My Math Forum predictor-corrector scheme

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March 20th, 2011, 03:31 AM   #1
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predictor-corrector scheme

Hi!
I'm asking for help to understand the realization of the iterative predictor-corrector scheme (I'm not a specialist in this field) for PDE system . The description of a method attached in article.
The question concerns algorithm realization under the formula (A5), resulted in article
$U^{new}_i=U^{old}_i+dtF_i(\overline{U}^{new},\ove rline{U}^{old})$ (A5)
where $F_i$ - function in the right part of the equations, and $\overline{U}$ - variables.
$\overline{U}^{old}$, as I have understood, these are predicted value of all variables from, for example, Euler's method. Then it is necessary to calculate $U^{new}_i$ variables and to compare them with predicted, repeating algorithm before achievement of necessary accuracy. But how to calculate $U^{new}_i$ if in the right part there are the same $\overline{U}^{new}$, for me not clearly? In any way I don't understand the dependence $F_i(U)$\$ of two variables means? How to use this formula in calculations? I will be glad to any concrete algorithms or the references to it.
Attached Images
 FDTD_PC.tif (155.5 KB, 1089 views)

 March 20th, 2011, 08:53 PM #2 Member   Joined: Nov 2007 Posts: 73 Thanks: 0 Re: predictor-corrector scheme Without more references (I can't see the attachment), I guess the corrector method is implicit. Roughly speaking, you have $U^{new}$ both sides. You can solve the equation for $U^{new}$ using any method for zeros, like Newton's method. Be happy!
March 21st, 2011, 07:49 AM   #3
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Re: predictor-corrector scheme

I have partial differential equations system so the using of zero methods is more complex, some help I have found in the article where the iterative PC method was used. Sorry for attachment, here it is http://zalil.ru/30711340
Attached Images
 FDTD_PC.tif (137.1 KB, 1059 views)

 March 21st, 2011, 07:59 AM #4 Member   Joined: Nov 2007 Posts: 73 Thanks: 0 Re: predictor-corrector scheme Yes, I know it is a PDE system. But when you have the discretized equation, this one on the form u^{new}... is not longer a PDE. That's the point in almost all numerical methods. Think for instance in an implicit Runge-Kutta method for ODE's.
 March 21st, 2011, 08:27 AM #5 Newbie   Joined: Mar 2011 Posts: 6 Thanks: 0 Re: predictor-corrector scheme The discretization of ODE and using predictor corrector method is more simple. For me not clear how I must use corrector cycle: for every equation separately or for all system in one cycle. For example the second equation discretization will be $U_1^{n+1}=U_1^{n}+dt\omega(U_2^{n+1}+U_2^{n})$ does it right to use such scheme $U_1^{new}=U_1^{n}+dt\omega(U_2^{predict}+U_2^{n})$ where $U_2^{predict}$ is calculate from other equation on previous step?
 March 22nd, 2011, 04:36 AM #6 Member   Joined: Nov 2007 Posts: 73 Thanks: 0 Re: predictor-corrector scheme No, I think your suggestion is not correct. If you use the U^{predict} instead of $U^{n+1}$ you're not going to obtain a correction as n increases. You have a system for the correction, and this system involves all components of the unknown. Let's see. Your unknown is $U=(U_1,U_2,U_3)$. Using the predictor you obtain $U^{predict}=(U_1^{predict},U_2^{predict},U_3^{pred ict})$. Let's take $U^0=U^{predict}$. The correction procedure is an iteration in the form: $U^{n+1}=U^n + \Delta t F(U^n,U^{n+1})$ More precisely: $U_1^{n+1}=U_1^n+\Delta t F_1(U^n,U^{n+1})$ $U_2^{n+1}=U_2^n+\Delta t F_2(U^n,U^{n+1})$ $U_3^{n+1}=U_3^n+\Delta t F_3(U^n,U^{n+1})$ The point is in the right hand side of every equation you have ALL the unknowns. This forces to solve ALL the system at the same time. One method to compute U^{n+1} is to write: $U_1^{n+1}-U_1^n-\Delta t F_1(U^n,U^{n+1})=0$ $U_2^{n+1}-U_2^n-\Delta t F_2(U^n,U^{n+1})=0$ $U_3^{n+1}-U_3^n-\Delta t F_3(U^n,U^{n+1})=0$ This is a problem about find a zero of a function $\mathbb{R}^3\longrightarrow\mathbb{R}^3$. You can use the Newton method (or any method for zeros). Be happy!
 March 22nd, 2011, 08:44 AM #7 Newbie   Joined: Mar 2011 Posts: 6 Thanks: 0 Re: predictor-corrector scheme Thank you for developed answer Please explain what you mean when use $F_i(U^{n+1},U^{n})$. For example for equation (A4c) $F_2(U)=\omega U_2$. What the dependence on two parametrs $F_2(U^{n+1},U^{n})$ means?
 March 22nd, 2011, 01:52 PM #8 Member   Joined: Nov 2007 Posts: 73 Thanks: 0 Re: predictor-corrector scheme I am not sure about what you meant. The F dependence on two parameters just meant (in this case) that F depends on the n and the n+1 approximated solutions.
 March 22nd, 2011, 08:37 PM #9 Newbie   Joined: Mar 2011 Posts: 6 Thanks: 0 Re: predictor-corrector scheme I don't understand how the expression for $U^{n+1}$ will look exactly (n - time step or number of iteration in time-spatial step ? ). If $F_1=\omega U_2$ and $U_1^{0}=U_1^{predict}, U_2^{0}=U_2^{predict}$ so if we use corrector scheme does it right to write $U_1^{n+1}=U_1^n+dt\omega (U_2^n + U_2^{n+1})/2$ and from where we get $U_2^{n+1}$ on the first corrector iteration for example $n=0$?
 March 23rd, 2011, 05:07 AM #10 Member   Joined: Nov 2007 Posts: 73 Thanks: 0 Re: predictor-corrector scheme That's the point. The method it's not explicit. It's an implicit method. That means you need to solve an equation (normal equation, not differential one) to compute every approximation U^{n+1} using U^n. Check this wikipedia article: http://en.wikipedia.org/wiki/Explicit_a ... it_methods I think it can helps you. Note in the wikipedia example the implicit formula is easy to solve, but in your case probably you need to solve it numerically too (Newton method or any other method for zeros).

### corrector predictor method with example for comp sci

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