Energy-preserving Finite Volume Method Of Surface Gravity Wave Equation Based On WENO Reconstruction

The surface gravity wave equation describes the ocean surface fluctuation under the action of gravity.At present,the discrete solutions of the conservation equation are mostly based on the finite difference method.Although the simulated numerical solutions have high accuracy,they may not satisfy the physical laws in the process of sea water movement and development.The energy-preserving algorithm of surface gravity wave equation studied in this paper can maintain the characteristics of the original system and be consistent with the physical laws in the process of ocean motion development,which is of great significance to the future study of ocean numerical model energy-preserving algorithm.First of all,based on the surface gravity wave equation and the idea of WENO reconstruction and the principle of finite volume method,this paper constructs a numerical method that can accurately maintain the energy conservation of the system.On the one hand,the energy evolution equation is used to estimate the "theoretical" change of energy on each grid.On the other hand,when the grid function is reconstructed,the real solution is estimated based on the WENO linear function,and two parameters are introduced to adjust the numerical solution,finally achieving the conservation of "numerical" energy and "theoretical" energy.Finally,it is proved that the convergence order of the constructed numerical method is 2.Secondly,in the case that quadratic polynomial is used as reconstruction function,two parameters are introduced into the WENO quadratic polynomial reconstruction function,and the surface gravity wave equation and its energy development equation are solved at the same time.Appropriate parameters are selected to make the numerical solution reach the conservation of "numerical" energy and "theoretical" energy in the iterative process.Finally,it is proved that the convergence order of the constructed numerical method is 2 order,which indicates that a certain precision cost is paid while maintaining the energy con servation.Numerical experiments show that the proposed method has some advantages in long time simulation.