自适应间断有限元方法求解双曲守恒律方程

研究自适应Runge-Kutta间断Galerkin (RKDG)方法求解双曲守恒律方程组,并提出两种生成相容三角形网格的自适应算法.第一种算法适用于规则网格,实现简单、计算速度快.第二种算法基于非结构网格,设计一类基于间断界面的自适应网格加密策略,方法灵活高效.两种方法都具有令人满意的计算效果,而且降低了RKDG的计算量.     

用有限元方法求解双曲守恒律

分片线性插值有限元给出了求解双曲守恒律的计算方法。有别于不连续有限元方法求解双曲守恒律在相邻单元边界上求Ｒｉｅｍａｎｎ解,利用双曲守恒律的Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程形式,直接应用有限元求解,在ＣＦＬ下,证明了计算格式满足极大值原理,并且是ＴＶＤ格式。数值例子在文后给出。此外,方法推广到流体力学方程组和高维问题,将在另文中邓以讨论。     

双曲守恒律的Taylor-Galerkin有限元方法

利用双曲守恒律的Hamilton-Jacobi方程形式,应用Taylor公式与Galerkin有限元给出了求解双曲守恒律的计算方法。采用TVD差分格式的构造思想,对数值通量作修正,在等距网格情形下有限元方法得到的计算格式满足TVD性质,并给出了数值例子。     

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.     

二维交错网格的GAUSS型格式

利用Gauss型求积公式在交错网格的情况下构造了一类不需解Riemann问题的求解二维双曲守恒律的二阶显式Gauss型差分格式,该格式在CFL条件限制下为MmB格式.并将格式推广到二维方程组,进行了数值试验.     

求解双曲型守恒律方程的一类自适应多分辨方法

针对双曲型守恒律方程问题,发展一种有效的自适应多分辨分析方法.通过对嵌套网格上的数值解构造离散多分辨分析,建立小波系数与多层嵌套网格点之间的对应关系.对于小波系数较大的网格点采用高精度WENO格式计算,其余区域则直接采用多项式插值.数值试验表明,该方法在保持原规则网格方法的精度和分辨率的同时,显著地减少计算的CPU时间.     

针对二维双曲守恒律方程求解方法的研究

对双曲守恒律方程进行数值求解是计算流体力学的重要研究内容。本文从物理概念出发,通过对计算流体力学和双曲守恒律方程研究现状及发展趋势进行引入,详细介绍了满足熵稳定条件的二维双曲守恒律方程的熵守恒、熵稳定、熵相容、高分辨率熵稳定格式,可将其格式应用于具体算例的数值求解中。     

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.      

龙格库塔间断有限元方法在计算爆轰问题中的应用

构造求解带源项守恒律方程组的龙格库塔间断有限元(RKDG)方法,并分别结合源项的Strang分裂法和无分裂法数值求解模型守恒律方程和反应欧拉方程.为了和有限体积型WENO方法进行比较,设计计算源项的WENO重构格式.对一维带源项守恒律的计算表明,对于非刚性问题,RKDG方法比有限体积型WENO方法的误差更小;对于刚性问题,RKDG方法对于间断面位置的捕捉更为精确.对于一二维爆轰波问题的计算结果表明,RKDG方法对爆轰波结构的分辨和爆轰波位置的捕捉能力更强.     

双曲型守恒律的一种高精度TVD差分格式

构造了一维双曲型守恒律方程的一个高精度高分辨率的守恒型TVD差分格式.其主要思想是:首先将计算区域划分为互不重叠的小单元,且每个小单元再根据希望的精度阶数分为细小单元;其次,根据流动方向将通量分裂为正、负通量,并通过小单元上的高阶插值逼近得到了细小单元边界上的正、负数值通量,为避免由高阶插值产生的数值振荡,进一步根据流向对其进行TVD校正;再利用高阶Runge KuttaTVD离散方法对时间进行离散,得到了高阶全离散方法.进一步推广到一维方程组情形.最后对一维欧拉方程组计算了几个算例.     

自适应间断有限元方法求解三维欧拉方程

将龙格库塔间断有限元方法(RDDG)与自适应方法相结合,求解三维欧拉方程.区域剖分采用非结构四面体网格,依据数值解的变化采用自适应技术对网格进行局部加密或粗化,减少总体网格数目,提高计算效率.给出四种自适应策略并分析不同自适应策略的优缺点.数值算例表明方法的有效性.     

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.     

