Efficient algorithm and convergence analysis of conservative SAV compact difference scheme for Boussinesq Paradigm equation

In this paper, we construct an efficient and conservative compact difference scheme based on the scalar auxiliary variable (SAV) approach for Boussinesq Paradigm (BP) equation. The compact difference scheme preserves the mass and discrete modified energy. We prove uniquely solvability of the compact difference scheme and analyze the bounded estimates of the numerical solution. The rates of convergence of second-order in temporal direction and fourth-order in spatial direction are given by using the discrete energy method in detail. Some numerical experiments are given to verify our theoretical analysis.