Numerical Advection Schemes in Two Dimensions

3.2 Directional splitting 3 ADVECTING IN TWO DIMENSIONS
0.707d
d
3.2 Directional splitting
Figure 1: 2D wave propagation
It is not possible to extend all numerical advection methods into two dimensions
quite so easily. An interesting example is the Lax-Wendroff method.
In one dimension the method is defined as
ϕ m+1
i = ϕ m i − U 2 (ϕm i+1 − ϕ m i−1) + U 2
2 (ϕm i+1 − 2ϕ m i + ϕ m i−1) + O(∆x 2 , ∆t 2 )
where U is the 1D Courant number u ∆t . For this case the Courant number is
∆x
stable between −1 and 1. However if we just add on the additional terms for
the y-direction the scheme becomes completely unstable. The explanation
for this lies within the derivation of the Lax-Wendroff method. The starting
point for the derivation is the Taylor series of ϕ about t
∂ϕ(x, y, t)
ϕ(x, y, t ± ∆t) = ϕ(x, y, t) ± ∆t + ∆t2 ∂ 2 ϕ(x, y, t)
+ ... (13)
∂t 2 ∂t 2
Using equation (1) each of the derivatives with respect to t can be written in
terms of x and y. For notational simplicity we shall use the convention that
can be rewritten ϕ t . As such we have that
∂ϕ(x,y,t)
∂t
ϕ t = −uϕ x − vϕ y (14)
ϕ tt = −u 2 ϕ xx − v 2 ϕ yy + 2uvϕ xy (15)
Substituting in the formulae for the first and second derivatives of secondorder
centred differences (Rood 1987)
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