Accelerated GPU simulation of compressible flow by the discontinuous evolution Galerkin method

The aim of the present paper is to report on our recent results for GPU accelerated simulations of compressible flows. For numerical simulation the adaptive discontinuous Galerkin method with the multidimensional bicharacteristic based evolution Galerkin operator has been used. For time discretization we have applied the explicit third order Runge-Kutta method. Evaluation of the genuinely multidimensional evolution operator has been accelerated using the GPU implementation. We have obtained a speedup up to 30 (in comparison to a single CPU core) for the calculation of the evolution Galerkin operator on a typical discretization mesh consisting of 16384 mesh cells.     

Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations

This article considers a posteriori error estimation and anisotropic mesh refinement for three-dimensional laminar aerodynamic flow simulations. The optimal order symmetric interior penalty discontinuous Galerkin discretization which has previously been developed for the compressible Navier–Stokes equations in two dimensions is extended to three dimensions. Symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model. Using duality arguments, an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients. Furthermore, residual-based indicators as well as adjoint-based indicators for goal-oriented refinement are derived. These refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous Galerkin (DG) discretization. Two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint-based error estimation are presented. The performance of the proposed discretization, error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows.     

A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions

We present a method for solving Poisson and heat equations with discontinuous coefficients in two- and three-dimensions. It uses a Cartesian cut-cell/embedded boundary method to represent the interface between materials, as described in Johansen and Colella (1998). Matching conditions across the interface are enforced using an approximation to fluxes at the boundary. Overall second order accuracy is achieved, as indicated by an array of tests using non-trivial interface geometries. Both the elliptic and heat solvers are shown to remain stable and efficient for material coefficient contrasts up to 10⁶, thanks in part to the use of geometric multigrid. A test of accuracy when adaptive mesh refinement capabilities are utilized is also performed. An example problem relevant to nuclear reactor core simulation is presented, demonstrating the ability of the method to solve problems with realistic physical parameters.     

RKDG有限元法求解一维拉格朗日形式的Euler方程

描述一种新的求解Euler方程的拉格朗日格式,该格式用Runge-Kutta Discontinuous Galerkin(RKDG)方法在拉格朗日坐标系求解Euler方程,剖分网格随流体运动.新格式不仅保证流体的质量、动量和能量守恒,而且能够在时间和空间上同时达到二阶精度.数值算例表明在一维情况,随着拉氏网格的移动和改变,格式在时间和空间上仍保持二阶精度,并且没有数值震荡.     

Unified phase-field model for epitaxial growth of multi-scale three-dimensional islands on insulating surfaces

In this paper we present a unified phase-field model for non-equilibrium growths of various three-dimensional metal islands on insulating surfaces. We introduce a phase-field variable to distinguish the island from the non-island regions and substrate and a density variable to describe local density of deposited adatoms. Two partial differential equations with appropriate boundary conditions, as the governing equations, are used to describe the evolution of the three-dimensional metal islands and the diffusion of adatoms. We solve the equations by using an adaptive mesh refinement method so that we can simulate the non-equilibrium growth of three-dimensional metal islands from tens of nanometers to several micrometers. We investigate the dependence of simulated results on the model parameters and experimental conditions. Equilibrium shape of such islands can be obtained through sufficient post-deposition relaxation. Experimental trends of island size and shape on various scales are obtained with reasonable parameters. This method should be a good approach to non-equilibrium growths of multi-scale three-dimensional metal islands.     

三维非匹配网格上的扩散方程数值求解

将子网格剖分的支撑算子方法,拓展应用于三维非匹配网格上的扩散方程求解.算例表明该方法在正交非匹配网格上能够精确获得线性解;在一般非匹配网格上可以达到二阶精度;在求解曲面网格和节点不共面网格时,精度比平面近似的方法要高,也可以达到2阶精度,同时也适合求解含有物质界面的混合介质网格.     

3维全电磁粒子软件NEPTUNE中的并行计算方法

 介绍了NEPTUNE软件采用的一些并行计算方法：采用“块-网格片”二层并行区域分解方法,使计算规模能够扩展到上千个处理器核。基于复杂几何特征采用自适应技术并行生成结构网格,在原有规则区域的基础上剔除无效网格,大幅降低了存储量和并行执行时间。在经典的Boris和SOR迭代方法基础上,采用红黑排序和几何约束,提出了非规则区域上的Poisson方程并行求解方法。采用这些方法后,当使用NEPTUNE软件模拟MILO器件时,可在1 024个处理器核上获得51.8%的并行效率。