2D finite-difference solution to the $15^\circ$ equation

The $15^\circ$ one-way wave fifteen.3d takes in two dimensions the simpler form:

$$\kqt \approx i \frac{\ct}{2\ctt} + \; k_o + \frac{i\cg}{2\ct... ...lp\frac{\cg}{2\ctt\; k_o}\rp^2-\frac{\cgg}{\ctt} \rb \kqg^2 \;,$$ (82)

where

$$\; k_o^2 = \frac{\lp\ww s\rp^2}{\ctt} - \lp\frac{\ct}{2\ctt}\rp^2 \;.$$ (83)

If we substitute the Fourier-domain wavenumbers by their equivalent space-domain partial derivatives, we obtain  

$$\done{\UU}{\qt} \approx -\frac{\ct}{2\ctt} + i\; k_o + \frac... ...g}{2\ctt \; k_o}\rp^2-\frac{\cgg}{\ctt} \rb \dtwo{\UU}{\qg} \;.$$ (84)

A finite-difference implementation of fifteen.2d.space involving the Crank-Nicolson method is  ^ _+1-^ _ i2 k_o ^+1_-^-1_+ ^+1_+1-^-1_+14
- i2 k_o 2 k_o^2- ^-1_-2^ _+^+1_+ ^-1_+1-2^ _+1+^+1_+12^2 .

If we make the notations

&=& i2 k_o 4
&=& - i2 k_o 2 k_o^2- 2^2 ,

we can write fifteen.2d.findif as

^ _+1-^ _&& ^+1_-^-1_+ ^+1_+1-^-1_+1
&+&^-1_-2^ _+^+1_+ ^-1_+1-2^ _+1+^+1_+1,

or, if we isolate the terms corresponding to the two extrapolation levels as:

^ _+1&-& ^+1_+1-^-1_+1- ^-1_+1-2^ _+1+^+1_+1=
^ _&+& ^+1_-^-1_+ ^-1_-2^ _+^+1_.

After grouping the terms, we obtain

$$-\lp \nu-\mu \rp \UU^{\qg-1}_{\qt+1}+ \lp 1 + 2\nu\rp\UU^{\q... ...1 - 2\nu\rp\UU^{\qg} _{\qt}+ \lp \nu+\mu\rp\UU^{\qg+1}_{\qt}\;,$$

which is a finite-difference representation of the $15^\circ$ solvable using fast tridiagonal solvers.