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I am simulating in COMSOL a system of 3 coupled PDEs (parabolic & elliptic) along with 10 stiff ODEs. In order to have the system working, I am downsizing the time step size too much to achieve convergence. I am lucky enough that I am doing 1D simulations which don't take too much time, but this drives me to think about more complex 3D cases in which the distretized system will result in hugh DOF to be solved, with the very low step size this takes too much time.

So, is there computationally efficient method to solve such system (e.g. downsizing step size for ODEs but having larger ones for PDEs. Just imagining)

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    $\begingroup$ Once you semi-discretize, you just have a system of ODEs. It sounds like some of yours will be stiff and others will not. It is common to apply IMEX (implicit-explicit) methods to such systems. I can't help you more without more details. $\endgroup$ – David Ketcheson Jun 18 '16 at 16:42
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    $\begingroup$ Given you discretize everything into ODEs, you could put the system of equations into some adaptive integrator and that may help you by increasing the step size when it makes sense to instead of remaining at the small initial time step you set. $\endgroup$ – spektr Jun 19 '16 at 14:34

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