Two high-precision compact schemes for the dissipative symmetric regular long wave (SRLW) equation by multiple varying bounds integral method

Abstract

This paper mainly focuses on the numerical study of fourth-order nonlinear dissipative symmetric regular long wave equation. We propose two new methods: the Multiple Varying Bounds Integral (MVBI) method and Taylor Function Fitted (TFF) method. With the multiple varying bounds integral method, all the derivatives in the space direction of the differential equation can be eliminated and we can get different numerical formats by adjusting the integral bound parameters. According to the physical properties of the original differential equation, we can choose an appropriate format from them. Meanwhile, with the Taylor function fitted method, the derivatives of the function at one point, such as first-order and second-order, can be approximated by the original function value at the points around it. Hence, with the MVBI method and TFF method, we can establish two compact and high-precision numerical schemes. In addition, we prove that these numerical schemes are consistent with the original equation on the energy property. Next, the convergence and stability of numerical solution U and P̃ are both proved. Finally, numerical experiments are carried out to verify the effectiveness of numerical schemes.