A High-Order Compact Finite Volume Scheme For One Dimensional Sine-Gordon Equations

This paper constructed a high order compact finite volume method to calculate the numerical solution of the Sine-Gordon equation. In recent years, with the improvement of computer performance, computational mathematics obtained huge development. In most instances, people use the finite difference method (FDM) and finite element method (FEM) to solve equations. And the study in another framework of the finite volume method (FVM)is little. But it’s flexible in the grid subdivision, and relatively easier to deal with the boundary problems. The FVM can keep the law of conservation, it is very important in the study of fluid equations. Further, we use a compact finite volume method to improve the order at the boundary by decreasing the necessary nodes.Before the discretization in time for Sine-Gordon equations, we need to convert it to a first-order ordinary differential equation set. Then we use third order strong stable Runge-Kutta scheme to complete the discretization. Then, according to the Sine-Gordon equation of two classic examples and the numerical examples, we calculate its numerical solution and compared with exact solutions and obtain good results. Finally, it shows Klein-Gordon equations by proper numerical examples. All of the examples shows that the scheme keeps the conservation of finite volume method, and has high precision and good calculation stability.