Speeding up Boussinesq equations solving?

Hi all,

I am working on Boussinesq equation. The notebook can run perfectly for only 0.6 steps and then the calculation starts running slowly after 0.7.
All boundary conditions seemed fine. I am unsure if I used the correct method to solve this problem.
This is the method I am currently using.

I have tried several methods. Unfortunately, it does not work. I can’t figure out the mistake.
Please let me know if you have any suggestions.
Thank you very much.

Empty box-question.nb (93.5 KB)

Crossposted here.

Yes, it’s the same person. I am looking forward to seeing the suggestion.