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1.
研究了三维对流扩散方程基于有限差分法的多重网格算法。差分格式采用一般网格步长下的二阶中心差分格式和四阶紧致差分格式,建立了与两种格式相适应的部分半粗化的多重网格算法,构造了相应的限制算子和插值算子,并与传统的等距网格下的完全粗化的多重网格算法进行了比较。数值研究结果表明,对于各向异性问题,一般网格步长下的部分半粗化多重网格算法比等距网格下的完全粗化多重网格算法具有个更高的精度和更好的收敛效率。  相似文献   

2.
非结构网格的并行多重网格解算器   总被引:2,自引:0,他引:2  
李宗哲  王正华  姚路  曹维 《软件学报》2013,24(2):391-404
多重网格方法作为非结构网格的高效解算器,其串行与并行实现在时空上都具有优良特性.以控制方程离散过程为切入点,说明非结构网格在并行数值模拟的流程,指出多重网格方法主要用于求解时间推进格式产生的大规模代数系统方程,简述了算法实现的基本结构,分析了其高效性原理;其次,综述性地概括了几何多重网格与代数多种网格研究动态,并对其并行化的热点问题进行重点论述.同时,针对非结构网格的实际应用,总结了多重网格解算器采用的光滑算子;随后列举了非结构网格应用的部分开源项目软件,并简要说明了其应用功能;最后,指出并行多重网格解算器在非结构网格应用中的若干关键问题和未来的研究方向.  相似文献   

3.
Cahn-Hilliard(CH)方程是相场模型中的一个基本的非线性方程,通常使用数值方法进行分析。在对CH方程进行数值离散后会得到一个非线性的方程组,全逼近格式(Full Approximation Storage, FAS)是求解这类非线性方程组的一个高效多重网格迭代格式。目前众多的求解CH方程主要关注数值格式的收敛性,而没有论证求解器的可靠性。文中给出了求解CH方程离散得到的非线性方程组的多重网格算法的收敛性证明,从理论上保证了计算过程的可靠性。针对CH方程的时间二阶全离散差分数值格式,利用快速子空间下降(Fast Subspace Descent, FASD)框架给出其FAS格式多重网格求解器的收敛常数估计。为了完成这一目标,首先将原本的差分问题转化为完全等价的有限元问题,再论证有限元问题来自一个凸泛函能量形式的极小化,然后验证能量形式及空间分解满足FASD框架假设,最终得到原多重网格算法的收敛系数估计。结果显示,在非线性情形下,CH方程中的参数ε对网格尺度添加了限制,太小的参数会导致数值计算过程不收敛。最后通过数值实验验证了收敛系数与方程参数及网格尺度的依赖关系。  相似文献   

4.
研究井间地震波场的形成过程以及波场的传播机理、规律,对于指导实际井间地震勘探有着重要的意义.从具有倾斜对称轴的横向各向同性介质(TTI)的二维三分量一阶速度-应力弹性波方程出发,采用高阶紧致交错网格差分算子对方程进行差分离散,得到了TTI介质中井间地震波场正演的高阶有限差分格式.并推导了TTI介质完全匹配层吸收边界条件公式和相应的紧致交错网格高阶差分格式,在此基础上实现了二维三分量TTI介质中井间地震波场模拟.数值算例表明:紧致交错网格高阶有限差分方法模拟的记录精度高,数值频散小,该方法能够精确的模拟复杂各向异性介质中的地震波传播过程,可以得到高精度的正演记录.完全匹配层吸收边界能有效地解决人工边界问题,是一种高效的边界吸收算法.  相似文献   

5.
一维非定常对流扩散方程的高阶组合紧致迎风格式   总被引:1,自引:0,他引:1  
通过将对流项采用四五阶组合迎风紧致格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的半离散格式在时间方向采用四阶龙格库塔方法求解,从而得到了一种求解非定常对流扩散方程问题的高精度组合紧致有限差分格式,其收敛阶为O(h~4+τ~4).经Fourier精度分析和数值验证,证实了格式的良好性能.三个数值算例包括线性常系数问题,矩形波问题和非线性问题,数值结果表明:该格式具有很高的分辨率,且适用于对高雷诺数问题的数值模拟.  相似文献   

6.
针对一类奇异摄动对流扩散问题,将粒子群算法与差分格式相结合,在Bakhvalov-Shishkin网格上进行求解。对于Bakhvalov-Shishkin网格中的网格参数,采用粒子群算法进行优化,构造了求误差范数最小值的目标函数。对两个算例进行了数值计算,实验结果表明,与选择固定的网格参数相比,采用粒子群算法计算能得到更好的数值结果,并且数值结果具有收敛性,验证了该方法的有效性和优越性。  相似文献   

7.
选取一对合适的步长使用中心差分格式离散半线性椭圆问题形成粗网格和细网格,使用三次样条插值算子将粗网格上高精度近似解插值到细网格为其提供初始值,结合牛顿法提出了牛顿-瀑布型两层网格法.数值实验表明该算法具有稳健性强、计算效率高的优点.  相似文献   

8.
通过将原方程变换为对流扩散方程,将所得方程的对流项采用四阶组合紧致迎风格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的空间半离散格式采用四阶龙格库塔方法进行时间推进,得到了一种求解非定常对流扩散反应问题的高精度方法,其收敛阶为O(h4+τ4).经数值实验并与文献结果进行对比,表明该格式适用于对流占优问题的数值模拟,验证了格式的良好性能.  相似文献   

