Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper,we study the superconvergence properties of the energy-conserving discontinuous Galerkin(DG)method in[18]for one-dimensional linear hyperbolic equations.We prove the approximate solution superconverges to a particular projection of the exact solution.The order of this superconvergence is proved to be k+2 when piecewise Pk polynomials with K≥1 are used.The proof is valid for arbitrary non-uniform regular meshes and for piecewise polynomials with arbitrary K≥1.Furthermore,we find that the derivative and function value approxi?mations of the DG solution are superconvergent at a class of special points,with an order of k+1 and R+2,respectively.We also prove,under suitable choice of initial discretization,a(2k+l)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages.Numerical experiments are given to demonstrate these theoretical results.

An H^1-Galerkin Expanded Mixed Element Method for Semi-linear Hyperbolic Wave Equation

An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variables: the scalar unknown, its gradient and its flux(coefficient times the gradient), simultaneously. We also prove the existence and uniqueness of semi-discrete solution. Finally, we obtain some numerical results to illustrate the efficiency of the method.

UNIFORM DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED LINEAR 2ND ORDER HYPERBOLIC PROBLEM WITH ZEROTH ORDER REDUCED EQUATION

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.

Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations

In this paper, we investigate the effect of weight function in the nonlinear part on global solvability of the Cauchy problem for a class of semi-linear hyperbolic equations with damping.

A PRIORI ESTIMATE FOR MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF QUASI-LINEAR ELLIPTIC EQUATIONS

Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions

EXISTENCE OF POSITIVE SOLUTIONS TO QUASI-LINEAR EQUATION INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT

This paper is concerned with the quasi-linear equation with critical Sobolev-Hardy exponent whereΩ(?)RN(N(?)3)is a smooth bounded domain,0∈Ω,0(?)s<p,1<p<N,p(s):=p(N-s)/N-p is the critical Sobolev-Hardy exponent,λ>0,p(?)r<p,p:=Np/N-p is the critical Sobolev exponent,μ>,0(?)t<p,p(?)q<p(t)=p(N-t)/N-p.The existence of a positive solution is proved by Sobolev-Hardy inequality and variational method.

THE ESTIMATION OF SOLUTION OF THE BOUNDARY VALUE PROBLEM OF THE SYSTEMS FOR QUASI-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equationswhere x,f, y , h, A, B and C all belong to Rn , and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.

Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.

THE FUNDAMENTAL EQUATIONS OF DYNAMICS USINGREPRESENTATION OF QUASI-COORDINATES IN THE SPACEOF NON-LINEAR NON-HOLONOMIC CONSTRAINTS

The dot product of the bases vectors on the super-surface of the non-linear nonholonomic constraints with one order, expressed by quasi-coorfinates, and Mishirskiiequalions are regarded as the fundamental equations of dynamics with non-linear andnon-holononlic constraints in one order for the system of the variable mass. From thesethe variant ddferential-equations of dynamics expressed by quasi-coordinates arederived. The fundamental equations of dynamics are compatible with the principle ofJourdain. A case is cited.

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.

An O(k²+kh²+h²) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

Interaction of Conormal Waves With Strong and Weak Singularities For Semi－Linear Equations

We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.

The Weak Galerkin Method for Linear Hyperbolic Equation

The linear hyperbolic equation is of great interest inmany branches of physics and industry.In this paper,we use theweak Galerkinmethod to solve the linear hyperbolic equation.Since the weak Galerkin finite element space consists of discontinuous polynomials,the discontinuous feature of the equation can be maintained.The optimal error estimates are proved.Some numerical experiments are provided to verify the efficiency of the method.

M-D Riemann problem for a class of quasilinear hyperbolic system and its perturbation

In this note the Cauchy problem of quasilinear hyperbolic system will be discussed U_t+F(U)_x+G(U)_y=0, (1)where U=~t(u, v), F=~t(u^2, uv), G=~t(uv, v^2). The system is nonstrictly hyperbolic andpossibly arises in some physical problems such as oil recovery and elastic theory. The initialdata of our porblem are given

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied.We prove that each weak solution can be decomposed into two parts near singular points,one of which is a finite sum of functions of the form cr~α log^m r(?)(θ),where the coefficients c depend on the H^1-norm of the solution,the C^(0,δ)-norm of the solution,and the equation only;and the other one of which is a regular one,the norm of which is also estimated.

DIRICHLET BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER QUASI-LINEAR DIFFERENTIAL EQUATIONS WITH CHANGING SIGN NONLINEARITIES

This paper is concerned with the existence of positive solutions of two-point Dirichlet singular and nonsingular boundary problems for second-order quasi-linear differential equations with changing sign nonlinearities.

A CLASS OF FREE BOUNDARY PROBLEMS TO QUASI-LINEAR PARABOLIC EQUATIONS

In this paper we deal with a class of free boundary problems (1.1)-(1.8), which comesfrom the 'secondary frost heave'' problems. We prove the global existence andw uniqueness ofsolutions to them, and the resuits corresponding to diffraction problems are also new.

线性双曲最优控制问题的P^(2)_(0)-P_(1)混合有限元方法的先验误差估计

针对线性双曲最优控制问题,通过非标准的P^(2)_(0)-P_(1)混合有限元方法,利用椭圆投影、标准L^(2)投影、标准L^(2)-正交投影算子等理论,其中状态和对偶状态采用P^(2)_(0)-P_(1)混合有限元逼近,控制变量采用分片常数逼近,给出问题模型中所有变量的先验误差估计.

含混合项的拟线性Schrodinger方程的正规化基态解

该文研究了一类含混合项拟线性Schrodinger方程正规化基态解的存在性.推广了文献[1-2]中的结果,与他们研究的情形相比,该文中方程对应能量泛函的结构更加复杂.

Lane-Emden型方程的广义Vieta-Fibonacci多项式迭代方法

基于广义Vieta-Fibonacci多项式的拟线性化矩阵配置方法,提出了一种求带有Dirichlet边界条件、Neumann边界条件和Neumann-Robin边界条件的一类Lane-Emden型微分方程的数值解的方法 .首先将Lane-Emden型方程拟线性化,然后利用广义Vieta-Fibonacci多项式展开得到矩阵形式,再用迭代方法进行求解.最后通过求不同边值条件下的Lane-Emden型方程的近似解,将数值结果与其他方法得到的近似解进行对比,验证了广义Vieta-Fibonacci多项式拟线性化迭代方法的有效性和准确性.