GENERALIZED UPWIND SCHEME WITH FRACTIONAL STEPS FOR 3-D PROBLEM OF CONVECTION DOMINATING GROUNDWATER TRANSPORT

A generalized upwind scheme with fractional steps for 3-D mathematical models of convection dominating groundwater quality is suggested. The mass transport equation is split into a convection equation and a dispersive equation. The generalized upwind scheme is used to solve the convection equation and the finite element method is used to compute the dispersive equation.These procedures which not only overcome the phenomenon of the negative concentration and numerical dispersion appear frequently with normal FEM or FDM to solve models of convection dominating groundwater transport but also avoid the step for computing each node velocity give a more suitable method to calculate the concentrations of the well points.

Central Upwind Scheme for Solving Multivariate Cell Population Balance Models

Microbial cultures are comprised of heterogeneous cells that differ according to their size and intracellular concentrations of DNA, proteins and other constituents. Because of the included level of details, multi-variable cell population balance models (PBMs) offer the most general way to describe the complicated phenomena associated with cell growth, substrate consumption and product formation. For that reason, solving and understanding of such models are essential to predict and control cell growth in the processes of biotechnological interest. Such models typically consist of a partial integro-differential equation for describing cell growth and an ordinary integro-differential equation for representing substrate consumption. However, the involved mathematical complexities make their numerical solutions challenging for the given numerical scheme. In this article, the central upwind scheme is applied to solve the single-variate and bivariate cell population balance models considering equal and unequal partitioning of cellular materials. The validity of the developed algorithms is verified through several case studies. It was found that the suggested scheme is more reliable and effective.

The uniform convergence of upwind schemes on layer-adapted meshes for a singularly perturbed Robin BVP

In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.

Velocity Projection with Upwind Scheme Based on the Discontinuous Galerkin Methods for the Two Phase Flow Problem

The upwind scheme is very important in the numerical approximation of some problems such as the convection dominated problem, the two-phase flow problem, and so on. For the fractional flow formulation of the two-phase flow problem, the Penalty Discontinuous Galerkin (PDG) methods combined with the upwind scheme are usually used to solve the phase pressure equation. In this case, unless the upwind scheme is taken into consideration in the velocity reconstruction, the local mass balance cannot hold exactly. In this paper, we present a scheme of velocity reconstruction in some H(div) spaces with considering the upwind scheme totally. Furthermore, the different ways to calculate the nonlinear coefficients may have distinct and significant effects, which have been investigated by some authors. We propose a new algorithm to obtain a more effective and stable approximation of the coefficients under the consideration of the upwind scheme.

PARALLELIZED UPWIND FLUX SPLITTING SCHEME FOR SUPERSONIC REACTING FLOWS ON UNSTRUCTURED HYBRID MESHES

A parallelized upwind flux splitting scheme for supersonic reacting flows on hybrid meshes is presented. The complexity of super/hyper-sonic combustion flows makes it necessary to establish solvers with higher resolution and efficiency for multi-component Euler/N-S equations. Hence, a spatial second-order van Leer type flux vector splitting scheme is established by introducing auxiliary points in interpolation, and a domain decomposition method used on unstructured hybrid meshes for obtaining high calculating efficiency. The numerical scheme with five-stage Runge-Kutta time step method is implemented to the simulation of combustion flows, including the supersonic hydrogen/air combustion and the normal injection of hydrogen into reacting flows. Satisfying results are obtained compared with limited references.

CONVERGENCE OF AN IMMERSED INTERFACE UPWIND SCHEME FOR LINEAR ADVECTION EQUATIONS WITH PIECEWISE CONSTANT COEFFICIENTS I:L^1-ERROR ESTIMATES

We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L^l-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L^l-error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].

A study of parameter-free shock capturing upwind schemes on low speeds' issues

Nowadays,the upwind schemes are in a rapid development to capture shock accurately.However,these upwind schemes’properties at low speeds,such as their reconstruction scheme dependencies,grid dependencies,and Mach number dependencies,are concerned by few people.In this paper,a systematic study on their low speeds’issues is conducted.Through a series of tests,we can find that most parameter-free upwind schemes,widely used in practice today,are not applicable to low speeds’simulations.In contrast,SLAU and SLAU2 can give reliable results.Also,the upwind scheme’s influence on the accuracy is stronger than the reconstruction scheme’s influence at low speeds.

