Linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries equations

Abstract

In this paper, second-order and fourth-order linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries (KdV) equations are proposed. First, the KdV equation is semi-discretized into an oscillatory Hamiltonian system of Ordinary Differential Equations (ODEs) by semi-discretizations of the skew adjoint operator and the Hamiltonian in the Hamiltonian form of the KdV equation. Then the resulting semi-discrete system is integrated by exponential average vector field integrator. The order of convergence and the conservation properties are established for the introduced full-discrete procedures. It shows that the proposed methods are energy-preserving, mass-preserving and qualitatively preserve the dispersive relation well. Numerical experiments are carried out to validate the efficiency and the accuracy of the proposed schemes.