A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

DISCONTINUOUS GALERKIN TIME DOMAIN METHOD FOR SOI THIN-RIDGE WAVEGUIDE PROBLEM

A novel high-order three-dimensional(3-D)discontinuous Galerkin time domain(DGTD)method based on a normalized formulation of Maxwell′s equations is developed for modeling and simulating silicon-on-insulator(SOI)thin-ridge waveguide.The DGTD method employs unstructured meshes and piecewise high-order polynomials for spatial discretization,and Runge-Kutta methods for time integration.It is found that the numerical results of the leakage loss of SOI thin-ridge waveguide agree well with those of analytical solutions,which proves that the proposed method is an ideal tool for the quantitative analysis for SOI thin-ridge waveguide.

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 flex- ible 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.

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

A high-order leap-frog based non-dissipative discontinuous Galerkin time-domain method for solving Maxwell's equations is introduced and analyzed.The proposed method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements,with a Nth-order leap-frog time scheme.Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes.The method is proved to be stable under some CFL-like condition on the time step.The convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided.Numerical experiments with high-order elements show the potential of the method.

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.The staggered DG scheme defines the discrete pressure on the primal triangular mesh,while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh.In this paper,a new pair of equal-order-interpolation velocity-pressure finite elements is proposed.On the primary triangular mesh(the pressure elements),the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle.On the dual mesh instead(the velocity elements),the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries,while they are continuous inside.In other words,the basis functions on the dual mesh arc built by continuous finite elements on the subtriangles.This choice allows the construction of an efficient,quadrature-free and memory saving algorithm.In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations,the arbitrary high order of accuracy in time is achieved through tire use of time-dependent test and basis functions,in combination with simple and efficient Picard iterations.Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes,not only from a computer memory point of view,but also concerning the computational time.

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.

Caputo型时间分数阶变系数扩散方程的局部间断Galerkin方法

提出一种带有Caputo导数的时间分数阶变系数扩散方程的数值解法.方程的解在初始时刻附近通常具有弱正则性,采用非一致网格上的L1公式离散时间分数阶导数,并使用局部间断Galerkin(local discontinuous Galerkin,LDG)方法离散空间导数,给出方程的全离散格式.基于离散的分数阶Gronwall不等式,证明了格式的数值稳定性和收敛性,且所得结果关于α是鲁棒的,即当α→1^(-)时不会发生爆破.最后,通过数值算例验证理论分析的结果.

A Krylov Space-Based Finite Element Time Domain Method for Broadband Frequency Domain Solutions

A Krylov space based time domain method for wave propagation problems is introduced. The proposed method uses the Arnoldi algorithm to obtain broad-band frequency domain solutions. This method is especially advantageous in cases where slow convergence is observed when using traditional time domain methods. The efficiency of the method is examined in several test cases to show its fast convergence in such problems.

ANALYSIS OF THE IMPLICIT-EXPLICIT ULTRA-WEAK DISCONTINUOUS GALERKIN METHOD FOR CONVECTION-DIFFUSION PROBLEMS

In this paper,we first present the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations.Then,coupling with a kind of Runge-Kutta type implicit-explicit time discretization which treats the convection term explicitly and the diffusion term implicitly,we analyze the stability and error estimates of the corresponding fully discrete schemes.The fully discrete schemes are proved to be stable if the time-stepτ≤τ0,whereτ0 is a constant independent of the mesh-size h.Furthermore,by the aid of a special projection and a careful estimate for the convection term,the optimal error estimate is also obtained for the third order fully discrete scheme.Numerical experiments are displayed to verify the theoretical results.

Optimal Error Estimates of the Semi-Discrete Local Discontinuous Galerkin Method and Exponential Time Differencing Schemes for the Thin Film Epitaxy Problem without Slope Selection

In this paper,we prove the optimal error estimates in L2 norm of the semidiscrete local discontinuous Galerkin(LDG)method for the thin film epitaxy problem without slope selection.To relax the severe time step restriction of explicit time marching methods,we employ a class of exponential time differencing(ETD)schemes for time integration,which is based on a linear convex splitting principle.Numerical experiments of the accuracy and long time simulations are given to show the efficiency and capability of the proposed numerical schemes.

探地雷达正演的间断伽辽金法影响因素分析

时域间断伽辽金(discontinuous Galerkin time-domain, DGTD)算法具有守恒性、稳定性、高精度性和间断性等优点,现已成为一种有效的探地雷达(Ground penetrating radar, GPR)正演方法.为了提高DGTD算法的计算效率和精度,作者详细分析了数值通量、时间离散格式、单元大小与局部基函数阶次、网格剖分方式等影响因素.数值实验表明,局部Lax-Friedrichs中τ=1/2的补偿数值通量既可以消除伪解,又可以提高计算精度;在精度相同的情况下,低存储显式Runge-Kutta方案(low-storage explicit Runge-Kutta, LSERK)的稳定性条件和低存储优势要明显优于其它两种时间离散格式,尤其是在大型复杂模型和三维正演模拟中更有优势.而提高基函数的阶次或增大网格数,均可以提高其误差的收敛性,局部基函数阶次N和单元大小d与电磁波波长λ的适用关系为d/N约等于λ/15;当单元数目大致相等时,网格剖分方式对于高阶DGTD算法的影响较小,说明DGTD算法对网格具有较好的适应性.最后,采用DGTD算法对火星乌托邦平原模型进行正演,验证了基于最优参数的DGTD算法模拟精度高,可为火星乌托邦平原GPR实测数据的解译奠定理论基础.

