time integration stability in modelica

Question

I am constructing a finite volume model in Dymola which evolves in time and space. The spatial discretization is hard coded in the equations section, the time evolution is implemented with a term consisting of der(phi).

Is the time integration of Dymola always numerically stable when using a variable step size algorithm? If not, can I do something about that?

Is the Euler integration algorithm from Dymola the explicit or implicit Euler method?

Answer 1

The Dymola Euler solver by default is explicit (if an in-line sovler is not selected).

Answer 2

The stability of time integration is going to depend on your integrator. Generally speaking, implicit methods are going to be much better than explicit ones.

But since you mention spatial and time discretization, I think it is worth pointing out that for certain classes of problems things can get pretty sticky. In general, I think elliptic and parabolic PDEs are pretty safe to solve in this way. But hyperbolic PDEs can get very tricky.

For example, the Courant-Friedrichs-Lewy condition will affect the overall stability of the solution method. But by discretizing in space first, you leave the solver with information only regarding time and it cannot check or conform to the CFL condition. My guess is that a variable time step integrator will detect the error being introduced by not following the CFL condition but that it will struggle to identify the proper time step and probably also end up permitting an unacceptably unstable solution.