A multi-symplectic algorithm for generalized Benjamin-Bona-Mahoney (BBM) equation with stable long-time numerical behavior

Aim. Many practical problems are nonlinear but linearization often brings poor long-time numerical behavior. To overcome this shortcoming, we propose constructing the multi-symplectic formulation of the generalized BBM equation. In the full paper, we explain our multi-symplectic algorithm in some detail; in this abstract, we just add some pertinent remarks to naming the first two sections of the full paper. Section 1 is: The multi-symplectic formulation of the generalized BBM equation and its conservation laws. In section 1, we derive eq. (6) as the multi-symplectic formulation and eqs. (7), (8) and (9) as its conservation laws. Section 2 is: The multi-symplectic Preissmann scheme and its equivalent formulation. In section 2, we rewrite the well-known Preissmann scheme as eq. (10) and derive its equivalent formulation as shown in eq. (11). Finally, we do the numerical simulation of the bell-shaped solitary wave solution of the generalized BBM equation. The simulation results, shown in Figs. 1 through 3 in the full paper, indicate preliminarily that our multi-symplectic algorithm does have stable long-time numerical behavior.