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This question pertains to stability analysis of finite difference methods using the discrete Fourier transform.

Suppose I have a convection diffusion equation of the form:

(1) $\hspace{.5in}u_t + \nabla\cdot(vu) = \nabla\cdot(k\nabla u) + F(t)$

which of course may be rewritten as

(2) $\hspace{.5in}u_t = \mathcal{L}(\nabla\cdot(vu), \nabla\cdot(k\nabla u)) + F(t).$

Provided some reasonable assumptions on $F$ it will, of course, not affect the well-posedness of (2). Similarly, when we make our problem discrete via finite difference methods it will not affect the stability of the scheme provided those same assumptions. Thus, for our stability analysis we restrict our-selves to the homogeneous problem.

To illustrate my question I give the following example:

(For relative simplicity of notation suppose that we have two spatial variables).
(3) $\hspace{.25in}\begin{cases}D_+^tU_{i,j}^n = \frac{3}{2}LU_{i,j}^n - \frac{1}{2}LU_{i,j}^{n-1} & \textrm{(predictor)} \\ D_+^tU_{i,j}^n = \frac{1}{2}(LU_{i,j}^p + LU_{i,j}^{n}) & \textrm{(corrector)}\end{cases} $

where $L = \mathcal{L}(\nabla_{h,k}(V_{i,j}^n U_{i,j}^n), \nabla_{h,k}\cdot(K_{i,j}^n\nabla_{h,k}U_{i,j}^n))$, and above are standard second order centered finite difference operators and: $$x_{j+1} = x_j + h, y_{j+1} = y_j + k, t_{j+1} = t_j + m.$$

From (3), after some algebra, we arrive at

(4) $\hspace{.25in}U_{i,j}^{n+1} - U_{i,j}^n = \frac{3}{4}m^2L^2U_{i,j}^n - \frac{1}{4}m^2L^2U_{i,j}^{n-1} + mLU_{i,j}^{n}.$

The next step is the to take the discrete Fourier Transform (alternatively may be thought of assuming that $U = e^{ihx}e^{iky}$), which yields

(5) $\hspace{.25in} G = 1 + mL\left[\left(\frac{3}{4}mL + 1\right)\hat{U}_n - \frac{1}{4}mL\hat{U}_{n-1} \right]$,

Ultimately, of course we want $|G| \leq 1$. From here the usual procedure is to examine the "worst case scenario", which I am capable of doing by brute force. However, this results in much gross algebra when calculating the composition of $L$ with itself.

My question, then is this: Is there a way of explicitly calculating $L^2\hat{U}_n$ without first explicitly calculating $L^2$?

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