1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Discretizing elliptical PDEs

  1. May 11, 2015 #1
    1. The problem statement, all variables and given/known data
    Hi - looking at 'discretizing elliptical PDEs'.
    I understand the normal lattice approach, but this approach uses the variational principle. I have a couple of questions please. The text says:
    $$ \: Given\: E=\int_{0}^{1} \,dx\int_{0}^{1} \,dy\left[\frac{1}{2}\left(\nabla \phi\right)^{2} - S\phi \right] $$
    "It is easy to show that E is stationary under all variations $\delta \phi$ that respect the Dirichlet boundary conditions imposed. Indeed the variation is $$ \delta E=\int_{0}^{1} \,dx\int_{0}^{1} \,dy\left[\nabla \phi .\nabla \delta \phi - S \delta \phi \right] $$
    ..which upon integrating the the second derivative by parts becomes...
    $$ \delta E=\int_{C} \,dl\: \delta \phi \vec{n} . \nabla \phi + \int_{0}^{1} \,dx \: \int_{0}^{1} \,dy\: \delta \phi \: \left[-\nabla^2 \phi - S \right] $$
    ... where the line integral is over the boundary of the region of interest (C) and n is the unit vector to the boundary."

    Sadly, while I recognize all the words and symbols, I cannot follow it at all; I also have no idea where they get the starting equation or what it means. (The sometimes problem with computational physics is that it assumes certain background knowledge). I am hoping someone can expand enough (or provide links) so that I can follow all the above steps in detail?

    The second question follows a short bit later in the text, apparently the above 'easily' leads to the following equation:
    $$ E= \frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n}\left[\left({\phi}_{ij} - {\phi}_{i-1,j}\right)^2 + \left({\phi}_{ij} - {\phi}_{i,j-1}\right)^2 -{h}^{2}\sum_{i=1}^{n}\sum_{j=1}^{n}{S}_{ij}{\phi}_{ij}\right] $$

    I probably don't need to understand how they get to the above equation - I'd just prefer to understand as much as I can, so again please help me to follow it.

    What I really MUST do is now take a differential of the above w.r.t. $$ {\phi}_{ij} ,\: IE\: \frac{\partial{E}}{{\partial{\phi}_{ij}}} $$

    2. Relevant equations
    $$ E= \frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n}\left[\left({\phi}_{ij} - {\phi}_{i-1,j}\right)^2 + \left({\phi}_{ij} - {\phi}_{i,j-1}\right)^2 -{h}^{2}\sum_{i=1}^{n}\sum_{j=1}^{n}{S}_{ij}{\phi}_{ij}\right] $$

    3. The attempt at a solution
    I don't even know where to start in terms of the text above, but I should be able to do the partial derivative of the relevant eqn - with a little help help on the following 2 queries first please:
    I think that I can just differentiate inside the summations, is that right?
    What do I do with the i-1 and j-1 terms when differentiating w.r.t. ∅ij?
    ------------------
    Thanks for reading.
     
  2. jcsd
  3. May 12, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Do you understand the concept of discretisation? Can you briefly describe what you think it means?
     
  4. Jun 2, 2015 #3
    Sorry Orodruin, for some reason I didn't get a notification that you'd replied. In the meantime I have made progress with the 2nd part on the actual discretisation (using a regular lattice with spacing h to approximate the derivatives, in this case using the 2-point difference formula).

    But I am stuck again at a later stage in the book. We are given an energy functional using cylindrical coords:
    $$ E=\int_{0}^{\infty} r.dr\left[\frac{1}{2}\left(\frac{d{\phi}}{{dr}}\right)^2 - S.\phi\right] $$
    First we are asked to discretize using an ri = (i - 1/2)h lattice. I correctly get:
    $$ E=\frac{1}{2h}\sum_{i=1}^{n}{r}_{i-\frac{1}{2}}\left({\phi}_{i} - {\phi}_{i-1}\right)^2 - h\sum_{i=1}^{n}{r}_{i}{S}_{i}{\phi}_{i} $$
    $$ Then \: setting\: \frac{\partial{E}}{{\partial{\phi}_{i}}}=0 \:I \: get: $$
    $$ 2{r}_{i}{\phi}_{i} - {r}_{i-\frac{1}{2}}{\phi}_{i-1} - {r}_{i+\frac{1}{2}}{\phi}_{i+1} = {h}^{2}{r}_{i}{S}_{i} $$
    Which is all good - just the background for where I get stuck.
    -------------
    Now I am asked to do the same as above, but for a ri=ih lattice, and I should end with the same equation as the last above, but I don't and can't quite figure out what to do differently. My workings follow, discretising for the ri=ih lattice:
    $$ E=\frac{1}{2h}\sum_{i=1}^{n}{r}_{i}\left({\phi}_{i} - {\phi}_{i-1}\right)^2 - h\sum_{i=1}^{n}{r}_{i}{S}_{i}{\phi}_{i} $$
    $$ \frac{\partial E}{\partial{\phi}}=\frac{1}{2h} \frac{\partial }{\partial{\phi}}[\sum_{i=1}^{n}{r}_{i}\left({\phi}_{i} - {\phi}_{i-1}\right)^2 - h\sum_{i=1}^{n}{r}_{i}{S}_{i}{\phi}_{i}] =0$$
    $$ \therefore \frac{1}{2h} \frac{\partial }{\partial{\phi}}[{r}_{1}\left({\phi}_{1} - {\phi}_{0}\right)^2...+{r}_{i}\left({\phi}_{i} - {\phi}_{i-1}\right)^2 + {r}_{i+1}\left({\phi}_{i+1} - {\phi}_{i}\right)^2 ...+{r}_{n}\left({\phi}_{n} - {\phi}_{n-1}\right)^2] = h {r}_{i}{S}_{i} $$
    $$ \therefore \frac{1}{2h} [2{r}_{i}\left({\phi}_{i} - {\phi}_{i-1}\right) + 2{r}_{i+1}\left({\phi}_{i+1} - {\phi}_{i}\right)(-1) ] = h {r}_{i}{S}_{i} $$
    $$ \therefore \left({r}_{i}+{r}_{i+1}\right){\phi}_{i}-{r}_{i}{\phi}_{i-1}-{r}_{i+1}{\phi}_{i+1}={h}^{2}{r}_{i}{S}_{i} $$
    $$ Now \: \frac{1}{2}\left({r}_{i}+{r}_{i+1}\right)={r}_{i+\frac{1}{2}} $$
    $$ \therefore 2{r}_{i+\frac{1}{2}}.{\phi}_{i}-{r}_{i}{\phi}_{i-1}-{r}_{i+1}{\phi}_{i+1}={h}^{2}{r}_{i}{S}_{i} $$
    As you can see I've ended up shifted half a lattice. I think I haven't understood the difference between the (i - 1/2)h and ih lattices?
     
  5. Jun 18, 2015 #4
    Final, desperate call - assignment is due tomorrow, so will appreciate anyone who can help me quickly ....I think I have the method right, there is probably just some piece of the puzzle I don't know ....if you need more info. please just ask.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Discretizing elliptical PDEs
  1. Algebra or PDEs? (Replies: 4)

  2. Elliptic integral (Replies: 2)

  3. Elliptical trajectory (Replies: 2)

  4. Elliptical Motion (Replies: 3)

Loading...