Finite difference equation - parabolic PDE

**math2011** · June 4th 2011, 07:37 AM

Consider the finite difference scheme
$\frac{3 U^n_p - 4U^{n-1}_p + U^{n-2}_p}{2 \Delta t} - a\frac{U^n_{p+1} - 2 U^n_p + U^n_{p-1}}{\Delta x^2} = F^n_p$
where, with the usual notation, $2 \leq n \leq N$ and $1 \leq p \leq P$ . Find $\rho$ such that $U^n_p = \rho^n \sin (k \pi x_p / L)$ is a solution of the equation in the case $f \equiv 0$ .

Let $\displaystyle u^0(x) = \phi_k(x) = \sin \frac{k \pi}{L} x$ , then

$\frac{3 U^n - 4U^{n-1} + U^{n-2}}{2 \Delta t} - a A_P U^n =& 0$

$3 U^n - 4U^{n-1} + U^{n-2} =& 2 a \Delta t A_P U^n$

$4U^{n-1} - U^{n-2} =& 3 U^n - 2 a \Delta t A_P U^n$

$4U^{n-1} - U^{n-2} =& (3 - 2 a \Delta t A_P) U^n$

$U^n =& \frac{4U^{n-1} - U^{n-2}}{3 - 2 a \Delta t A_P}$

where $A_P$ is the tridiagonal coefficient matrix. I am stuck here. How can I derive $\rho$ in the question?

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