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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!

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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.

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  • $\begingroup$ Hi, thanks for the answer! I have a couple of questions before accepting this: i) why does the convergence hold for any initial condition? I read somewhere the truncation error demands “suitable smoothness conditions”, how do we quantify this? ii) similarly how do we judge the accuracy of the second initial condition compared to the first? how do we quantify the effect of the “kinks” of the second function on convergence? $\endgroup$
    – user107224
    Commented Dec 16, 2020 at 21:28
  • $\begingroup$ @user107224 Stability and convergence are independent from smoothness. But the notion of order of accuracy requires some smoothness due to the Taylor series expansion in the error analysis. $\endgroup$
    – EditPiAf
    Commented Dec 17, 2020 at 8:19

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