Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014     

On the stability of two new two-step explicit methods for the numerical integration of second order initial value problem on a variable mesh

We present two novel two-step explicit methods for the numerical solution of the second order initial value problem on a variable mesh. In the case of a constant mesh the method is superstable in the sense of Chawla (1985). Numerical experimentation is provided to verify the stability analysis.     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Feature-driven Cartesian adaptive mesh refinement for vortex-dominated flows

We develop locally normalized feature-detection methods to guide the adaptive mesh refinement (AMR) process for Cartesian grid systems to improve the resolution of vortical features in aerodynamic wakes. The methods include: the Q-criterion [1], the λ₂ method [2], the λ_ci method [3], and the λ₊ method [4]. Specific attention is given to automate the feature identification process by applying a local normalization based upon the shear-strain rate so that they can be applied to a wide range of flow-fields without the need for user intervention. To validate the methods, we assess tagging efficiency and accuracy using a series of static vortex-dominated flow-fields, and use the methods to drive the AMR process for several theoretical and practical simulations. We demonstrate that the adaptive solutions provide comparable accuracy to solutions obtained on uniformly refined meshes at a fraction of the computational cost. Overall, the normalized feature detection methods are shown to be effective in driving the AMR process in an automated and efficient manner.     

Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type

Methods based on exponential finite difference approximations of h⁴ accuracy are developed to solve one and two‐dimensional convection–diffusion type differential equations with constant and variable convection coefficients. In the one‐dimensional case, the numerical scheme developed uses three points. For the two‐dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri‐diagonal systems. The methods are applied on a number of linear and non‐linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non‐uniform mesh is also illustrated. The h⁴ accuracy of the schemes is also computationally demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.     

Hybrid mesh adaptation applied to industrial numerical combustion

This paper presents an anisotropic mesh adaptation method applied to industrial combustion problems. The method is based on a measure of the distance between two Riemannian metrics called metric non‐conformity. This measure, which can be used to build a cost function to adapt meshes comprising several types of mesh elements, provides the basis for a generic mesh adaptation approach applicable to various types of physical problems governed by partial differential equations. The approach is shown to be applicable to industrial combustion problems, through the specification of a target metric computed as the intersection of several Hessian matrices reconstructed from the main variables of the governing equations. Numerical results show that the approach is cost effective in that it can drastically improve the prediction of temperature and species distributions in the flame region of a combustor while reducing computational cost. The results can be used as a basis for pollutant prediction models. Copyright © 2011 John Wiley & Sons, Ltd.     

On the construction of interpolation mesh surfaces

A method for constructing two-dimensional interpolation mesh functions is proposed that is more flexible than the classical cubic spline method because it makes it possible to construct interpolation surfaces that fit the given function at specified points by varying certain parameters. The method is relatively simple and is well suited for practical implementation.     

Validation of adaptive unstructured hexahedral mesh computations of flow around a wind turbine airfoil

In this paper we investigate local adaptive refinement of unstructured hexahedral meshes for computations of the flow around the DU91 wind turbine airfoil. This is a 25% thick airfoil, found at the mid‐span section of a wind turbine blade. Wind turbine applications typically involve unsteady flows due to changes in the angle of attack and to unsteady flow separation at high angles of attack. In order to obtain reasonably accurate results for all these conditions one should use a mesh which is refined in many regions, which is not computationally efficient. Our solution is to apply an automated mesh adaptation technique. In this paper we test an adaptive refinement strategy developed for unstructured hexahedral meshes for steady flow conditions. The automated mesh adaptation is based on local flow sensors for pressure, velocity, density or a combination of these flow variables. This way the mesh is refined only in those regions necessary for high accuracy, retaining computational efficiency. A validation study is performed for two cases: attached flow at an angle of 6° and separated flow at 12°. The results obtained using our adaptive mesh strategy are compared with experimental data and with results obtained with an equally sized non‐adapted mesh. From these computations it can be concluded that for a given computing time, adapted meshes result in solutions closer to the experimental data compared to non‐adapted meshes for attached flow. Finally, we show results for unsteady computations. Copyright © 2005 John Wiley & Sons, Ltd.     

黏性不可压流体的自适应网格技术和基本特性方程分离算法的联合分析

组合基本特性方程分离算法和自适应网格技术,分析二维黏性不可压流体.该方法使用3节点三角单元,对速度分量和压力等变量分析,使用等阶次的插值函数.组合解法的主要优点在于,在自适应网格技术中,对解梯度变化大的区域,通过耦合误差估计生成小的单元,利于提高解的精度,在其它区域生成大单元,可以节省时间.最后,通过对一个黏性流体圆柱体绕流问题的瞬态和稳态特性分析,给出了组合解法性能的评价.     

高精度TVD有限差分格式的自由梯度条件

1自由流条件守恒性气动方程组的高精度TVD有限差分格式一般通过使用通量限制器加入非线性的人工粘性来构成。在曲线坐标系中，通量限制器比较的两个因子与各自所在单元的界面雅可比转换行列式相关。为了消除畸变网格单元界面雅可比转换行列式的变化对通量限制器比较结果的影响，本文提出了高精度TVD格式的自由梯度性质，并分别就Osher和HartenTVD格式讨论了同时满足自由流和自由梯度的度量张量计算方法，使高精度TVD格式在畸变网格上得以实现。任意曲线坐标系中守恒型二维欧拉方程表示为其中，E、F为笛卡尔坐标系中的通量；J为坐标变…