# Which numerical scheme should be used?

Trying to find a way to solve exponentially nonlinear elliptic equation with complex source term i manage to know about such schemes like Godunov, Lax-Friedrich, MUSCL. I still seraching for literature about them.

The main question is could this schemes do something with $$\dfrac{\partial}{\partial x}\left(ce^{\varphi_n-\varphi_c}\dfrac{\partial\varphi_n}{\partial x}\right) = R$$

This schemes traditionally for parabolic/hyperbolic equation, but i could add some time derivative (i erase it because i need stationary solution). Or it is possible to use them with elliptic one? (why not)

I also know that it could possible if only the scheme with it initial condition will be stable. For example, MUSCL scheme deal with discontinueties, large gradient etc. (from Wiki)

Is it possible when $ce^{\varphi_n-\varphi_c}$ change very fast (1e18 in a small area)? Could you adviced me some literature?

P.S. sorry for my english, i working on it.

• I almost trying some simple aproaches which i know. Newton linearization, Pikard iteration (with FVM). This approach working good for a much more lover exponenta. – Sergey Konoplev Jan 5 '17 at 21:43
• Are you sure the problem is well-posed and is well conditioned? – Paul Jan 5 '17 at 23:23
• The problem is well posed because it has unique solution, its solution a continuous and it exist (i assume that because it modeling some physics and it almost solved in some commercial soft, unfornetly i couldnt prof its existance now ). But the matrixes i recieve till now (by using Pikard etc) is ill conditioned because of changing of coefficient from 1e18 till 1. – Sergey Konoplev Jan 6 '17 at 9:52
• Floating point arithmetic only has 16 digits. Coefficients that vary by 18 orders of magnitude on small scales can not be resolved computationally -- for all practical purposes, your coefficient is zero in some areas, and you will have to model it as such. – Wolfgang Bangerth Jan 6 '17 at 12:54
• The schemes you mention (Godunov, Lax-Friedrichs, MUSCL) are for hyperbolic problems, not elliptic problems. It seems to me that you're very confused and it's unlikely that we can help. Do you have a mentor you can speak to? – David Ketcheson Jan 6 '17 at 17:06