Multi-symplectic quasi-interpolation method for the KdV equation

In this paper, we propose a multi-symplectic quasi-interpolation method for solving the KdV equation. Based on the multi-symplectic formulation, we discretize the equation in space with the quasi-interpolation approach and integrate the time with the implicit midpoint scheme to derive the full-discrete system. The local conservation properties of the KdV equation are also investigated, including multi-symplectic conservation laws and local energy conservation laws. The quasi-interpolation method is advantageous in that it gives the approximation directly without the need to solve any linear system. It is very simple and easy to implement. Moreover, our multi-symplectic quasi-interpolation method can preserve the multi-symplecticity and has local energy conservation laws on nonuniform grids. To demonstrate the accuracy and conservation properties of our method after long time simulation, we consider a single solitary wave and several interactive solitary waves in the numerical part.