Least-Squares及Galerkin谱元方法求解环形区域内的泊松方程

为研究基于Least-Squares变分及Galerkin变分两种形式的谱元方法的求解特性,推导了极坐标系中采用两种变分方法求解环形区域内Poisson方程时对应的弱解形式,采用Chebyshev多项式构造插值基函数进行空间离散,得到两种谱元方法对应的代数方程组,由此分析了系数矩阵结构的特点。数值计算结果显示:Least-Squares谱元方法为实现方程的降阶而引入新的求解变量,使得代数方程组形式更为复杂,但边界条件的处理比Galerkin谱元方法更为简单;两种谱元方法均能求解极坐标系中的Poisson方程且能获得高精度的数值解,二者绝对误差分布基本一致;固定单元内的插值阶数时,增加单元数可减小数值误差,且表现出代数精度的特点,误差降低速度较慢,而固定单元数时,在一定范围内数值误差随插值阶数的增加而减小的速度更快,表现出谱精度的特点;单元内插值阶数较高时,代数方程组系数矩阵的条件数急剧增多,方程组呈现病态,数值误差增大,这一特点限制了单元内插值阶数的取值。研究内容对深入了解两种谱元方法在极坐标系中求解Poisson方程时的特点、进一步采用相关分裂算法求解实际流动问题具有参考价值。

一种基于局部间断Galerkin方法的IC互连线电容提取策略

求解椭圆方程的局部间断Galerkin(LDG)方法具有精度高、并行效率高的优点,且能适用于各种网格。文章提出采用LDG方法来求解IC版图中电势分布函数满足的Laplace方程,从而给出了一个提取互连线电容的新方法。该问题的求解区域需要在矩形区域内部去掉数量不等的导体区域,在这种特殊的计算区域上,通过数值测试验证了LDG方法能达到理论的收敛阶。随着芯片制造工艺的发展,导体尺寸和间距也越来越小,给数值模拟带来新的问题。文章采用倍增网格剖分方法,大幅减小了计算单元数。对包含不同数量和形状导体的七个电路版图,用新方法提取互连线电容,得到的结果与商业工具给出的结果非常接近,表明了新方法的有效性。

基于Galerkin截断的薄膜-床面耦合振动响应分析

将废塑料薄膜进行分选回收是目前最为高效节能的塑料垃圾处理方式,废旧塑料薄膜及床面的振动会直接影响分选的效率。提出了将薄膜模型和床面模型结合建立薄膜-床面耦合系统动力学模型的方法。并通过受力分析,利用Galerkin截断将床面的变形表达为模态函数的线性组合,建立了薄膜-床面非线性耦合振动微分方程。研究了不同截断阶数对薄膜-床面耦合非线性振动动态响应的影响,确定了保证薄膜-床面耦合系统振动收敛性的Galerkin截断阶数。通过床面位移响应对此方法进行了验证和对比,结果表明Galerkin截断法适用于求解耦合系统振动分析,且计算速度较快。

ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS

The Galerkin and least-squares methods are two classes of the most popular Krylov subspace methOds for solving large linear systems of equations. Unfortunately, both the methods may suffer from serious breakdowns of the same type: In a breakdown situation the Galerkin method is unable to calculate an approximate solution, while the least-squares method, although does not really break down, is unsucessful in reducing the norm of its residual. In this paper we first establish a unified theorem which gives a relationship between breakdowns in the two methods. We further illustrate theoretically and experimentally that if the coefficient matrix of a lienar system is of high defectiveness with the associated eigenvalues less than 1, then the restarted Galerkin and least-squares methods will be in great risks of complete breakdowns. It appears that our findings may help to understand phenomena observed practically and to derive treatments for breakdowns of this type.

