Numerical methods for differential equations

[SarthakC]

Hi,
Are there any numerical techniques I can use to solve differential equations with periodic boundary conditions? I know of several techniques for other kinds of boundary conditions (such as Runge-Kutta method, Euler method etc.), but I am interested in knowing how to numerially solve differential equations with periodic boundary conditions..

Any help would be appreciated!

 

[pasmith]

You will want a Spectral method.

 

[SarthakC]

Umm... Maybe I should make my specific scenario a little clearer. My differential equation is of the form $$f(x) \partial_x^2 y + g(x) \partial_x y + h(x) y =0$$

In this scenario, replacing $y$ with a Fourier series in $x$ is not really useful, because the functions $f, g$ and $h$ do not let me compare coefficients anymore, because in order to do that I would have to multiply by some element of the orthogonal Fourier basis, but these functions will not let me extract the information as I want. For example if i just had $\partial_x y + \delta(x) y=0$, then replacing with $y$ with $\sum_n c_n e^{inx}$ would give me $$\sum_n c_n (i n + \delta(x)) e^{inx}=0$$
The only procedure I'm aware of would proceed from here by mulitplying by $e^{-imx}$ and integrating, but that would give me $c_m (i m) + \sum_n c_n = 0$. This may not be the best example, but in general I would not get equations from where I could calculate the coefficients directly, right?Wouldn't a spectral method work nicely only when I had a simpler equation with constant coefficients or something?

 

[HallsofIvy]

The spectral method works as long as you can write the coefficients themselves as Fourier series or Fourier Transforms.

 

[SarthakC]

Could someone give me an example? For example let's just take a simple first order equation of $$y^\prime + \Theta(x) y = 0$$ where $\Theta(x)$ is the Heaviside step function, with a domain from $(-\pi,\pi)$, with peridic boundary equations. Could anyone just show me a few steps from where I can proceed?

 

[the_wolfman]

Actually, for most numerical methods, applying periodic boundary conditions is pretty straight forward.

Are you familiar with finite difference?

In finite difference we break the domain into a finite number of points:
x_0, x_1, \dots, x_i, \dots, x_N

Then we approximate the differential to create an algebraic equation for the interior points x_1 \dots x_{N-1}. For second order equations this equations is often of the form a x_{i-1} + bx_i + cx_{i+1} = f(x_i)

Then we have to apply the correct boundary conditions to the points at the edge of the domain x_0, x_N

For Dirichlet B.C we might set x_0 =0 and x_N =5

For periodic boundary conditions we can use the interior equation for x_0. Noting that x_{-1} = x_{N-1} gives the equations a x_{N-1} + bx_0 + cx_{1} = f(x_0).

For the end point we can use x_N =x_0. We often don't solve this last equations. Instead we enforce this equality in our the algebraic system of equations by replacing all x_N with x_0 just like we replaced x_{-1} with x_{N-1}. This reduces the error due to finite precision math and slightly reduces the computation time.

 

[SarthakC]

Hmm. So the idea is I use the finite difference method and the boundary conditions to give me an algebraic problem which I then solve numerically to get all the $x_i$s?

 

[the_wolfman]

Hmm. So the idea is I use the finite difference method and the boundary conditions to give me an algebraic problem which I then solve numerically to get all the $x_i$s?

That's the general idea.

There are a few simple examples in the second half of this lecture:
http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf

 

[SarthakC]

Thanks!
I'll give both the spectral method and the finite difference method a shot and try to solve my original problem. I'll get back to you guys if I have any more clarifications!

Thanks!