A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.

二维非均匀介质内辐射与瞬态导热耦合传热分析

充分利用间断谱元法对非规则区域良好适应性、数值稳定、局部高阶收敛性等优点,开展了二维非均匀介质内辐射与瞬态导热耦合传热分析。在求解过程中,将空间计算域离散为多个不重合的单元,并在单元内采用高阶谱节点进行离散。对于非均匀介质内离散坐标形式的辐射传递方程,根据离散角度上的辐射能传递方向,确定单元间的上下游关系,并采用迎风格式来实现单元间辐射信息的有效传递;对于瞬态能量守恒方程,采用全隐格式进行瞬态项离散。通过多组非均匀介质内辐射与瞬态导热耦合传热算例,发现了间断谱元法具有随单元数增加的线性收敛特性,以及随单元内节点数增加的指数收敛特性;通过与文献中结果对比,发现最大相对误差均小于1%,验证了间断谱元法的正确性和有效性;随着辐射与导热耦合参数的增加,导热在耦合传热所占比重会加大,非均匀介质带来的温度场非均匀性会减弱,达到稳态的时间变长。

Spectral/hp element methods:Recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials,modified to accommodate a C~0-continuous expansion. Computationally and theoretically, by increasing the polynomial order p,high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

角度-空间间断谱元法的遮蔽条件下二维热辐射特性研究

航空发动机燃烧室温度分布对燃烧室结构设计、燃烧过程检测均具有重要意义。燃烧室结构封闭且复杂、存在大面积的辐射遮挡,这给计算其内部由热辐射引起的温度变化分析带来了一定困难。本文提出采用角度-空间间断谱元法求解遮蔽条件下封闭腔内热辐射问题。在求解过程中,通过黎曼迎风方案对空间单元边界上的辐射信息进行分割,实现单元间信息的有效传递。通过与文献结果进行对比发现:角度上采用间断谱元法可有效抑制辐射遮蔽带来的数值振荡,相同角度离散单元数下,可以获得比离散坐标法更光滑的计算结果。通过分析网格数和基函数阶数对数值结果的影响,发现该方法具有hp收敛特性。此外,控制角度计算节点相同,在与离散坐标法相当的计算精度下,消耗更少的计算时间。

Performance Analysis of a High-Order Discontinuous Galerkin Method Application to the Reverse Time Migration

This work pertains to numerical aspects of a finite element method based discontinuous functions.Our study focuses on the Interior Penalty Discontinuous Galerkin method(IPDGM)because of its high-level of flexibility for solving the full wave equation in heterogeneousmedia.We assess the performance of IPDGMthrough a comparison study with a spectral element method(SEM).We show that IPDGM is as accurate as SEM.In addition,we illustrate the efficiency of IPDGM when employed in a seismic imaging process by considering two-dimensional problems involving the Reverse Time Migration.

基于非结构/混合网格的高阶精度格式研究进展

尽管以二阶精度格式为基础的计算流体力学(CFD)方法和软件已经在航空航天飞行器设计中发挥了重要的作用,但是由于二阶精度格式的耗散和色散较大,对于湍流、分离等多尺度流动现象的模拟,现有成熟的CFD软件仍难以给出满意的结果,为此CFD工作者发展了众多的高阶精度计算格式.如果以适应的计算网格来分类,一般可以分为基于结构网格的有限差分格式、基于非结构/混合网格的有限体积法和有限元方法,以及各种类型的混合方法.由于非结构/混合网格具有良好的几何适应性,基于非结构/混合网格的高阶精度格式近年来备受关注.本文综述了近年来基于非结构/混合网格的高阶精度格式研究进展,重点介绍了空间离散方法,主要包括k-Exact和ENO/WENO等有限体积方法,间断伽辽金(DG)有限元方法,有限谱体积(SV)和有限谱差分(SD)方法,以及近来发展的各种DG/FV混合算法和将各种方法统一在一个框架内的CPR(correction procedure via reconstruction)方法等.随后简要介绍了高阶精度格式应用于复杂外形流动数值模拟的一些需要关注的问题,包括曲边界的处理方法、间断侦测和限制器、各种加速收敛技术等.在综述过程中,介绍了各种方法的优势与不足,其间介绍了作者发展的基于"静动态混合重构"的DG/FV混合算法.最后展望了基于非结构/混合网格的高阶精度格式的未来发展趋势及应用前景.

