[Slyphlamen]

Hello,


For a project of mine I am aiming at simulating 2D compressible Euler flow in spherical polar symmetry with photo-chemistry. As photo-chemicals reactions are incorporated, I end up in conservative form with a split operator of the form : $\frac{\partial \textbf{U}}{\partial t} = \textbf{F} + \textbf{S}$; where S is a stiff operator and F is not.


As I am trying to follow an author, the only information I have for a numerical scheme that can do the job is : "The two-dimensional time-dependent multi-fluid model is integrated in time by using the fourth-stage Runge-Kutta method. For the space derivative, central difference scheme is used. [...] we must handle those source terms in the continuity and energy equations because of their stiffness. An explicit method could result in a long time-marching and incorrect results. Thus, we choice a semi-implicit method [...]".


Therefore I looked for such semi-implicit schemes, and found Section 4 of https://www.researchgate.net/publica...ion_simulation.


Since I have a bit of a hard time with terminology in the field of semi-implicit schemes, am I right in thinking that semi-implicit and IMEX (implicit-explicit) schemes are the same ?
Likewise I am right in considering the scheme in the above link as a semi-implicit fourth order Runge-Kutta scheme (for the explicit part) ?


And finally, for the implicit integration I was wondering if someone could recommend some litterature on this topic as I don't have many clues on how to go at it.

For now I tried to implement it by looking for the roots of equation (17) of the article, using the scipy.optimize.root with a Newton-Krylov method (https://www.sciencedirect.com/scienc...340?via%3Dihub) but somehow my implemented scheme does not seem to be satisfying. When I try to reproduce the Lokta-Volterra plot found here by feeding my IERK45 scheme the Lokta-Volterra function to the implicit part, and a null function to the explicit part my results "lose in amplitude" over time (with the same conditions).




Sorry if anything is unclear, or if other topics already discuss this (which doesn't seem to be the case).