
Solving PDEs
Hi,
Given a PDE of this form
http://i661.photobucket.com/albums/u...U/Capture1.jpg
then one way to solve it iteratively is
http://i661.photobucket.com/albums/u.../Capture7.jpg
(Equations from IISc).
The update equation has me confused for the following reason. $\displaystyle u(x, t)$ appears in the PDE as function of both time $\displaystyle t$ and distance $\displaystyle x$. However, the update equation only gives you $\displaystyle u$ as a function of time ($\displaystyle u^{n+1}_{j}$), and what's more, it seems to assume that $\displaystyle u$ is already known as a function of distance because it has $\displaystyle u^{n}_{j+1}$ terms.
In short, the solver seems to assume that you already know the solution :confused:.
Anyone familiar with this who can enlighten me?
Thanks.

Re: Solving PDEs
Hey algorithm.
What is the definition of u_j given in your book?

Re: Solving PDEs
chiro,
Thanks for your reply.
$\displaystyle u^n_{j}$ is defined as the numerical approximation of $\displaystyle u(x, t)$ at time index $\displaystyle n$, and distance index $\displaystyle x$. So basically it represents the $\displaystyle n^{th}$ time iteration and the $\displaystyle j^{th}$ distance iteration of the solver algorithm.

Re: Solving PDEs
What is the taylor series expansion look like and how similar is it to your approximation (where dx's, dt's, etc are approximated by delta_x, delta_t etc)?