The Benjamin-type equations are typical types of non-local partial differential equations usually describing long internal waves along the interface of two vigorously different fluid layers. In this work, we propose two kinds of novel linear-implicit and energy-preserving algorithms for the Benjamin-type equations. These algorithms are based on the invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, respectively. The IEQ and SAV are originally developed to construct energy stable schemes for the class of gradient flows. Herein, we innovate such schemes to the Benjamin-type equations and, essentially, verify them to be effective to construct energy-preserving schemes for the Hamiltonian structures. Meanwhile, numerical experiments are presented to demonstrate the efficiency of these schemes eventually.

中文翻译：

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本杰明型方程的线性隐式和能量保持方案

Benjamin 型方程是典型的非局部偏微分方程类型，通常描述沿两个剧烈不同的流体层界面的长内波。在这项工作中，我们为 Benjamin 型方程提出了两种新颖的线性隐式和能量保持算法。这些算法分别基于不变能量二次方 (IEQ) 和标量辅助变量 (SAV) 方法。IEQ 和 SAV 最初是为构建梯度流类的能量稳定方案而开发的。在这里，我们将这些方案创新到 Benjamin 型方程，并从本质上验证它们对于构建哈密顿结构的能量保持方案是有效的。同时，通过数值实验最终证明了这些方案的有效性。