对流扩散方程的隐式全离散局部间断Galerkin方法

研究了对流扩散方程的隐式全离散局部间断Galerkin方法的稳定性和误差分析.将三阶隐式Runge-Kutta时间离散和具有广义交替数值流通量的LDG方法相结合得到全离散LDG格式,通过广义交替数值流通量,建立数值解和辅助解内积之间的关系,证明了全离散LDG格式的无条件稳定,同时引入广义Gauss-Radau投影,通过投影的逼近性质和一些基本不等式建立了数值方法的最优误差估计,最后通过数值实验验证该方法理论分析的正确性.

间断Galerkin有限元隐式算法GPU并行化研究

为了提高间断伽辽金(discontinuous Galerkin,DG)有限元方法的计算效率,围绕求解Euler方程,构建了基于图形处理器(graphics processing unit,GPU)并行加速的隐式DG算法。算法结合Roe格式进行空间离散,采用人工黏性法处理激波等间断问题,时间推进选用下上对称高斯-赛德尔(lower-upper symmetric Gauss-Seidel,LU-SGS)隐式格式。为了克服传统隐式格式固有的数据关联依赖问题,借助于本文提出的面向任意网格的单元着色分组技术,先给出了LUSGS隐式格式的并行化改造,使得隐式时间推进能按颜色组别依次并行,由于同一颜色组内算法已不存在数据关联,可以据此实现并行化。在此基础上,再结合DG算法局部紧致等特点,基于统一计算设备架构(compute unified device architecture,CUDA)编程模型,设计了依据单元的核函数,并构建了对应的线程与数据结构,给出了DG有限元隐式GPU并行算法。最后,发展的算法通过了多个二维和三维典型流动算例考核与性能测试,展示出隐式算法GPU加速的效果,且获得的计算结果能与现有的文献或实验数据接近。

Least-squares reverse time migration in visco-acoustic media based on symplectic stereo-modeling method

The authors proposed a symplectic stereo-modeling method(SSM)in the Birkhoffian dynam-ics and apply it to the visco-acoustic least-squares reverse time migration(LSRTM).The SSM adopts ste-reo-modeling operator in space and symplectic Runge-Kutta scheme in time,resulting in great ability in suppressing numerical dispersion and long-time computing.These advantages are further confirmed by numerical dispersion analysis,long-time computation test and computational efficiency comparison.After these theoretical analyses and experiments,acoustic and visco-acoustic LSRTM are tested and compared between SSM method and the conventional symplectic method(CSM)using the fault and marmousi models.Meanwhile,dynamic source encoding and exponential decay moving average gradients method are adopted to reduce the computation cost and improve the convergence rate.The imaging results show that LSRTM based on visco-acoustic wave equations effectively takes into account the influence of viscosity can therefore compensate for the amplitude attenuation.Besides,SSM method not only has high numerical accuracy and computational efficiency,but also performs effectively in LSRTM.

变系数Volterra型积分微分方程的2种Legendre谱Galerkin数值积分方法

为了进一步提高求解Volterra型积分微分的数值精度,针对一种变系数Volterra型积分微分方程,提出了2种Legendre谱Galerkin数值积分法。采用Galerkin Legendre数值积分对Volterra型积分微分方程的积分项进行预处理,对其构造Legendre tau格式,同时用Chebyshev-Gauss-Lobatto配置点对变系数和积分项部分进行计算,并通过对方程的定义区间进行分解,提出了一种多区间Legendre谱Galerkin数值积分法。该方法的格式对于奇数阶模型具有对称结构。此外,通过引入Volterra型积分微分方程的最小二乘函数,构造了Legendre谱Galerkin最小二乘数值积分法。该方法对应的代数方程系数矩阵是对称正定的。数值算例验证了这2种Legendre谱Galerkin数值积分方法的高阶精度和有效性。

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.

