# Solving PDEs

• October 20th 2012, 03:56 AM
algorithm
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.../Capture-7.jpg

(Equations from IISc).

The update equation has me confused for the following reason. $u(x, t)$ appears in the PDE as function of both time $t$ and distance $x$. However, the update equation only gives you $u$ as a function of time ( $u^{n+1}_{j}$), and what's more, it seems to assume that $u$ is already known as a function of distance because it has $u^{n}_{j+1}$ terms.

In short, the solver seems to assume that you already know the solution :confused:.

Anyone familiar with this who can enlighten me?

Thanks.
• October 20th 2012, 05:46 PM
chiro
Re: Solving PDEs
Hey algorithm.

What is the definition of u_j given in your book?
• October 21st 2012, 02:42 AM
algorithm
Re: Solving PDEs
chiro,

$u^n_{j}$ is defined as the numerical approximation of $u(x, t)$ at time index $n$, and distance index $x$. So basically it represents the $n^{th}$ time iteration and the $j^{th}$ distance iteration of the solver algorithm.
• October 21st 2012, 05:55 PM
chiro
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)?