Modified Burgers’ equation by the local discontinuous Galerkin method

In this paper,we present the local discontinuous Galerkin method for solving Burgers’ equation and the modified Burgers’ equation.We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail.The method is applied to the solution of the one-dimensional viscous Burgers’ equation and two forms of the modified Burgers’ equation.The numerical results indicate that the method is very accurate and efficient.     

Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations

In the current work, we extend the local discontinuous Galerkin method to a more general application system. The Burgers and coupled Burgers equations are solved by the local discontinuous Galerkin method. Numerical experiments are given to verify the efficiency and accuracy of our method. Moreover the numerical results show that the method can approximate sharp fronts accurately with minimal oscillation.     

基于广义交替数值通量的局部间断Galerkin方法求解二维波动方程

波的传播往往在复杂的地质结构中进行,如何有效地求解非均匀介质中的波动方程一直是研究的热点.本文将局部间断Galekin(local discontinuous Galerkin, LDG)方法引入到数值求解波动方程中.首先引入辅助变量,将二阶波动方程写成一阶偏微分方程组,然后对相应的线性化波动方程和伴随方程构造间断Galerkin格式;为了保证离散格式满足能量守恒,在单元边界上选取广义交替数值通量,理论证明该方法满足能量守恒性.在时间离散上,采用指数积分因子方法,为了提高计算效率,应用Krylov子空间方法近似指数矩阵与向量的乘积.数值实验中给出了带有精确解的算例,验证了LDG方法的数值精度和能量守恒性;此外,也考虑了非均匀介质和复杂计算区域的计算,结果表明LDG方法适合模拟具有复杂结构和多尺度结构介质中的传播.     

Direct discontinuous Galerkin method for the generalized Burgers—Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge-Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.     

Solving coupled nonlinear Schrodinger equations via a direct discontinuous Galerkin method

In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system.The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method.Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.     

基于间断有限元方法的并列圆柱层流流动特性

基于高阶的间断有限元方法, 数值模拟低马赫数下并列圆柱的可压缩层流流动, 捕捉并列圆柱流场中的漩涡结构, 以便分析并列圆柱尾流的流动特性. 针对二维圆柱的边界形式, 采用曲边三角形单元构造二维圆柱的曲面边界, 以适应高阶离散格式的精度. 在验证方法合理性的基础上, 分析圆柱间距及雷诺数对漩涡脱落及受力特性的影响规律. 研究结果表明: 并列圆柱的间距是影响流场流动特性的一个主要因素, 它会改变圆柱漩涡脱落的形式. 随着圆柱间距的增加, 上下圆柱的平均阻力系数及平均升力系数的绝对值随之显著下降. 雷诺数对于平均阻力系数的影响相对较小. 但随着雷诺数的增加, 上下圆柱的平均升力系数会随之降低, 而漩涡的脱落频率会随之增大.     

A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation

To develop and analyze new computational techniques for the Boltzmann equation based on model or approximation adaptivity, it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the Boltzmann equation and that admits the extraction of highly accurate reference solutions. For standard collision processes, the Boltzmann equation itself fails to meet the second requirement for d = 2, 3 spatial dimensions, on account of its setting in 2d, while for d = 1 the first requirement is violated because the Boltzmann equation then lacks the convergence-to-equilibrium property that forms the substructure of simplified models. In this article we present a numerical investigation of a new one-dimensional prototype of the Boltzmann equation. The underlying molecular model is endowed with random collisions, which have been fabricated such that the corresponding Boltzmann equation exhibits convergence to Maxwell–Boltzmann equilibrium solutions. The new Boltzmann model is approximated by means of a discontinuous Galerkin (DG) finite-element method. To validate the one-dimensional Boltzmann model, we conduct numerical experiments and compare the results with Monte-Carlo simulations of equivalent molecular-dynamics models.     

Direct perturbation method for perturbed complex Burgers equation

So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear Schr?dinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou's direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.     

Solving coupled nonlinear Schrödinger equations via a direct discontinuous Galerkin method

In this work, we present the direct discontinuous Galerkin (DDG) method for the one-dimensional coupled nonlinear Schrödinger (CNLS) equation. We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system. The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method. Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.     

