On the Numerical Simulation of Unsteady Solutions for the 2D Boussinesq Paradigm Equation

For the solution of the 2D Boussinesq Paradigm Equation (BPE) an implicit, unconditionally stable difference scheme with second order truncation error in space and time is designed. Two different asymptotic boundary conditions are implemented: the trivial one, and a condition that matches the expected asymptotic behavior of the profile at infinity. The available in the literature solutions of BPE of type of stationary localized waves are used as initial conditions for different phase speeds and their evolution is investigated numerically. We find that, the solitary waves retain their identity for moderate times; for larger times they either transform into diverging propagating waves or blow-up.