A linearity‐preserving cell‐centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes

In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell‐centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell‐centered unknowns, which reduces the scheme to a completely cell‐centered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity‐preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity‐preserving cell‐centered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Copyright © 2010 John Wiley & Sons, Ltd.     

Exponential spline interpolation in characteristic based scheme for solving the advective–diffusion equation

This paper demonstrates the use of shape‐preserving exponential spline interpolation in a characteristic based numerical scheme for the solution of the linear advective–diffusion equation. The results from this scheme are compared with results from a number of numerical schemes in current use using test problems in one and two dimensions. These test cases are used to assess the merits of using shape‐preserving interpolation in a characteristic based scheme. The evaluation of the schemes is based on accuracy, efficiency, and complexity. The use of the shape‐preserving interpolation in a characteristic based scheme is accurate, captures discontinuities, does not introduce spurious oscillations, and preserves the monotonicity and positivity properties of the exact solution. However, fitting exponential spline interpolants to the nodal concentrations is computationally expensive. Exponential spline interpolants were also fitted to the integral of the concentration profile. The integral of the concentration profile is a smoother function than the concentration profile. It requires less computational effort to fit an exponential spline interpolant to the integral than the nodal concentrations. By differentiating the interpolant, the nodal concentrations are obtained. This results in a more efficient and more accurate numerical scheme. Copyright © 2000 John Wiley & Sons, Ltd.     

污染扩散方程高精度紧致格式及其稳定性分析

针对污染扩散方程提出了时间任意阶精度的显式格式,并对该格式的稳定性和精度进行了分析,理论结果表明:一阶精度的计算格式是传统的显格式,其稳定条件为:s≤1/2(s=D.Δt/Δx2,D为扩散系数,Δt为时间步长,Δx为空间步长),随着保留精度阶数的增加,稳定性范围也会随之增大;当保留无穷阶精度时,格式是无条件稳定的。这也就从一个侧面揭示了稳定性与时间精度之间的关系,为高性能数值计算格式的构思提供了可以借鉴的原则。数值算例的结果表明,本文格式具有一定的实用性。     

An entropy fixed cell‐centered Lagrangian scheme

On the basis of the work [P.‐H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput. 29 (2007), 1781–1824], we present an entropy fixed cell‐centered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell‐centered. And using the nodal solver, we obtain the nodal viscous‐velocity, viscous‐pressures, antidissipation velocity, and antidissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous‐velocity and antidissipation velocity, so do nodal pressures, whereas these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum, and energy; preserves entropy for isentropic flows, and satisfies a local entropy inequality for nonisentropic flows. One‐ and two‐dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.Copyright © 2013 John Wiley & Sons, Ltd.     

An improved second moment method for solution of pure advection problems

The second moment numerical method (SMM) of Egan and Mahoney [Numerical modeling of advection and diffusion of urban area source pollutant. Journal of Applied Meteorology 1972; 11 : 312–322] is adapted to solve for the pure advection transport equation in a variety of flow fields. SMM eliminates numerical diffusion by employing a procedure that takes into account the first and second moments of mass distribution in each grid element. For pure translational flow fields, the method is conservative, positive definite and shape‐preserving. In rotational and/or shear flows, the accuracy of SMM is significantly reduced. Two improvements are presented to make the SMM applicable to a wider range of flow problems. It is shown that the improved SMM (ISMM) is less diffusive and more shape‐preserving than the SMM in rotational and/or deformational flows. The ISMM can also be used to solve for a color function in compressible flow fields. The computational efficiency of this method is compared with that of other methods and, for a given accuracy, it is shown that ISMM is a cost‐effective, non‐diffusive and shape‐preserving method. Copyright © 2000 John Wiley & Sons, Ltd.     

A swept intersection‐based remapping method for axisymmetric ReALE computation

We use here a reconnection ALE (ReALE) strategy to solve hydrodynamic compressible flows in cylindrical geometries. The main difference between the classical ALE and the ReALE method is the rezoning step where we allow change in the topology. This leads for ReALE to a polygonal mesh, which follows more efficiently the flow. We present here a new displacement of generators in order to keep the Lagrangian features, which are usually lost using ALE with fixed topology. The reconnection capability allows to deal with complex geometries and high‐vorticity problems contrary to ALE method. The main difficulty of ReALE is the remapping step where we have to remap physical variables on a mesh with a different topology. For this step, a new remapping method based on a swept intersection algorithm has been developed in the case of planar geometries. We present here the extension of the swept intersection‐based remapping method to cylindrical geometries. We demonstrate that our method can be applied to several numerical examples up to problem representative of hydrodynamic experiments. Copyright © 2015 John Wiley & Sons, Ltd.     

A swept‐intersection‐based remapping method in a ReALE framework

A complete reconnection‐based arbitrary Lagrangian–Eulerian (ReALE) strategy devoted to the computation of hydrodynamic applications for compressible fluid flows is presented here. In ReALE, we replace the rezoning phase of classical ALE method by a rezoning where we allow the connectivity between cells of the mesh to change. This leads to a polygonal mesh that recovers the Lagrangian features in order to follow more efficiently the flow. Those reconnections allow to deal with complex geometries and high vorticity problems contrary to ALE method. For optimizing the remapping phase, we have modified the idea of swept‐integration‐based. The new method is called swept‐intersection‐based remapping method. We demonstrate that our method can be applied to several numerical examples representative of hydrodynamic experiments.Copyright © 2012 John Wiley & Sons, Ltd.     

三维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解三维非定常变系数对流扩散方程的一种高精度全隐紧致差分格式,该格式在空间上具有四阶精度,时间具有二阶精度.为了克服传统迭代法在每一个时间步上迭代求解隐格式时收敛速度慢的缺点,采用多重网格加速技术,建立了适用于本文高精度全隐紧致格式的多重网格算法,从而大大加快了迭代收敛速度.数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性.     

对流扩散方程的迎风变换及相应有限差分方法

本文提出所谓迎风变换,将对流扩散方程分解为对流迎风函数和扩散方程,并构造相应的有限差分格式。对流迎风函数以简明的指数解析形式反映对流扩散现象的迎风效应,原则上消除了源于不对称对流算子的困难,能够便利对流扩散方程的数值求解。有限差分格式具有二阶精度和无条件稳定性,算例表明其准确性、收敛速度及对边界层效应的适应能力均明显优于中心差分格式和迎风差分格式。     

THREE HIGH-ORDER SPLITTING SCHEMES FOR 3D TRANSPORT EQUATION

IntroductionWith the development of modern industry, various pollutants discharge into the air,rivers, lakes and oceans, which makes the environmental qualities worse and has bad effectson the mankind’s health and the sustained development of industry an…     

二维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解二维非定常变系数对流扩散方程的一种时间二阶、空间四阶精度的三层全隐紧致差分格式。为了加快迭代求解隐格式时在每一个时间步上的收敛速度,采用多重网格加速技术,建立了适用于本文高精度金隐紧致格式的多重网格算法。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

Explicit characteristic‐based finite volume element method for level set equation

Accurate modeling of interfacial flows requires a realistic representation of interface topology. To reduce the computational effort from the complexity of the interface topological changes, the level set method is widely used for solving two‐phase flow problems. This paper presents an explicit characteristic‐based finite volume element method for solving the two‐dimensional level set equation. The method is applicable for the case of non‐divergence‐free velocity field. Accuracy and performance of the proposed method are evaluated via test cases with prescribed velocity fields on structured grids. By given a velocity field, the motion of interface in the normal direction and the mean curvature, examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time. Copyright © 2010 John Wiley & Sons, Ltd.