Stable spectral collocation solutions to a class of Benjamin Bona Mahony initial value problems

We are concerned with stable spectral collocation solutions to non-periodic Benjamin Bona Mahony (BBM), modified BBM and Benjamin Bona Mahony-Burgers (BBM-B) initial value problems on the real axis. The spectral collocation is based alternatively on the scaled Hermite and sinc functions. In order to march in time we use several one step and linear multistep finite difference schemes such that the method of lines (MoL) involved is stable in sense of Lax. The method based on Hermite functions ensures the correct behavior of the solutions at large spatial distances and in long time periods. In order to prove the stability we use the pseudospectra of the linearized spatial discretization operators. The extent at which the energy integral of BBM model is conserved over time is analyzed for Hermite collocation along with various finite difference schemes. This analysis has been fairly useful in optimizing the scaling parameter. The effectiveness of our approach has been confirmed by some numerical experiments.