An Improved Algorithm -- the Lax method

We see from (2.20) that our simple method is in fact unstable for the advection equation, for all finite values of $\delta x$ and $\delta t$. How might we improve on this? Let us consider a minor (?) modification of (2.11)

$$u_j^{n+1} = {\mathchoice{{\textstyle{\frac12}}}{{\textstyle... ...c \delta t\over 2\delta x}\left(u_{j+1}^n - u_{j-1}^n\right).$$ (2.21)

in which the term in $u^n_j$ has been replaced by an average over its 2 neighbours. When we apply the same (von Neumann) analysis to this algorithm we find

$$v^{n+1} = \left[\cos(k \delta x) - i{c\delta t\over\delta x}\sin(k \delta x)\right]v^n$$ (2.22)

so that

$$\left\vert g\right\vert^2 = \cos^2(k\delta x) + \left(c \delta t\over \delta x\right)^2\sin^2(k\delta x)$$ (2.23)

$$= 1 - \sin^2(k\delta x) \left\{1 - \left(c \delta t\over \delta x\right)^2\right\}$$ (2.24)

which is stable for all $k$ as long as

$${\delta x\over\delta t} \ge c$$ (2.25)

which is an example of the Courant-Friedrichs-Lewy condition applicable to hyperbolic equations. There is a simple physical explanation of this condition: if we start with the initial condition such that $u(x,t=0) =0$ everywhere except at one point on the spatial grid, then a point $m$ steps away on the grid will remain zero until at least $m$ time steps later. If, however, the equation is supposed to describe a physical phenomenon which travels faster than that then something must go wrong. This is equivalent to the condition that the time step, $\delta t$, must be smaller than the time taken for the wave to travel the distance of the spatial step, $\delta x$; or that the speed of propagation of information on the grid, $\delta x/\delta t$, must be greater than any other speed in the problem.