# Finite difference equation - parabolic PDE

• Jun 4th 2011, 07:37 AM
math2011
Finite difference equation - parabolic PDE
Consider the finite difference scheme
$\displaystyle \frac{3 U^n_p - 4U^{n-1}_p + U^{n-2}_p}{2 \Delta t} - a\frac{U^n_{p+1} - 2 U^n_p + U^n_{p-1}}{\Delta x^2} = F^n_p$
where, with the usual notation, $\displaystyle 2 \leq n \leq N$ and $1 \leq p \leq P$. Find $\displaystyle \rho$ such that $\displaystyle U^n_p = \rho^n \sin (k \pi x_p / L)$ is a solution of the equation in the case $\displaystyle f \equiv 0$.

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

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

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

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

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

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

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