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solitons & computational fluid dynamics 513We next substitute these derivatives into the Taylor expansion (19.10) to obtainu(x, t +∆t)=u(x, t) − ∆tɛ ∂∂x( ) u2+ (∆t)222ɛ 2∂ [u ∂∂x ∂x( u22)].We now replace the outer x derivatives by central differences of spacing ∆x/2:u(x, t +∆t)=u(x, t) − ∆tɛ2× 1 [u2∆xu 2 (x +∆x, t) − u 2 (x − ∆x, t)2∆x(x + ∆x2 ,t ) ∂∂x u2 (x + ∆x2 ,t )− u(∂∂x u2 x − ∆x )]2 ,t .+ (∆t)2 ɛ 22(x − ∆x2 ,t )Next we approximate u(x ± ∆x/2,t) by the average of adjacent grid points,u(x ± ∆x2u(x, t)+u(x ± ∆x, t),t) ≃ ,2and apply a central-difference approximation to the second derivatives:∂u 2 (x ± ∆x/2,t)∂x= u2 (x ± ∆x, t) − u 2 (x, t).±∆xFinally, putting all these derivatives together yields the discrete formu i,j+1 = u i,j − β 4(u2i+1,j − u 2 βi−1,j) 2 [+ (ui+1,j + u i,j ) ( u 2 i+1,j − u 2 i,j)8−(u i,j + u i−1,j ) ( u 2 i,j − u 2 i−1,j)], (19.12)where we have substituted the CFL number β. This Lax–Wendroff scheme isexplicit, centered upon the grid points, and stable for β