Convergence of Lax-Wendroff scheme for various initial conditions

Question

I am considering the Lax-Wendroff discretisation $$U_j^{(n+1)}=U_j^{(n)}-\frac{\Delta t}{2\Delta x}\left(U_{j+1}^{(n)}-U_{j-1}^{(n)}\right)+\frac{\Delta t^2}{2\Delta x^2}\left(U_{j+1}^{(n)}-2U_j^{(n)}+U_{j-1}^{(n)}\right)$$ of the simple advection equation $u_t+u_x=0$. I have proven that the truncation error is of $\mathcal{O}(\Delta t^2,\Delta x^2)$, and I have deduced by von Neumann stability analysis that we demand $\Delta t\leq\Delta x$ for $L_2$-stability. However, I am not sure how to interpret these results. In particular, I am not sure what these imply for convergence of the scheme dependent on the initial conditions such as for the cases i) $\mathrm{e}^{-x^2}$,  and ii) $\max\{1-|x|,0\}$. What should I be looking out for/arguing for? Thanks!

Answer 1

The accuracy of the Lax-Wendroff scheme is presented in this post, and the stability analysis is performed here. Convergence results hold for any initial condition, but the notion of order of accuracy requires that the solution is sufficiently smooth (the Taylor series expansion used for the error analysis requires some smoothness). The main differences between (i) and (ii) are:

• smoothness $C^\infty$ vs. smoothness $C^0$;
• infinite support vs. bounded support.

In computational applications where the order of accuracy is measured 'experimentally', it is recommended to solve problems with very smooth solutions. Otherwise, the estimation of the order of accuracy might be inaccurate.