Implicit Euler equations
 

explicit euler equations usually have long convergence times. Is it possible to have an implicit time stepping instead to make the time step better? I am not meaning runge kutta methods. My own numerical method is finite volume with Roe' scheme.

I suggest to use a BDF method in where the solution of the implicit non-linear system is obtained by means of a modified Newton-Raphson scheme, with a dual time step integration technique (*)

(*) A. Jameson, Time dependent calculations using multigrid with applications to unsteady flows past airfoils and wings, in: 10th AIAA
Computational Fluid Dynamics Conference, Honolulu, HI, pp. 1–13. AIAA Paper 91-1596.