Analysis of an invariant preserving numerical method for nonlinear long wave equation

Abstract

In this thesis, we propose a fourth-order implicit finite difference scheme for the Benjamin–Bona-Mahony-Burger (BBMB) equation, which is a nonlinear long wave equation describing dynamics of water waves, acoustic-gravity waves and hydromagnetic waves in cold plasma. The solution of the BBMB model is known to preserve some invariant, namely momentum and energy, as time increase. The proposed method is also proven to preserve the invariant in the discrete sense. Our method achieves fourth-order of accuracy in space while requiring significantly fewer neighboring points for the solution approximation compared to existing fourth-order difference schemes. This advantage comes at the cost of using a larger coefficient matrix to invert during the computation, but the matrix is sparse. Numerical experiments are done in order to demonstrate the efficiency and high accuracy of the scheme.