High-order Linearly Energy-preserving Compact Finite Difference Schemes for the Korteweg-de Vries Equation

In this paper, the discrete variational derivative method (DVDM) and the compact difference method are combined to construct linearly energy-preserving schemes for the Korteweg-de Vries equation. The sixth-order compact difference method is used in the spatial direction, and the discrete variational derivative method is used in the temporal direction. The resulting fully discrete schemes are linear, unconditionally stable, uniquely solvable, and can precisely conserve the discrete mass and energy. At last, some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the proposed schemes. Numerical results show that the proposed schemes are more advantageous than the existing methods.