Third-Order Upwind Schemes for Convection Equations

Aim To construct a third order upwind scheme for convection equation. Methods Upwind Lagrange interpolation was used. Results and Conclusion The schemes L p stability for p∈ is proved. Numerical examples show that performance of the third order upwind scheme is better than that of most second order schemes.

Further Study on Errors in Metric Evaluation by Linear Upwind Schemes with Flux Splitting in Stationary Grids

The importance of eliminating errors in grid-metric evaluation for highorder difference schemes has been widely recognized in recent years,and it is known from the proof by Vinokur and Yee(NASA TM 209598,2000)that when conservative derivations of grid metric are used by Thomas,Lombard and Neier(AIAA J.,1978,17(10)and J.Spacecraft and rocket,1990,27(2)),errors caused by metric evaluation could be eliminated by linear schemes when flux splitting is not considered.According to the above achievement,central schemes without the use of flux splitting could fulfill the requirement of error elimination.Difficulties will arise for upwind schemes to attain the objective when the splitting is considered.In this study,further investigations are made on three aspects:Firstly,an idea of central scheme decomposition is introduced,and the procedure to derive the central scheme is proposed to evaluate grid metrics only.Secondly,the analysis has been made on the requirement of flux splitting to acquire free-stream preservation,and a Lax-Friedrichs-type splitting scheme is proposed as an example.Discussions about current study with that by Nonomura et al.(Computers and Fluids,2015,107)have been made.Thirdly,for halfnode-or mixed-type schemes,interpolations should be used to derive variables at half nodes.The requirement to achieve metric identity on this situation is analyzed and an idea of directionally consistent interpolation is proposed,which is manifested to be indispensable to avoid violations of metric identity and to eliminate metric-caused errors thereafter.Two numerical problems are tested,i.e.,the free-stream and vortex preservation onwavy,largely randomized and triangular-like grids.Numerical results validate aforementioned theoretical outcomes.

A High-Order NVD/TVD-Based Polynomial Upwind Scheme for the Modified Burgers’ Equations

A bounded high order upwind scheme is presented for the modified Burgers’equation by using the normalized-variable formulation in the finite volume framework.The characteristic line of the present scheme in the normalizedvariable diagram is designed on the Hermite polynomial interpolation.In order to suppress unphysical oscillations,the present scheme respects both the TVD(total variational diminishing)constraint and CBC(convection boundedness criterion)condition.Numerical results demonstrate the present scheme possesses good robustness and high resolution for the modified Burgers’equation.

Numerical Analysis of Upwind Difference Schemes for Two-Dimensional First-Order Hyperbolic Equations with Variable Coefficients

In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.

UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

WT5,5”BX] A new class of numerical schemes is proposed to solve convection diffusion equations by combining the upwind technique and the method of operator splitting. For every time step, the multi dimensional approximation is performed in several independent directions alternatively, while the upwind technique is applied to treat the convection term in every individual direction. This scheme possesses maximum principle. Stability and convergence are analysed by energy method.[WT5,5”HZ]

一维平流方程迎风格式的最优时空步长

在考虑舍入误差影响的情况下,研究一维平流方程迎风格式最优时空步长的选取.首先,分析每一时间层产生的离散误差和舍入误差,以及2种误差向高时间层传播的累积,得到数值解总误差的理论上界;然后推导出最优时间步长和最优空间步长的理论公式,进而得到2种不同机器精度下最优时间步长之比满足的一个仅与机器精度有关的普适关系;最后通过数值算例验证了结论的可靠性.

