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Thread: Solving PDEs

  1. #1
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    Solving PDEs

    Hi,

    Given a PDE of this form



    then one way to solve it iteratively is



    (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 .

    Anyone familiar with this who can enlighten me?

    Thanks.
    Last edited by algorithm; Oct 20th 2012 at 04:09 AM.
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  2. #2
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    Re: Solving PDEs

    Hey algorithm.

    What is the definition of u_j given in your book?
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  3. #3
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    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.
    Last edited by algorithm; Oct 21st 2012 at 06:24 AM.
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  4. #4
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    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)?
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