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The Lax-Friedrichs scheme for the 1D linear advection equation $$\frac{}{}+ _\frac{}{} = 0$$ with a constant flow speed of $_$ is $$(+Δ,)=\frac{1}{2}[(,+Δ)+(,−Δ)]−\frac{Δ_}{2Δ}[(,+Δ)−(,−Δ)]$$ where $Δ$, $Δ$ and $_$ are the time step, the cell width of a 1D uniform mesh and the flow speed respectively. Applying Taylor series expansions of $(+Δ,)$, $(,+Δ)$ and $(,−Δ)$ about $(,)$, show that the leading error term (...) of the Lax-Friedrichs scheme is $$...=\frac{1}{2}\left[\frac{Δ^2}{Δ}− Δ^2_\right]\frac{^2}{^2}$$

Could someone please explain how to do this as I can't find out how to work out the Taylor series

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user855901
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1 Answers1

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Let us use the notation $U_j^n$ for approximations of $U(x_j, t_n)$ with $x_j = j\Delta x$ and $t_n = n\Delta t$. Using Taylor series, one shows that for smooth solutions \begin{aligned} \frac{U_j^{n+1} - \frac12( U_{j+1}^n + U_{j-1}^n )}{\Delta t} &= (U_t)_j^n + \frac{\Delta t}{2}(U_{tt})_j^n - \frac{\Delta x^2}{2\Delta t} (U_{xx})_j^n + O(\Delta t^2, \Delta x^2) \\ \frac{U_{j+1}^{n} - U_{j-1}^n}{2\Delta x} &= (U_x)_j^n + O(\Delta x^2) . \end{aligned} Now let us compute the difference between the PDE and its approximation deduced from the Lax-Friedrichs method. Up to higher-order terms, the scheme produces $$ \begin{aligned} (U_t + v_x U_x)_j^n & - \frac{U_j^{n+1} - \frac12( U_{j+1}^n + U_{j-1}^n )}{\Delta t} - v_x \frac{U_{j+1}^{n} - U_{j-1}^n}{2\Delta x} \\ &= \underbrace{-\frac{\Delta t}{2}(U_{tt})_j^n + \frac{\Delta x^2}{2\Delta t} (U_{xx})_j^n}_\text{l.e.t.} \end{aligned} $$ Then, the transport equation yields the relationships $U_{tx} = -v_x U_{xx}$ and $U_{tt} = -v_x U_{xt}$ between temporal and spatial derivatives, which lead to $U_{tt} = v_x^2 U_{xx}$. Finally, $$ \text{l.e.t} = \frac{1}{2}\left(\frac{\Delta x^2}{\Delta t} - \Delta t\, v_x^2\right) (U_{xx})_j^n \, , $$ where $j$ and $n$ can be removed.

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