The LTE for the ADI 3.13 in Morton & Mayers

The numerical scheme is

δxSq[
        U[i_, j_, n_]] := (U[i + 1, j, n] - 2 U[i, j, n]  +  
          U[i - 1, j, n])
δySq[ 
        U[i_, j_, n_]] := (U[i, j + 1, n] - 2 U[i, j, n]  +  
          U[i, j - 1, n])
δxSq[α_×
          U[i_, j_, n_]] := α (U[i + 1, j, n] - 2 U[i, j, n]  +  
            U[i - 1, j, n])
δySq[α_ × 
          U[i_, j_, n_]] := α (U[i, j + 1, n] - 2 U[i, j, n]  +  
            U[i, j - 1, n])

NumericalScheme  = 
      U[i, j, n + 1] -  (1/2) νx δxSq[U[i, j, n + 1]] -  
        (1/2) νy δySq[U[i, j, n + 1]] +  
        (1/4) νx νy Map[δySq, δxSq[
              U[i, j, n + 1]]] - (U[i, j, n] +  
            (1/2) νx δxSq[U[i, j, n]] +  
            (1/2) νy δySq[U[i, j, n]] +  
            (1/4) νx νy Map[δySq, δxSq[
                  U[i, j, n]]])

Fred  =  
        NumericalScheme /. {
U[i_, j_, n_] -> 
              u[i Δx,  j Δy,  
                n Δt], 
νx -> α Δ(t/Δ)(x^2), 
νy -> α Δ(t/Δ)(y^2)
};

    
Barney  =  
      Fred /. {
i -> 0, 
j -> 
            0, 
n -> 0}

Wilma  =  
      Expand[
Normal[
Series[
            Barney,  
{Δx,  0,  
              3}, 
{Δy,  0,  
              3}, 
{Δt,  0,  
              3}
]
]
]

BaMBam=Expand[
Wilma/.{
Derivative[0, 0, 1][u][0, 0, 0]->α(Derivative[0, 2, 0][u][0, 0, 0]+Derivative[2, 0, 0][u][0, 0, 0])
}
]

Pebbles =Flatten[
Table[
{
Derivative[i_, j_, n][u][0, 0, 0]->α(Derivative[i, j + 2, n - 1][u][0, 0, 0]+Derivative[i + 2, j, n - 1][u][0, 0, 0])
},
{n,  1,  3}]
]

BaMBam = 
      Expand[
Wilma //. 
          Pebbles
]

Stability Ciomputation

FullSimplify[ (E^(I kx Δx)) - 2  + (E^(-I kx Δx)) ]