PDE discretization for semi-infinite boundary?

[maistral]

Hi. Been a while since I logged in here, I missed this place.

Anyway, I have a question (title). Is that even possible?

Say for example I have the standard heat equation (PDE) subject to the boundary conditions:
T(0,t) = To
T(∞,t) = Ti

And the initial condition:
T(0,t) = Ti

I am aware of how to solve this analytically. I would like to try solving it numerically - using finite difference methods. However, I have no idea how will I discretize something that is infinite...

What I did is assume x to be a gigantic value (which would become very unwieldy if my spatial step is something like 0.0001). Then, at the final spatial point, either I assume that the derivative is zero (because apparently if I compare it with the analytical solution by having t = a very large value, then run the numerical simulation for a very long time, both dependent values will become To) or just set the value to be equal to Ti (but I run at the difficulty of having the last point still equal to Ti at large time values).

So my question is: How is it possible to discretize infinite boundaries effectively? Thanks!

PS: I know that this is graduate-level math, but please talk to me in layman's language. I easily get inebriated if you guys bring out complex terms suddenly. Thanks.

 

[bigfooted]

You have to use a coordinate transformation of the PDE such that the domain $[0,\infty)$ maps to $[0,1]$. The early paper of Grosh and Orszag on this issue mention some other ideas as well:
https://www.researchgate.net/publication/222460408_Numerical_solution_of_problems_in_unbounded_regions_Coordinate_transforms
And in this paper they call this technique compactification:
https://philippelefloch.files.wordpress.com/2010/07/2010-july-anil-zenginoglu.pdf

 

[maistral]

I knew it. Coordinate transformation :|

That's the thing I'm trying to avoid actually. Thanks a lot.

 

[Chestermiller]

As time progresses, the solution propagates into the semi-infinite slab more and more deeply. So, unless you just want to accept the idea that, beyond a certain amount of time, the solution is going to become inaccurate, you need to use a coordinate transformation. If you want to do it analytically, then you can develop the so-called similarity solution. If you are doing it numerically, then you need to map the spatial coordinates so they stretch with respect to time in proportion to the square root of time.