Stability and Accuracy of a Meshless Finite Difference Method for the Stokes Equations

IF 2.7 3区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
Alexander Westermann, Oleg Davydov, Andriy Sokolov, Stefan Turek
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Abstract

We study the behavior of the meshless finite difference method based on radial basis functions applied to the stationary incompressible Stokes equations in two spatial dimensions, with the velocity and the pressure discretized on their own node sets. We demonstrate that the main condition for the stability of the numerical solution is that the distribution of the pressure nodes has to be coarser than that of the velocity both globally and locally in the domain, and there is no need for any more complex assumptions similar to the Ladyzhenskaya-Babuška-Brezzi condition in the finite element method. Optimal stability is achieved when the relative local density of the velocity to pressure nodes is about 4:1. The convergence rates of the method correspond to the convergence rates of numerical differentiation for both low and higher order discretizations. The method works well on both mesh-based and irregular nodes, such as those generated by random or quasi-random numbers and on nodes with varying density. There is no need for special staggered arrangements, which suggests that node generation algorithms may produce just one node set and obtain the other by either refinement or coarsening. Numerical results for the benchmark Driven Cavity Problem confirm the robustness and high accuracy of the method, in particular resolving a cascade of multiple Moffatt Eddies at the tip of the wedge by using nodes obtained from the quasi-random Halton sequence.

Abstract Image

斯托克斯方程无网格有限差分法的稳定性和准确性
研究了基于径向基函数的无网格有限差分法在二维空间中求解平稳不可压缩Stokes方程时,速度和压力在各自的节点集上离散化的行为。我们证明了数值解的稳定性的主要条件是压力节点的分布必须比整体和局部区域的速度分布更粗,并且不需要任何类似于有限元方法中Ladyzhenskaya-Babuška-Brezzi条件的更复杂的假设。当速度与压力节点的相对局部密度约为4:1时,稳定性最佳。该方法的收敛速度对应于低阶和高阶离散的数值微分的收敛速度。该方法在基于网格和不规则节点(如随机或准随机数生成的节点)以及密度变化的节点上都能很好地工作。不需要特殊的交错排列,这表明节点生成算法可以只产生一个节点集,并通过细化或粗化获得另一个节点集。基准驱动空腔问题的数值结果证实了该方法的鲁棒性和高精度,特别是利用准随机Halton序列中获得的节点来求解楔尖处的多个Moffatt涡级联。
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来源期刊
CiteScore
5.70
自引率
6.90%
发文量
276
审稿时长
5.3 months
期刊介绍: The International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems. The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome.
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