二维波动方程数值反演的迭代方法

本文针对二维波动方程的反演问题，提出了一种新的迭代方法。该方法是一种牛顿型迭代，并且每步迭代都利用吉洪诺夫正则化方法克服反问题的不适定性，因此具有良好的数值稳定性。文中给出了数值仿真实例说明了本方法的可行性及有效性。     

非线性Schrdinger方程的直接间断Galerkin方法

讨论一维和二维非线性Schrdinger(NLS)方程的数值求解.基于扩散广义黎曼问题的数值流量,构造一种直接间断Galerkin方法(DDG)求解非线性Schrdinger方程.证明该方法L2稳定性,并说明DDG格式是一种守恒的数值格式.对一维NLS方程的计算表明,DDG格式能够模拟各种孤立子形态,而且可以保持长时间的高精度.二维NLS方程的数值结果显示该方法的高精度和捕捉大梯度的能力.     

局部时间步长间断有限元方法求解三维欧拉方程

使用间断有限元方法求解三维流体力学方程.空间剖分采用非结构四面体网格,为了克服显格式在单元网格尺寸差别较大时计算效率低下的问题,在格式中采用局部时间步长技术(LTS),即控制方程在空间、时间上积分得到一种单步格式,既可以局部计算每个单元又避免了Runge-Kutta高精度格式处理三维问题时存储量过大的问题.为了提高流体力学方程计算精度,在计算单元边界的数值流通量时使用任意高阶精度方法(ADER).数值算例表明格式稳定有效.     

隐-显积分因子间断Galerkin方法求解二维辐射扩散方程

提出一种求解二维非平衡辐射扩散方程的数值方法.空间离散上采用加权间断Galerkin有限元方法,其中数值流量的构造采用一种新的加权平均;时间离散上采用隐-显积分因子方法,将扩散系数线性化,然后用积分因子方法求解间断Galerkin方法离散后的非线性常微分方程组.数值试验中在非结构网格上求解了多介质的辐射扩散方程.结果表明:对于强非线性和强耦合的非线性扩散方程组,该方法是一种非常有效的数值算法.     

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.     

一种基于高阶节点间断Galerkin方法的新型限制器设计

使用高阶间断Galerkin(discontinuous Galerkin, DG)方法求解双曲守恒律方程组时, 非物理效应常常导致计算过程的中断, 这在很大程度上制约着该方法在计算流体力学中的应用.文章结合局部单元上原始流动变量的Taylor展开, 设计了一种新型的限制器, 通过对各阶空间导数的重构, 有效地消除了非物理振荡的不利影响.对二维Euler方程的计算结果表明, 该限制器不仅能够捕捉高质量的激波, 而且能够保证残值的有效收敛.      

求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法

构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数,包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组,再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用,如孤立子的传播及相互碰撞等.     

求解Kuramoto-Sivashinsky方程的平移基无单元Galerkin方法

Kuramoto-Sivashinsky方程是一种可以描述复杂混沌现象的高阶非线性演化方程.方程中高阶导数项的存在,使得传统无单元Galerkin方法采用高次多项式基函数构造形函数时,形函数违背了一致性条件.因此,本文提出了一种采用平移多项式基函数的无单元Galerkin方法.与传统无单元Galerkin方法相比,该方法在方程离散时依然采用Galerkin进行离散,但形函数的构造采用了基于平移多项式基函数的移动最小二乘近似.通过对具有行波解和混沌现象的Kuramoto-Sivashinsky方程的数值模拟,验证了本文方法的有效性.     

求解扩散方程的一类交替分组四点方法

给出了一种交替分组四点格式,该方法绝对稳定,具有并行本性,便于在并行计算机上直接使用.并进行了数值试验.     

Lax-Wendroff时间离散的自适应间断有限元方法求解三维可压缩欧拉方程

应用自适应LWDG方法求解三维双曲守恒律方程组,与传统的二阶RKDG方法相比,该方法具有计算量小和精度高的特点.给出一种自适应策略,其中均衡折中策略适用于非相容四面体网格.将二维情形下的后验误差指示子推广到三维双曲守恒律方程组中,数值实验证明了方法的有效性.     

Modified Burgers equation by 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.     

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.     

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.     

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.     

An improved element-free Galerkin method for solving generalized fifth-order Korteweg-de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.     

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.     

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.     

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.     

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 meshless method for the nonlinear generalized regularized long wave equation

This paper presents a meshless method for the nonlinear generalized regularized long wave(GRLW) equation based on the moving least-squares approximation.The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method.A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm.Compared with numerical methods based on mesh,the meshless method for the GRLW equation only requires the scattered nodes instead of meshing the domain of the problem.Some examples,such as the propagation of single soliton and the interaction of two solitary waves,are given to show the effectiveness of the meshless method.