Design And Analysis Of Discontinuous Galerkin Methods For The High Dimensional Nonlinear Equations

In this thesis,we mainly develop the local discontinuous Galerkin method to study some kinds of nonlinear equations with high-order derivatives,including two-dimensional Davey-Stewartson system,two-dimensional Camassa-Holm equation,two-dimensional(μ)-Camassa-Holm equation,one-dimensional Camassa-HolmNovikov equation.Considering the characteristics of the above systems,we design the high-precision discontinuous Galerkin method and the high-precision coupling of the continuous Galerkin method and discontinuous Galerkin method.Besides,we analyze the stability and theoretical errors based on the conserved quantities.Numerical experiments in different circumstances are provided to verify the theoretical analysis results.The thesis is mainly divided into the following three parts.In the first part,we propose a coupled discontinuous Galerkin method and continuous Galerkin method for solving the nonlinear evolution Davey-Stewartson system in dimensionless form.The main advantage of this coupled numerical scheme is that it combines the advantages of the discontinuous Galerkin method and the continuous Galerkin method.The discontinuous Galerkin method is used to deal with the nonlinear Schrodinger equation to obtain high parallelism and high order accuracy,and the continuous Galerkin method is used to solve the velocity to maintain the conservation of total energy.A rigorous stability analysis is carried out to verify that the numerical scheme satisfies two types of energy conservation.In the process of theoretical error analysis of nonlinear terms,the priori hypothesis and linearization hypothesis are avoided as far as possible.An operator suitable for error analysis of general nonlinear terms is proposed and an important conclusion of operator estimation is given.By the property of conserved quantity and projection property,we prove that the theoretical precision results of the first partial derivative of velocity potential and amplitude can reach the optimal convergence with k-th order of accuracy under L2 norm,where k is the highest degree of approximation to the space polynomial.Finally,several numerical examples are given to verify the theoretical analysis results,including the elliptic-elliptic Davey-Stewartson system and the hyperbolic-elliptic Davey-Stewartson.In the second part,we apply the local discontinuous Galerkin method to solve the two-dimensional Camassa-Holm equation and two-dimensional μ-Camassa-Holm equation and give the theoretical error analysis.These two types of shallow water wave models have high similarity in structure.Both of them are fully integrable models with a double Hamiltonian structure and an infinite number of conserved quantities,which can be used to describe both the soliton and the wave burst phenomenon.Because of the property of conserved quantity,the μ-Camassa-Holm equation can be regarded as a partial linear degradation of the Camassa-Holm equation.Therefore,we consider the unified error analysis of the two kinds of equations under the same framework.The different terms are explained separately.Compared with one-dimensional equations,two-dimensional(μ)-Camassa-Holm equations contain many nonlinear highorder derivative coupling terms,which brings great challenges to numerical scheme design and theoretical error analysis.When dealing with a large number of high-order nonlinear terms,we cleverly classify them and give an important conclusion of generality.Theoretically,we prove that the H1 norm has an optimal convergence order on uniform Cartesian meshes when piecewise tensor product polynomials,Qk(each component is a tensor product space of polynomial of degree k),are used on overlapping meshes.We avoid the prior assumption of numerical error in the process of proof,which increases the credibility of the proof.Then,several numerical experiments with different initial values and boundary conditions are performed to verify the stability and convergence of the numerical scheme.In the third part,we design a locally discontinuous finite element discrete scheme for the one-dimensional Camassa-Holm-Novikov equation.This equation is a partial differential equation with a linear combination of the Camassa-Holm equation with quadratic nonlinearity and the Novikov equation with cubic nonlinearity.It is an integrable model with good properties such as multi-solitonic solutions.When the coefficients are not zero,the Camassa-Holm-Novikov equation is rewritten into an equivalent first-order system.Then we obtain a non-conserved high-precision locally discontinuous finite element numerical scheme by selecting the specific numerical flux.The stability of the numerical scheme is analyzed based on the physical conserved quantity of the equation.Finally,we get the optimal convergence order through the smooth solution precision test and verify the effectiveness of the numerical method.