Numerical solution of the two-dimensional regularized long-wave equation with a conservative linearized high order finite difference scheme

In this article, a finite difference scheme is investigated to solve a two-dimensional regularized long-wave equation. The original equation is deformed into another formulation. The fourth order compact difference scheme and the fourth order Padé scheme are used for the treatment of the second and first derivatives of the deformation equation in space direction, and the fourth order backward difference formula is used for the discretization of the time derivative. The nonlinear term in the proposed scheme is linearized by Taylor series expansion method. The new scheme is shown to be of the fourth order accuracy in both temporal and spatial dimensions. The existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the new scheme is shown to be conservative and unconditionally stable. Finally, numerical experiments are provided to verify our theoretical analysis results of the new scheme.