9.
并行多重网格计算:各向异性扩散问题   总被引:5,自引:0,他引:5  
1.引言本文讨论典型各向异性扩散问题在分布式存储环境的并行多重网格计算,其中Ω为d(d=2,3)维空间中规则有界区域,系数D(x)正定对称,σ(x)≤0和f(x)在Ω中连续,且具有Dirichlet边界条件g(X)在Ω上连续.传统处理问题(1)的有效多重网格算法主要有:1)采用标准网格粗化策略,线性延拓,FW残差限制以及同时松弛所有强耦合变量的块松弛方法l‘,’,’,“].或者网格粗化仅沿某个方向进行,另一方向采用块松弛l‘l;幻多粗网格层校正算法,如Wederickson和McBryan的并行超收敛算法l‘],Hackbush的频率分解算法[‘…  相似文献   

10.
多重网格方法求解两类Helmholtz方程   总被引:1,自引:0,他引:1  
详细给出了多重网格方法的实现过程,借助正定Helmholtz方程及不定Helmholtz方程的求解来探讨多重网格方法的特性。对多重网格V环、W环以及F环三种不同迭代格式的收敛效果进行了对比。通过正定Helmholtz方程的求解,发现多重网格的确有很高的计算效率。对于不定Helmholtz方程,随着波数的增加,利用多重网格方法得到结果不收敛,原因出在细网格光滑和粗网格矫正过程。如何针对此问题对多重网格进行有效改进还有待进一步研究。  相似文献   

11.
An efficient parallel multigrid pressure correction algorithm is proposed for the solution of the incompressible Navier–Stokes equations on computing architectures with acceleration devices. The pressure correction procedure is based on the numerical solution of a Poisson-type problem, which is discretized using a fourth-order finite difference compact scheme. Since this is the most time-consuming part of the solver, we propose a parallel pressure correction algorithm using an iterative method based on a block cyclic reduction solution method combined with a multigrid technique. The grid points are numbered with respect to the red–black ordering scheme for the parallel Gauss–Seidel smoother. These parallelization techniques allow the execution of the entire simulation computations on the acceleration device, minimizing memory communication costs. The realization is developed using the OpenACC API, and the numerical method is demonstrated for the solution of two classical incompressible flow test problems. The first is the two-dimensional lid-driven cavity problem over equal mesh sizes while the other is the Stokes boundary layer, which is a decent benchmark problem for unequal mesh spacing. The effect of several multigrid components on modern and legacy acceleration architectures is examined. Eventually the performance investigation demonstrates that the proposed parallel multigrid solver achieves an acceleration of more than 10\(\times \) over the sequential solver and more than 4\(\times \) over multi-core CPU only realizations for all tested accelerators.  相似文献   

12.
We propose a method with sixth-order accuracy to solve the three-dimensional (3D) convection diffusion equation. We first use a 15-point fourth-order compact discretization scheme to obtain fourth-order solutions on both fine and coarse grids using the multigrid method. Then an iterative mesh refinement technique combined with Richardson extrapolation is used to approximate the sixth-order accurate solution on the fine grid. Numerical results are presented for a variety of test cases to demonstrate the efficiency and accuracy of the proposed method, compared with the standard fourth-order compact scheme.  相似文献   

13.
Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the \(L^2\)-norm and \(L^{\infty }\)-norm for the solution and its gradient when the exact solution belongs to \(C^6\). Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.  相似文献   

14.
A partial semi-coarsening multigrid method based on the high-order compact (HOC) difference scheme on nonuniform grids is developed to solve the 2D convection–diffusion problems with boundary or internal layers. The significance of this study is that the multigrid method allows different number of grid points along different coordinate directions on nonuniform grids. Numerical experiments on some convection–diffusion problems with boundary or internal layers are conducted. They demonstrate that the partial semi-coarsening multigrid method combined with the HOC scheme on nonuniform grids, without losing the high-order accuracy, is very efficient and effective to decrease the computational cost by reducing the number of grid points along the direction which does not contain boundary or internal layers.  相似文献   

15.
In this paper a family of fourth-order and sixth-order compact difference schemes for the three dimensional (3D) linear Poisson equation are derived in detail. By using finite volume (FV) method for derivation, the highest-order compact schemes based on two different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outline for the nonlinear problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference scheme.  相似文献   

16.
A high-order compact finite difference scheme combined with the temporal extrapolation technique is investigated for the fourth-order fractional diffusion-wave system in this paper. The solvability, stability and convergence of the scheme are analyzed simultaneously by the energy method. Numerical experiments show that the proposed compact scheme is more accurate and efficient than the Crank–Nicolson scheme.  相似文献   

17.
Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) [4], the fourth-order ADI method of Karaa and Zhang (2004) [5] and the fourth-order ADI method of Tian and Ge (2007) [23], but also proves more effective than the fourth-order Padé ADI method of You (2006) [6], in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.  相似文献   

18.
Multiscale multigrid (MSMG) method is an effective computational framework for efficiently computing high accuracy solutions for elliptic partial differential equations. In the current MSMG method, compared to the CPU cost on computing sixth-order solutions by applying extrapolation and other techniques on two fourth-order solutions from different scales grids, much more CPU time is spent on computing fourth-order solutions themselves on coarse and fine grids, particularly for high-dimensional problems. Here we propose to embed extrapolation cascadic multigrid (EXCMG) method into the MSMG framework to accelerate the whole process. Numerical results on 3D Poisson equations show that the new EXCMG–MSMG method is more efficient than the existing MSMG method and the EXCMG method for sixth-order solution computation.  相似文献   

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