非线性抛物型积分微分方程Galerkin有限元方法超收敛分析

主要研究非线性抛物型积分微分方程的协调Galerkin有限元方法Crank-Nicolson(CN)全离散格式。通过对非线性项的精细估计,采用插值与投影相结合的估计技巧,导出了L^(∞)(H^(1))模意义下具有O(h^(2)+τ^(2))阶的超逼近性质。进一步利用插值后处理技术得到了整体超收敛结果,弥补了以往文献的不足。同时,通过数值例子验证了理论分析的正确性和方法的高效性。

椭圆域上二阶/四阶变系数问题有效的谱Galerkin逼近

本文提出了椭圆域上二阶/四阶变系数问题的一种有效的谱Galerkin逼近.首先,我们将原问题化为极坐标下的等价形式,并建立其弱形式及相应的离散格式.其次,针对二阶情形,我们证明了弱解和逼近解的存在唯一性及它们之间的误差估计.另外,根据极条件和勒让得多项式的正交性,我们构造了一组有效的径向基函数,并在θ方向作截断的傅立叶展开,推导了离散格式等价的矩阵形式.最后,我们给出了大量的数值算例,数值结果表明了我们算法的收敛性和谱精度.

四阶线性方程极弱局部间断Galerkin法傅里叶分析

主要研究了四阶线性方程极弱局部间断Galerkin方法的傅里叶误差分析问题。首先,给出四阶线性方程的极弱局部间断Galerkin空间离散格式,并在周期边界条件及一致网格的条件下将离散格式表示为差分形式,然后,在k=2的情况下,利用傅里叶分析方法分析其稳定性及其误差估计问题,最后,利用数值实验,分别对得到的结果进行验证。

一类四阶方程基于降阶格式的谱Galerkin逼近及误差估计

本文针对一类四阶方程提出了一种基于降阶格式的有效谱Galerkin逼近.首先,引入一个辅助函数,将四阶方程化为两个耦合的二阶方程,并推导了它们的弱形式及其离散格式.其次,利用Lax-Milgram引理和非一致带权Sobolev空间中正交投影算子的逼近性质,严格地证明了弱解和逼近解的存在唯一性及它们之间的误差估计.最后,通过一些数值算例,数值结果表明该算法是收敛和高精度的.

High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.

Adaptive Sparse Grid Discontinuous Galerkin Method:Review and Software Implementation

This paper reviews the adaptive sparse grid discontinuous Galerkin(aSG-DG)method for computing high dimensional partial differential equations(PDEs)and its software implementation.The C++software package called AdaM-DG,implementing the aSG-DG method,is available on GitHub at https://github.com/JuntaoHuang/adaptive-multiresolution-DG.The package is capable of treating a large class of high dimensional linear and nonlinear PDEs.We review the essential components of the algorithm and the functionality of the software,including the multiwavelets used,assembling of bilinear operators,fast matrix-vector product for data with hierarchical structures.We further demonstrate the performance of the package by reporting the numerical error and the CPU cost for several benchmark tests,including linear transport equations,wave equations,and Hamilton-Jacobi(HJ)equations.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.

A Local Macroscopic Conservative(LoMaC)Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics

In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system.The LoMaC property refers to the exact local conservation of macroscopic mass,momentum,and energy at the discrete level.The recently developed LoMaC low rank tensor algorithm(arXiv:2207.00518)simultaneously evolves the macroscopic conservation laws of mass,momentum,and energy using the kinetic flux vector splitting;then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables.This paper is a generalization of our previous work,but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term.The algorithm is developed in a similar fashion as that for a finite difference scheme,by observing that the DG method can be viewed equivalently in a nodal fashion.With the nodal DG method,assuming a tensorized computational grid,one will be able to(i)derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms,and(ii)define a weighted inner product space based on the nodal DG grid points.The algorithm can be extended to the high dimensional problems by hierarchical Tucker(HT)decomposition of solution tensors and a corresponding conservative projection algorithm.In a similar spirit,the algorithm can be extended to DG methods on nodal points of an unstructured mesh,or to other types of discretization,e.g.,the spectral method in velocity direction.Extensive numerical results are performed to showcase the efficacy of the method.