广义Benjamin-Bona-Mahoney方程的多辛算法

A Multi-Symplectic Algorithm for Generalized Benjamin-Bona-Mahoney(BBM) Equation with Stable Long-Time Numerical Behavior

文章基于Bridges意义下的多辛理论构造了广义Benjamin-Bona-Mahoney方程的多辛偏微分方程组,利用变分原理得到了多种守恒律,构造了一种等价于Preissmann格式的隐式多辛格式。钟状孤波解的数值模拟结果表明该多辛格式具有较好的长时间数值稳定性。

Aim. Many practical problems are nonlinear but linearization often brings poor long-time numerical behavior. To overcome this shortcoming, we propose constructing the muhi-symplectic formulation of the generalized BBM equation. In the full paper, we explain our muhi-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 muhi-symplectic formulation of the generalized BBM equation and its conservation laws. In section 1, we derive eq. （6） as the muhi-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 muhi-symplectic algorithm does have stable long-time numerical behavior.