[mprinkey]

Well, just superficially, Eq3 shows no "wave-like" character...it cannot be viewed in terms of a wave-equation. And Eq2 is Poisson equation for w, so no wave character there. My thinking is that particular splitting of the problem does some violence to that wave character. I'd try another. Keep Eq2 as is, but rewrite Eq3 as:

d^2phi/dt^2 + phi - w_zz = 0

Then, at least, the time integration step of Eq3 will exhibit wave solutions rather than parabolic solutions. Your problem then becomes de facto a wave equation solution for phi with a convoluted way of computing the "forcing" function w_zz. This should be easy to change and test.

I don't have any other explanation here, probably because your problem lies well outside of the normal problems most of us deal with in transport phenomena.

[soes]

Thank you for your reply, I will try this way as well and will keep you updated. Yes, it is possible that this splitting causes this issue.

[soes]

Yesssss, it works now, it was not the issue of schemes initial condition boundary condition, only the issue was with the splitting method, thank you so much for the continuous feedbacks and thanks for you time and considerations