Aerodynamic/stealth design of S-duct inlet based on discrete adjoint method

It is a major challenge for the airframe-inlet design of modern combat aircrafts,as the flow and electromagnetic wave propagation in the inlet of stealth aircraft are very complex.In this study,an aerodynamic/stealth optimization design method for an S-duct inlet is proposed.The upwind scheme is introduced to the aerodynamic adjoint equation to resolve the shock wave and flow separation.The multilevel fast multipole algorithm(MLFMA)is utilized for the stealth adjoint equation.A dorsal S-duct inlet of flying wing layout is optimized to improve the aerodynamic and stealth characteristics.Both the aerodynamic and stealth characteristics of the inlet are effectively improved.Finally,the optimization results are analyzed,and it shows that the main contradiction between aerodynamic characteristics and stealth characteristics is the centerline and crosssectional area.The S-duct is smoothed,and the cross-sectional area is increased to improve the aerodynamic characteristics,while it is completely opposite for the stealth design.The radar cross section(RCS)is reduced by phase cancelation for low frequency conditions.The method is suitable for the aerodynamic/stealth design of the aircraft airframe-inlet system.

迎风格式在接触间断的数值耗散及其诱导误差

在不同流场参数条件下对三种迎风格式在接触间断中的数值耗散问题进行了数值实验,并对数值耗散产生的机理进行了分析。数值计算结果和理论分析表明,矢通量分裂格式计算接触间断问题时,若流场静止或流场内存在亚声速区域,密度耗散的产生会诱导出以特征速度运动的数值扰动误差,该误差对数值耗散的大小无影响,但会影响流场的速度及压力参数分布,从而改变流场的结构。在二维问题中,诱导误差相互干扰会产生大量的复杂小尺度结构,给流场结构分析带来困难。同时,在密度参数线性分布的流场中,若空间离散格式重构的对象为对流通量,使用矢通量分裂格式计算流场会产生数值误差,使计算精度难以到达二阶。

Uniform Convergence Analysis of Finite Difference Scheme for Singularly Perturbed Delay Differential Equation on an Adaptively Generated Grid

Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.

迎风格式在水滴欧拉方程数值求解中的应用

数值模拟是研究飞机结冰的重要手段,为获取水滴撞击特性,通常采用欧拉法求解水滴运动方程。液态水含量在空间分布上存在间断,会导致水滴欧拉方程的求解极易发散;若采用一阶格式,虽然稳定性得到提高,但计算精度难以保证满足工程设计需求;有研究者通过添加扩散项以提升收敛性,但会人为增加数值耗散。针对这一问题,本文采用一阶迎风格式对水滴欧拉方程的对流通量进行离散,并通过线性重构与限制器函数保证计算精度达到二阶,建立一种基于迎风离散格式的水滴欧拉方程二阶求解方法。通过对比NACA0012、NACA23012和NLF0414翼型水滴撞击特性计算结果与FENSAP-ICE软件的结果,验证了该方法的有效性与计算精度。

GROUP VELOCITY CONTROL SCHEME WITH LOW DISSIPATION

In order to prevent smearing the discontinuity, a modified term is added to the third order Upwind Compact Difference scheme to lower the dissipation error. Moreover, the dispersion error is controled to hold back the non physical oscillation by means of the group velocity control. The scheme is used to simulate the interactions of shock density stratified interface and the disturbed interface developing to vortex rollers. Numerical results are satisfactory.

NUMERICAL SCHEMES WITH HIGH ORDER OF ACCURACY FOR THE COMPUTATION OF SHOCK WAVES

High order accurate scheme is highly desirable for Slow computation with shocks. After analysis has been made for the reason of the generation of non-physical oscillations around the shock in numerical computations, a third-order, upwind biased, shock capturing scheme was proposed. Also, a new shock fitting method, called pseudo shock fitting method, was suggested, which in principle can be with any order of accuracy. Test cases for one dimensional flows show that the new method is very satisfactory.

Comments on Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

The explicit compact difference scheme, proposed in Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD by Lin et al., published in Applied Mathematics and Mechanics （English Edition）, 2007, 28（7）, 943-953, has the same performance as the conventional finite difference schemes. It is just another expression of the conventional finite difference schemes. The proposed expression does not have the advantages of a compact difference scheme. Nonetheless, we can more easily obtain and implement compared with the conventional expression in which the coefficients can only be obtained by solving equations, especially for higher accurate schemes.