电磁场2.5维时域间断有限元方法

介质沿空间固定方向均匀分布的结构在电磁导波器件中有十分广泛的应用,对这类器件的分析通常被称为2.5D电磁问题。利用器件在固定方向介质分布均匀的特点,将电磁场量沿该方向进行空间傅里叶变换,可以把对三维问题的分析转化为两维问题求解,从而极大地减小计算开销。针对传统基于差分的2.5D电磁场算法在弯曲形状逼近上有阶梯误差的缺陷,本文提出了基于三角形网格的2.5D时域间断有限元方法（DGTD）,并用它模拟了电偶极子与光纤的耦合效率和光子晶体光纤的色散特性。与基于规则网格的2.5D差分方法进行对比。结果表明,文中建立的2.5D DGTD方法对弯曲形状的模拟更加逼真,计算内存占用最大减少10.4%,计算精度最大相差0.011%,计算时间缩短74.9%,计算效率提高。     

Element-free Galerkin method for a kind of KdV equation

The present paper deals with the numerical solution of the third-order nonlinear KdV equation using the element-free Galerkin (EFG) method which is based on the moving least-squares approximation. A variational method is used to obtain discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for KdV equations needs only scattered nodes instead of meshing the domain of the problem. It does not require any element connectivity and does not suffer much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for the KdV equation is investigated by two numerical examples in this paper.     

A leap-frog discontinuous Galerkin time-domain method of analyzing electromagnetic scattering problems

Several major challenges need to be faced for efficient transient multiscale electromagnetic simulations, such as flexible and robust geometric modeling schemes, efficient and stable time-stepping algorithms, etc. Fortunately, because of the versatile choices of spatial discretization and temporal integration, a discontinuous Galerkin time-domain(DGTD) method can be a very promising method of solving transient multiscale electromagnetic problems. In this paper, we present the application of a leap-frog DGTD method to the analyzing of the multiscale electromagnetic scattering problems. The uniaxial perfect matching layer(UPML) truncation of the computational domain is discussed and formulated in the leap-frog DGTD context. Numerical validations are performed in the challenging test cases demonstrating the accuracy and effectiveness of the method in solving transient multiscale electromagnetic problems compared with those of other numerical methods.     

统一坐标系下二维气动方程组的Runge-Kutta间断Galerkin有限元方法

构造了统一坐标系下二维可压缩气动方程组的Runge-Kutta 间断Galerkin(RKDG)有限元格式. 文中将流体力学方程组和几何守恒律统一求解, 所有计算都在固定的网格上进行, 在计算过程中不需要网格节点的速度信息. 文中对几个数值算例进行了数值模拟, 得到了较好的数值模拟结果.     

Interactions among special embed-solitons for the (3+1)-dimensional Burgers equation

With the help of a modified mapping method and a new mapping method, we re-study the (3+1)-dimensional Burgers equation, and derive two families of variable separation solutions. By selecting appropriate functions in the variable separation solution, we discuss the interaction behaviors among taper-like, plateau-type rings, and rectangle-type embed-solitons in the periodic wave background. All the interaction behaviors are completely elastic, and no phase shift appears after interaction.     

Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method

In this paper,an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions.Regularized long wave equation(RLW)is integrated fully by using an exponential B-spline Galerkin method in space together with Crank–Nicolson method in time.Three numerical examples related to propagation of single solitary wave,interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method.Obtained results are compared with some early studies.     

Homotopic mapping solutions for generalized method of solitary wave complex Burgers equation

A class of generalized complex Burgers equation is considered. First, a set of equations of the complex value functions are solved by using the homotopic mapping method. The approximate solution for the original generalized complex Burgers equation is obtained. This method can find the approximation of arbitrary order of precision simply and reliably.     

Homotopic mapping method of solitary wave solutions for generalized complex Burgers equation

A class of generalized complex Burgers equation is considered. First, a set of equations of the complex value functions are solved by using the homotopic mapping method. The approximate solution for the original generalized complex Burgers equation is obtained. This method can find the approximation of arbitrary order of precision simply and reliably.     

The element-free Galerkin method of numerically solving a regularized long-wave equation

The element-free Galerkin (EFG) method is used in this paper to find the numerical solution to a regularized long-wave (RLW) equation. The Galerkin weak form is adopted to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. The effectiveness of the EFG method of solving the RLW equation is investigated by two numerical examples in this paper.     

On the use of the discontinuous Galerkin method for numerical simulation of two-dimensional compressible turbulence with shocks

In this paper, the discontinuous Galerkin (DG) method combined with localized artificial diffusivity is investigated in the context of numerical simulation of broadband compressible turbulent flows with shocks for under-resolved cases. Firstly, the spectral property of the DG method is analyzed using the approximate dispersion relation (ADR) method and compared with typical finite difference methods, which reveals quantitatively that significantly less grid points can be used with DG for comparable numerical error. Then several typical test cases relevant to problems of compressible turbulence are simulated, including one-dimensional shock/entropy wave interaction, two-dimensional decaying isotropic turbulence, and two-dimensional temporal mixing layers. Numerical results indicate that higher numerical accuracy can be achieved on the same number of degrees of freedom with DG than high order finite difference schemes. Furthermore, shocks are also well captured using the localized artificial diffusivity method. The results in this work can provide useful guidance for further applications of DG to direct and large eddy simulation of compressible turbulent flows.     

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast . If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast .