Some New Energy-Preserving Finite Volume Element Methods For Wave Equations

In the last few decades, people have proposed various numerical methods for the numerical approximation of the partial differential equations (PDEs), such as two-grid methods, structure-preserving methods, and so on. This thesis mainly discusses the applications of these methods within certain PDEs. In this thesis, we first discuss the application of two-grid finite volume element method in the nonlinear Sobolev equa-tion, and then introduce how to design integral-preserving methods using structure-preserving methods, for example, discrete variational derivative method, Hamiltonian boundary value methods, and so on.First of all, we propose a class of two-grid finite volume element methods for the nonlinear Sobolev equation. The method is a numerical method that is based on a coarse grid space and a fine grid space. On the coarse grid, we first conduct a nonlinear iteration (with grid size H) to obtain a rough approximation UH, and then solve a linearized equation (with grid size h) based on UH to produce a corrected solution. A priori error estimate for the two-grid finite volume element solution with respect to H1-norm is also proceeded, and we find that the optimal convergence rate is O(H3|ln H|\) when h= O(H3|ln H|). The theory results also show that the two-grid finite volume element methods are more efficient than the classical finite volume element method.Secondly, we propose a class of conservative finite volume element schemes for the Hamiltonian PDEs. The numerical methods are based on the discrete vari-ational derivative method and the finite volume element method. In this chap-ter, we first introduce the designing methods of the energy-preserving scheme and the momentum-preserving scheme, and then analyse their stability and conservation properties. The numerical results show that the energy-preserving scheme and the momentum-preserving scheme can precisely conserve the invariants in the discrete setting. However, the energy-preserving scheme have higher accuracy and better sta-bility than the momentum-preserving scheme.Finally, we propose a family of high-order energy-preserving schemes for the Hamiltonian PDEs. The proposed methods in space and time respectively adopt the Fourier pseudospectral method and the Hamiltonian boundary value methods. The numerical results show that the proposed schemes can reach spectral precision in s-pace, while in time can reach second-order and fourth-order accuracy, respectively. In addition, the proposed schemes can precisely conserve the discrete mass and energy to within the machine precision.