各向异性介质波导不连续问题的半解析谱单元法分析

将谱单元法与精细积分法相结合求解各向异性介质的波导不连续问题.从矢量波动方程的单变量变分形式出发,采用基于Gauss-Lobatto-Legendre多项式零点作为插值结点的谱单元,对含有各向异性介质波导的横截面进行离散,然后将问题导入哈密顿体系利用精细积分法进行求解.由于采用了谱单元法,在单元网格数较少时,可获得高准确度的计算结果;又由于利用了精细积分法,结构的纵向长度可以任意设定,克服了当人工边界设置在离介质块较远处时,计算量不断增加的缺点.研究表明半解析谱单元法可有效地应用于各向异性介质的波导不连续问题,在提高准确度的同时可大量节省计算时间.

一类非线性反应-扩散方程的间断Galerkin谱元方法

提出了一类非线性反应-扩散方程的间断Galerkin谱元方法,在每个子区间上,基本格式采用Legendre-Galerkin方法,非线性项采用Chebyshev-Gauss-Lobatto插值,跳跃项利用中心数值流量处理,时间方向应用4阶低存储Runge-Kutta格式离散.该方法处理某些初值间断问题有效,并可并行实现;给出了该方法半离散格式下的稳定性和收敛性分析,利用Chebyshev-Gauss-Lobatto插值算子在不带权意义下的逼近结果,获得了按L2-模的最优误差估计;最后,给出了连续问题和间断问题的数值算例.

求解一类交界面问题的模态基函数谱元法数值实验

采用模态基函数形式下的谱元法,对一类一维和二维交界面问题进行了数值计算研究。不仅考虑了具有间断系数的方程,还涉及了右端带有奇性源项的方程。对于二维问题的数值计算,研究了四边形单元谱元法和三角单元谱元法的离散格式。通过一些具有精确解的数值算例,验证了基于模态基函数谱元法求解交界面问题的可行性和有效性。与其他数值方法相比,谱元法对于一维问题可以实现指数阶收敛精度,对于二维问题也得到了较高的计算精度。

一阶线性双曲方程的间断耗散谱元法

讨论了一阶线性双曲方程的间断耗散谱元法,建立了Crank-Nicolson全离散格式的稳定性和误差估计.选取适当的基函数,设计了有效的并行算法,并给出了数值结果,验证了该方法的有效性.与传统的谱元法、耗散谱元法以及间断谱元法进行比较,显示出该方法的某些优势.

最优控制问题弱间断解的一个自适应算法

针对最优解有弱间断的最优控制问题提出一个自适应算法.时间区间被划分为若干子区间,使用分段多项式逼近最优控制问题的解,在每个子区间内,最优控制问题被拟谱方法离散,使用的配置点是Chebyshev-Gauss-Lobatto点.根据计算出的数值解提供的后验信息,该自适应算法既能剖分产生新的子区间,又能在子区间内增加逼近多项式的次数.最后通过若干例子表明了所提出算法的高精度和有效性.

二维非线性反应扩散方程的局部间断Galerkin谱元法

本文提出了二维非线性反应扩散方程的局部间断Galerkin谱元法.在空间方向上采用了Legendre-Galerkin Chebyshev谱配置法,即在每个子区域上,该格式按Legendre-Galerkin谱方法形成,子区域交界面处的跳跃项利用数值流量进行处理,非线性项采用在Chebyshev-Gauss-Lobatto点上的插值进行计算.时间方向上采用四阶低存储Runge-Kutta方法.文中给出了半离散格式下的稳定性和收敛性分析,以及单区域和多区域算法的数值算例,并与间断Galerkin有限元方法进行比较.

基于比例边界问断谱元法的无穷域亚音速圆柱绕流数值模拟

将比例边界坐标插值方法引入谱元法,构成比例边界谱单元;为了增加计算的稳定性,将节点布置在单元内部;用若干无限谱元离散计算域,用间断有限元方法对无穷域Euler方程亚音速圆柱绕流问题进行了数值模拟;计算结果的误差很小,显示了计算方法的可行性.

迎风间断谱元法分析非均匀介质内热辐射

非均匀介质内热辐射与均匀介质内热辐射相比存在差异,辐射传递方程也将变得复杂难以求解。在本工作中,以高效、快速获取多维非均匀介质内热辐射强度为目的,发展了迎风间断谱元法求解辐射传递方程。通过与文献结果对比,证明了该方法在求解非均匀介质内热辐射问题的可行性。此外,在求解过程中采用逐单元计算方案。与传统整体求解域组装方案相比,计算效率大大提升。对于光学厚度较小的算例,迎风间断谱元法与文献中基准解对比吻合较好,不存在数值振荡现象。此外,通过多组非均匀介质内辐射热平衡和辐射非热平衡算例,验证了该方法的准确性和有效性,并进一步分析了变折射率对热辐射的影响规律。