二维DGTD方法中UPML吸收边界的实现

针对二维情形单轴各向异性完全匹配层吸收边界条件,研究了横磁波情形时域离散伽略金算法单轴各向异性完全匹配层吸收边界的理论基础和实现方案.借鉴时域有限差分方法——时域离散伽略金算法中吸收边界阻抗匹配、各向异性介质参数设置和匹配矩阵等思想,结合时域离散伽略金算法空间离散网格的非结构特性和离散单元之间场量传递的特点,给出了在单轴各向异性完全匹配层中电磁场量时域迭代公式,进一步离散成为离散时域迭代计算式.由于时域离散伽略金算法网格的非结构特性,一阶SM吸收边界条件一般对入射电磁波的反射率在-24dB左右.仿真算例说明,给出的时域离散伽略金算法单轴各向异性完全匹配层吸收边界对电磁波的双重衰减达130dB,表明单轴各向异性完全匹配层具有良好的吸收效果.

DGTD用于RCS计算的初步研究

时域离散伽辽金法(Discontinuous Galerkin Time Domain,DGTD)同时具有时域有限元算法(Finite Element Time Domain,FETD)非结构网格剖分和时域有限差分算法(Finite Difference Time Domain,FDTD)显式迭代的优点,是一种非常有前途的电磁计算方法,该文首先描述了基于矢量基函数的时域离散伽辽金法的基本原理。然后,给出了DGTD处理散射问题时平面波入射加入的具体实现方法。最后,给出了金属球、介质球和金属弹头宽带散射的算例,算例结果的比较表明了该文算法的正确性和有效性。该文的研究,为复杂目标雷达散射截面RCS的准确预估打下了坚实的基础。

时域间断Galerkin有限元法在激光热加工过程中应用

针对半无限体和薄膜结构受多种激光热源作用下的非傅里叶热传导过程,采用时域间断Galerkin有限元法进行数值仿真.其主要特点是在时域内对温度及其时间导数分别进行三次Hermite插值和线性插值.对于半无限激光热源热传导问题,其计算结果与解析解吻合良好.算例表明,时域间断Galerkin有限元法在高频激光脉冲问题中,没有虚假的数值振荡,具有广泛的工程应用性.

抛物问题间断Galerkin区域分解方法

对一类抛物问题给出了间断Galerkin区域分解并行算法.该方法兼有间断Galerkin方法和区域分解算法的优点.文章证明了算法的稳定性和收敛性.给出的数值试验表明了算法的稳定性和精度.

层合压电材料冲击问题的时域间断Galerkin有限元方法求解

构建了压电耦合动力学问题的时域间断Galerkin有限元方法,在此基础上针对冲击作用下压电材料耦合问题进行了数值模拟。强间断、高梯度的冲击荷载作用下,传统的时域连续Galerkin有限元方法在压电耦合动力学问题模拟时,间断的波阵面及层合界面处会出现强烈的虚假数值振荡,这类振荡使得问题求解的精度大大降低。为了消除这类高频数值振荡,依据所发展的时域Galerkin有限元方法原理,在最终的有限元方法求解公式中引入了比例刚度阻尼。冲击作用下一维、二维压电动力学问题算例结果表明,所发展的时域间断Galerkin有限元方法,较好地消除了这类虚假的数值振荡,并捕捉了应力波及电势波的波前波后间断特性。

Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations

Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these methods have the flexibilitywhich is not shared by typical finite element methods, such as the allowance of ar-bitrary triangulation with hanging nodes, less restriction in changing the polynomialdegrees in each element independent of that in the neighbors (p adaptivity), and localdata structure and the resulting high parallel efficiency. In this paper, we give a generalreview of the local DG (LDG) methods for solving high-order time-dependent partialdifferential equations (PDEs). The important ingredient of the design of LDG schemes,namely the adequate choice of numerical fluxes, is highlighted. Some of the applica-tions of the LDG methods for high-order time-dependent PDEs are also be discussed.

Effect of channel shape on selection of time marching scheme in the discontinuous Galerkin method for 1-D open channel flow

One-dimensional open channel flows are simulated using the discontinuous Galerkin finite element method. Three different explicit time marching schemes, including multistep/multistage schemes, are evaluated for different channel shapes for accuracy and efficiency. The Forward Euler, second-order Adam-Bashforth （multistep）, and second-order total variation diminishing （TVD） Runge-Kutta （multistage） time marching schemes are utilized. The role of monotonized central, minmod, and zero TVD slope limiters for each of the time marching scheme is investigated. The numerical flux is approximated using HLL function. The accuracy and robustness of different time marching schemes are evaluated for steady and unsteady flows using analytical and measured data. The unsteady flows include dam break tests with wet and dry beds downstream of the dam in prismatic （rectangular, trapezoidal, triangular, and parabolic cross-sections） and non-prismatic （natural river） channels. The steady flow test involves simulation of hydraulic jump in a diverging rectangular channel. The various schemes are evaluated by comparing accuracy using statistical measures and efficiency using maximum possible time step size as well as CPU runtime. The second-order Adam-Bashforth time marching scheme is found to have the best accuracy and efficiency among the time stepping schemes tested.

基于Rosenbrock型指数积分的一维间断Galerkin有限元方法

提出基于Rosenbrock型指数积分的一维间断Galerkin有限元方法.该方法在空间上使用间断有限元方法离散,在时间上采用Rosenbrock型指数积分方法.这样不仅可以保持空间离散上的高精度,而且继承了指数时间积分方法具有显式大步长时间推进的优点.数值试验的结果表明,对于一维双曲守恒律问题,这种方法是一种有效的数值算法.