Local Discontinuous Galerkin Methods For The Non-Local Water Wave Equations

Compared with classical diffusion equation,the nonlocal diffusion model has great advantages in describing anomalous diffusion and fracture problem and so on.At present,the development of high-order numerical schemes for nonlocal diffusion models is still limited.The local discontinuous Galerkin（LDG）method is a highly accurate and flexible method,which has a good effect on solving non-local problems.In this thesis,we develop local discontinuous Galerkin methods to solve some kinds of nonlocal water wave equations.The main contents are as follows:（1）Local discontinuous Galerkin method for one-way water wave equation.Based on the features of fractional derivative,the considered model is first coupled into a classical first order differential equation and a nonlocal fractional integral equation.Then LDG method is used in space discretization by properly choosing the numerical fluxes.Numerical examples are provided to show the accuracy and effectiveness.（2）Local discontinuous Galerkin method for nonlocal Fowler equation.The LDG scheme is designed by splitting the fractional derivative into two classical first-order derivatives and a weak singular integral.We prove that the proposed scheme is stability with the optimal order for linear case,and suboptimal order for the nonlinear case.We further develop a LDG algorithm for a nonlocal phase transition model with Riesz potential operator.Finally,we demonstrate the accuracy and efficiency of our scheme by the linear and nonlinear numerical examples.（3）Local discontinuous Galerkin method for space fractional Allen-Cahn equation.Based on the features of fractional derivative,we design and analyze a semi-discrete LDG scheme for the initial boundary problem of space fractional Allen-Cahn equation.Error estimates in L² norm is proved for the semi-discrete LDG scheme.Finally,we test the accuracy and efficiency of designed numerical scheme by some numerical examples.（4）Local discontinuous Galerkin method for the nonlocal viscous conservation laws.Non-local models with fractional power law kernel and Riesz potential kernel are investigated,respectively.We develop the LDG method to solve the considered model.We prove the stability and convergence of semi-discrete LDG method in L² norm.The numerical results show that our numerical scheme is robust and effective.（5）Local discontinuous Galerkin method for a nonlocal Kd V-Burgers equation.the LDG scheme is derived for the considered equation.We prove stability and optimal order of convergence O(h^k+1)for linear case,and order of convergence ofO(h^k+1/2)for the nonlinear case.Finally,some numerical experiments are given to simulate the Kd V-Burgers equation.