Local Discontinuous Galerkin Methods And Theoretical Analysis For Nonlinear Wave Equations

In this thesis,we devote to develop the local discontinuous Galerkin(LDG)method to some partial differential equations.One of the advantages of the LDG method is that it can achieve the high order of accuracy to capture some delicated details when the equations encounter the non-smooth solutions or the discontinuous shock solutions.The equations considered in this thesis consist of the KdV-type dispersive nonlinear system,the one-dimensional ?-Camassa-Holm(?CH)equation,the one-dimensional?-Degasperis-Procesi(?DP)equation and the two-dimensional ?CH equation.Such equations/system possess their own essential invariants,and based on these invariants,we design some LDG numerical schemes,including the conservative ones and dissipa-tive ones,by choosing different types of numerical fluxes.Besides,we give the stability analysis of these schemes and present some error esitimates among them.This thesis consists of the following four parts.Firstly,we apply the LDG method to solve a kind of nonlinear dispersive KdV-type system.Based on a cardinal invariant of this system,we design and discuss two different types of numerical fluxes,including the conservative and dissipative ones for the linear and nonlinear terms respectively.Thus,one conservative together with three dissipa-tive LDG schemes for the KdV-type system are developed.The invariant preserving property for the conservative scheme and corresponding dissipative properties for the other three dissipative schemes are all presented and proven.Additionally,the error estimates for two schemes are given,whose numerical fluxes for linear terms are cho-sen as the dissipative-type.We find that there exists some symmetrizable property for the nonlinear terms,and then we introduce the definition and properties of the so-called E-flux,and we present a(k+2/1)-th order of accuracy for the scheme with dissipative nonlinear numerical fluxes,yet a k-th order for the conservative one,here k is the or-der of polynomials in the finite element space.Numerical experiments for this system in different circumstances are provided,and it transposes that:the choice of numeri-cal fluxes for the linear terms takes important effect on the accuracy of the numerical schemes;in the long-time simulation of solitary waves,the numerical solutions in the conservative scheme perform better than that in dissipative schemes,while refining the meshes or increasing the degree of polynomials are both efficient ways to improve the numerical schemes.Secondly,we design and analysis some LDG methods for the one-dimensional?CH equation.The ?CH equation is a completely integrable system with a bi-Hamiltonian structure and an infinite series of Hamiltonian invariants.We then design the conserva-tive and dissipative LDG schemes according to two important invariants,and give the corresponding stability analysis.We can show that,the conservative scheme can pre-serve both two invariants,yet the dissipative one only preserve one invariant but keep another non-increasing.Additionlly,we give the detailed error estimates for both two schemes.Besides,we use some numerical examples to verify the stability and conver-gence of both two schemes,and check their capability in simulating the(multi-)peakon problems of the ?CH equation.Thirdly,we develop the LDG method to solve the ?DP equation which is also a completely integrable system with a bi-Hamiltonian structure and infinite Hamiltonian invariants.Although there are only two different parameters between the ?DP equation and the ?CH equation,another important form of the ?DP equation is essentially needed when designing and analyzing the numerical schemes.We will design two different LDG schemes for the ?DP equation,including a conservative one and a dissipative one,and present the detailed stability analysis.What's more,we give the error estimate with(k+2/1)-th order of accuracy for the dissipative scheme,yet the ?-th order for the conservative scheme with the degree of polynomials being even and the number of cells in the mesh being odd.A series of numerical experiments are design to check our LDG schemes for the ?DP equation,including the accuracy tests,and the simulation about the multi-peakon or multi-shock travelling waves.Finally,we turn to develop and analyze the conservative and dissipative LDG methods to solve the two-dimensional ?CH equation.Although the two-dimensional?CH equation is an extension from the one-dimensional ?CH equation,it does possess its own characteristics in design of schemes and implementation of algorithm.We give a set of notations and definitions and generalize an important relation among its non-linear coupled terms,then we present the LDG schemes and stability analysis.What's more,In the implementation of this method,we find that the global mean operator ?(u)destroys the sparse property of matrices,thus we cannot use the normal linear system solver.As a consequence,we modify the algorithm and use the Least squares iterative method to solve this problem.Some numerical experiments also are designed to verify the accuracy of these two LDG schemes.