\ 0& \frac{-1 - \sqrt{13}}{2}&0 \\ 0&0&\frac{-1 + \sqrt{13}}{2}\end{pmatrix} . $$ One can show that $V = R^{-1}U$ satisfies the diagonal linear system of conservation laws $\partial_t V + \Lambda \partial_x V = 0$. Therefore, the stability of the Lax-Wendroff method for $\partial_t U + A \partial_x U = 0$ is deduced from the stability of the Lax-Wendroff method for all scalar conservation laws $\partial_t V_p + \lambda_p \partial_x V_p = 0$. The von Neumann analysis gives the Courant-Friedrichs-Lewy (CFL) stability condition $$ \left|\frac{\lambda_p \Delta t}{\Delta x}\right| \leq 1 $$ for all $p$, i.e. ${\Delta t} \leq \frac{1}{3}{\Delta x}$.