I'm new here and looking for help - so, hello! I tried to find something related to this topic but couldn't find anything that fits my needs. I try to make it as easy, short and clear as possible.
Okay, I am writing on my thesis right now and I got a problem regarding the fundamental understanding. Trying to get my answer from books lead to further confusion. So I also try to clarify the ideas I have right now.
Basically I am working on the numerical simulation of flow in a gas pipe. With an gas tank on the left side and an actuator (valve) on the right side. Withing the tube I work with simplified euler-equations which eventually pose a 2x2 PDE system. So all together I have an PDE system, coupled with an ODE on the tank side and a controllable input at the valve. The main goal will be to control the pressure within the tank on the left side
My current situation is that I am able to simulate the flow (and the pressure in the tank) for two different models (different grades of simplification) via lax-friedrichs and maccormack schemes. Works like a charm. I also got a diagonalized version of my models, which is where the questions ans problems start.
First: The direction of information within the pipe of my original system is from left to right for one state, and from right to left for the other state. I can confirm this by the anti-diagonal structure of the 2x2 system, right?
Okay, so I tried to diagonalize the problem, which worked for the linear model (the other is a quasilinear one). That means I transformed the system in a form where the information of both (now no more physical) states travels in only one direction, right? The simulation works quite well with lax-friedrichs and a bit shabby with maccormack - no question here, since this would lead to far. Anyway, since my information travels only one way now I can apply an upwind scheme for simulation, am I right? This doesn't work at all tho. Is there anything I need to consider?
Or am I completely off with the direction thing. It may be interesting for you to know that my coupling matrix may be diagonal now, but the eigenvalues have now alternating signs.
Since I am even more confused than before I will leave it here and hope someone can help or refer to according texts.
Thanks in advance and each of you have a nice start for the week!