# Semi-infinite domain transformation

Question is mostly related to literature or suggestions.

Given a semi infinite domain: $$x=[0; +\infty);y=[0; +\infty)$$. Willing to transform it to computational domain of: $$[0,1]\times[0,1]$$. I did find the $$(-\infty, +\infty)$$ to $$[-1,1]$$ transformations, though looking for mapping operators from semi infinite to finite domain, thank you in advance!

I know two papers that investigate infinite mapping layers and apply them to examples:

[1] Schoder, Stefan, et al. "Revisiting infinite mapping layer for open domain problems." Journal of computational physics 392 (2019): 354-367.

[2] Toth, Florian, Stefan Schoder, and Manfred Kaltenbacher. "An infinite mapping layer for deep water waves." PAMM 17.1 (2017): 689-690.

The first one, which is pretty good in my optinion, is free accesible.

Additionally to the $$arctan$$ mapping that nicoguaro already introduced they define two more mappings:

Let $$x\in[0,\infty]$$ your seimi-infinite domain and $$\tilde{x}\in[0,L]$$ the finite domain of length $$L$$ you want to map to. Then we can define the following two infinite mapping layers

\begin{align} \tilde{x} = \frac{xL}{x+\kappa}; \quad \tilde{x}=L(1-e^{-x/\kappa}) \end{align}

where $$\kappa$$ is a constant.

You choose the appropriate mapping layer by investigating your problems solution. For example in the paper [2] about deep-water waves they show that the exponential function yields the best results, since the wave amplitudes decay exponentially or for example in electrostatics the rational mapping from above is the best, since we have a rational decay of the solution here.

• thank you! just a quick question, for the incompressible Navier Stokes do you recommend checking both transformations behaviour? or trying to find further mappings Sep 26 '21 at 7:16
• Im not sure about which one will work best for your specific case or even work at all. Note that the problems in the papers I referenced are all elliptic PDE. So for the Stokes equation you might get good results. And keep in mind that the solution at infinity should be easy, meaning going to zero or to a simple laminar flow.
– Pepe
Sep 26 '21 at 20:50
• great, any suggestions on symbolic tools to apply the transform? Sep 27 '21 at 17:28
• Since the simulations in the papers above were done in opencfs, you might check out the sourcecode on gitlab to see how an implementation can be done. They also provide many examples in their testsuit. Examples for the three infinite mapping layers can be found in the WaterWaves directory of the testsuit. You could also try to implement it in ngsolve, which allows an impementation of many PDE's in Python.
– Pepe
Sep 27 '21 at 19:18

You could use the following transformation

\begin{align} &u = \tanh(x)\, ,\\ &v = \tanh(y)\, . \end{align}

Another option is to use $$2/\pi \arctan(x)$$, but I have had better results with the hyperbolic tangent in the past.