Conservative finite difference methods for the Boussinesq paradigm equation

Abstract

In this paper, we concentrate on developing and analyzing two types of finite difference schemes with energy conservation properties for solving the Boussinesq paradigm equation. Firstly, by introducing the auxiliary potential function ∂u∂t=Δv and lim|x|→∞v=0, the BPE is reformulated as an equivalent system of coupled equations. Then a class of efficient difference schemes are proposed for solving the resulting system, where one proposed scheme is a two-level nonlinear difference scheme and the other is a three-level linearized difference scheme using the invariant energy quadratization technique. Subsequently, we present the theoretical analysis of the proposed energy conservative finite difference schemes, which involves the discrete energy conservation properties, unique solvability and optimal error estimates. By using the discrete energy method, it is proven that the proposed schemes can achieve the optimal convergence rates of O(Δt2+hx2+hy2) in discrete L2-, H1- and L∞-norms. At last, numerical experiments illustrate the physical behaviors and efficiency of the proposed schemes.