Analysis of ADER and ADER-WAF schemes

^{We study stability properties and truncation errors of the finite-volumeADER schemes on structured meshes as applied to the linear advectionequation with constant coefficients in one-, two- and three-spatialdimensions. Stability of linear ADER schemes is analysed bymeans of the von Neumann method. For nonlinear schemes, we deducethe stability region from numerical experiments. The truncationerror analysis is carried out for linear ADER schemes in one-,two- and three-space dimensions and for nonlinear ADER schemesin one-space dimension.}     

对积分恒等式的一点补充和有限元的局部校正结果

１．积分恒等式简介其中ｈ＝ｍａｘ｛ｈｅ,ｋｅ｝, 易见用 Ｑ１（ｅ）表示 ｅ上全体双线性函数,ｕＩ∈ Ｑ１（ｅ）为ｕ在 ｅ上的双线性插值．文［１, ２］曾给出了一系列的积分恒等式,列举如下： Ａ：二阶恒等式,用于超收敛估计：对任给ｖ∈Ｑ１（ｅ）,有以及由于Ｉｅ１２的展开式中含有因子б１б２ｖ,将破坏超收敛估计,故对此项关于ｙ分部积分,得展开式其中τ１,τ２分别为ｅ的上下边.（１．３）,（１．４）和（１．６）是作导数超收敛估计的基本展开式．当然我们更有０次项的展开式．在下一节将讨论林氏积分表． 为了进一步…     

ADER discontinuous Galerkin schemes for aeroacoustics

In this paper we apply the ADER approach to the Discontinuous Galerkin (DG) framework for the two-dimensional linearized Euler equations. The result is an efficient high order accurate single-step scheme in time which uses less storage than Runge–Kutta DG schemes, especially for very high order of accuracy. The aim is to obtain an arbitrarily accurate scheme in space and time on unstructured grids for accurate noise propagation in the time domain in very complex geometries. We will present numerical convergence rates for ADER-DG methods up to 10th order of accuracy in space and time on structured and unstructured meshes. To cite this article: M. Dumbser, C.-D. Munz, C. R. Mecanique 333 (2005).     

薄体位势问题边界元法中的解析积分算法

薄体结构的数值分析是边界元法的难点问题之一。该文导出了一种完全解析积分算法，用这种算法计算了薄体平面位势问题边界元法中出现的几乎弱奇异、强奇异和超奇异积分。当边界离散为一系列线性单元，边界积分方程离散计算的积分可归纳为三种形式。对薄体问题，源点与积分单元距离通常相距很近，这些积分产生显著几乎奇异性，直接采用常规高斯积分不能有效计算。为此该文导出了这些几乎奇异积分的全解析计算公式。按源点与单元的距离是否为零，公式分两种情况。新算法采用全解析积分公式处理几乎奇异积分，首先精确计算出薄体问题边界未知位势和法向位势梯度，然后再进一步计算了域内点的物理参量。算例表明该文算法可处理狭长比为1．E-08的薄体问题，显示了边界元法分析薄体问题具有独特的优势。     

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tⁿ is connected to the new one at t^n + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.     

8英寸导电型4H-SiC单晶衬底制备与表征

使用物理气相传输法(PVT)通过扩径技术制备出直径为209 mm的4H-SiC单晶,并通过多线切割、研磨和抛光等一系列加工工艺制备出标准8英寸SiC单晶衬底。使用拉曼光谱仪、高分辨X射线衍射仪、光学显微镜、电阻仪、偏光应力仪、面型检测仪、位错检测仪等设备,对8英寸衬底的晶型、结晶质量、微管、电阻率、应力、面型、位错等进行了详细表征。拉曼光谱表明8英寸SiC衬底100%比例面积为单一4H晶型;衬底(004)面的5点X射线摇摆曲线半峰全宽分布在10.44″~11.52″;平均微管密度为0.04 cm^-2;平均电阻率为0.020 3 Ω·cm。使用偏光应力仪对8英寸SiC衬底内部应力进行检测表明整片应力分布均匀,且未发现应力集中的区域;翘曲度(Warp)为17.318 μm,弯曲度(Bow)为-3.773 μm。全自动位错密度检测仪对高温熔融KOH刻蚀后的8英寸衬底进行全片扫描,平均总位错密度为3 293 cm^-2,其中螺型位错(TSD)密度为81 cm^-2,刃型位错(TED)密度为3 074 cm^-2,基平面位错(BPD)密度为138 cm^-2。结果表明8英寸导电型4H-SiC衬底质量优良,同比行业标准达到行业先进水平。     

Optimization of the ADER‐DG method in GPU applied to linear hyperbolic PDEs

We optimized the Arbitrary accuracy DErivatives Riemann problem (ADER) ‐ Discontinuous Galerkin (DG) numerical method using the CUDA‐C language to run the code in a graphic processing unit (GPU). We focus on solving linear hyperbolic partial–differential equations where the method can be expressed as a combination of precomputed matrix multiplications becoming a good candidate to be used on the GPU hardware. Moreover, the method is arbitrarily high order involving intensive work on local data, a property that is also beneficial for the target hardware. We compare our GPU implementation against CPU versions of the same method observing similar convergence properties up to a threshold where the error remains fixed. This behavior is in agreement with the CPU version, but the threshold is slightly larger than in the CPU case. We also observe a big difference when considering single and double precisions where in the first case, the threshold error is significantly larger. Finally, we did observe a speed‐up factor in computational time that depends on the order of the method and the size of the problem. In the best case, our novel GPU implementation runs 23 times faster than the CPU version. We used three partial–differential equation to test the code considering the linear advection equation, the seismic wave equation, and the linear shallow water equation, all of them considering variable coefficients. Copyright © 2015 John Wiley & Sons, Ltd.     

Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.     

High-resolution algorithms of sharp interface treatment for compressible two-phase flows

In this paper, a kind of arbitrary high order derivatives (ADER) scheme based on the generalised Riemann problem is proposed to simulate multi-material flows by a coupling ghost fluid method. The states at cell interfaces are reconstructed by interpolating polynomials which are piece-wise smooth functions. The states are treated as the equivalent of the left and right states of the Riemann problem. The contact solvers are extrapolated in the vicinity of contact points to facilitate ghost fluids. The numerical method is applied to compressible flows with sharp discontinuities, such as the collision of two fluids of different physical states and gas–liquid two-phase flows. The numerical results demonstrate that unexpected physical oscillations through the contact discontinuities can be prevented effectively and the sharp interface can be captured efficiently.     

新型薄层色谱内标法测定葛根素含量

以大豆甙元为内标物在高效硅胶薄层板上采用新型内标法测定了葛根素注射液、中药葛根及中成药玉泉丸中葛根素的含量,建立了新型TLC内标法测定葛根素的新方法。结果表明：用大豆甙元作为内标物测定葛根素可以满足TLC内标物的一般条件,新型TLC内标法测定葛根素的含量具有测定结果准确,精密度好,回收率高等优点。