The High Resolution Numerical Methods For Nonlinear High Order Dispersive Equations

The main work of this thesis is to construct high resolution numerical methods for a series of nonlinear high order dispersive equations,including the conservative discon-tinuous Galerkin(DG)methods for the one-dimensional Serre equations,two differ-ent weighted essentially non-oscillatory(WENO)schemes for the one-dimensional ?-Camassa-Holm(?CH)and ?-Degasperis-Procesi(?DP)equations,including the finite volume and the finite difference methods,and the local discontinuous Galerkin(LDG)methods for the two-component ?-Camassa-Holm equations.This thesis consists of the following three parts.In the first part,we propose two conservative DG methods for the Serre equa-tions in conservative form,and construct the Hamiltonian conserved DG method for the Serre equations in non-conservative form.One of the schemes owns the well-balanced property via constructing a high order approximation to the source term for the Serre equations with a non-flat bottom topography.By virtue of the Hamiltonian structure of the Serre equations,we introduce an Hamiltonian invariant and then develop a DG scheme which can preserve the discrete version of such an invariant.Furthermore,we give many kinds of numerical experiments to validate our methods.The examples of smooth solitary wave and periodic cnoidal wave show that these numerical schemes are highly accurate and stable.To be more specific,for the velocity,all DG schemes can achieve the optimal order of accuracy.For the water depth,the first two DG methods can achieve the optimal order of accuracy,while the Hamiltonian conserved scheme can reach the optimal order of convergence rate when k?2,but only suboptimal or-der of accuracy when k=1.In order to demonstrate the robustness and effectiveness,we apply these schemes on the interaction of solitary waves,the breakup of a Gaussian hump,and the dispersive shock wave problems of the shallow water model.From the contrastive experimental figures,we can see that all three conserved schemes proposed in this paper are successfully validated.In the second part,we introduce and design two high resolution weighted essen-tially non-oscillatory(WENO)schemes for the one-dimensional ?CH and ?DP equa-tions,including the finite volume and the finite difference methods.Both the ?CH and?DP equations are completely integrable systems with bi-Hamiltonian structures and infinite conserved quantities.They also contains high order dispersive terms.The ?CH equation has multi-peaked solutions,while the ?DP equation possesses not only multi-peaked waves but also multi-shock waves.The WENO schemes can achieve high order accuracy for these two equations and capture the discontinuities in the specific fea-tures of their solution.In order to verify the accuracy and capability of the WENO schemes,we give some numerical examples in different cases,including the accuracy test of smooth traveling wave solution,the simulation experiment of multi-peakon and multi-shock solutions.In the third part,we analyze and study the numerical methods of the two-component ?CH equations.The two-component ?CH equations are a completely in-tegrable system with infinite conserved quantities.Based on the Hamiltonian of the equations and different numerical fluxes,we develop conservative and dissipative LDG schemes respectively.The conservative scheme can ensure that the Hamiltonian in the semi-discrete scheme is still conserved;the dissipative scheme can only guarantee that the Hamiltonian does not increase.In addition,we give the corresponding error analysis to prove that both numerical schemes have suboptimal convergence order in theory with the L2 projection operator.Finally,numerical examples are adopted to verify the sta-bility and accuracy of the schemes.Both schemes can achieve the optimal convergence rate numerically.Multi-peakon problems are also used to validate the schemes.