Linear-implicit and energy-preserving schemes for the Benjamin-type equations

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.