Generalization of the Spline Interpolation Based on the Principle of the Compact Schemes

In this work the authors extend the high order compact difference schemes to the matching technique to develop a Local Matched Reconstruction theory that can be also considered as a generalization of the spline theory. The problem of the high order reconstructions correlated to an optimal matching in overlapping regions for contiguous expansions in one or more dimensions is stressed; some new generalized matched interpolations and their related numerical schemes are presented together with Fourier analysis of errors. Finally, some relevant aspects of the computational efforts associated to the various approaches are discussed.     

High-order Compact Schemes for Nonlinear Dispersive Waves

High-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge–Kutta scheme are applied to simulate nonlinear dispersive wave propagation problems described the Korteweg-de Vries (KdV)-like equations, which involve a third derivative term. Several examples such as KdV equation, and KdV-Burgers equation are presented and the solutions obtained are compared with some other numerical methods. Computational results demonstrate that high-order compact schemes work very well for problems involving a third derivative term.     

基于非等距网格高阶紧致差分格式的多重网格算法研究

本文结合非等距网格高精度紧致差分格式的优越性与多重网格方法的快速收敛性,求解二维对流扩散方程。研究结果表明,对于处理物理量在不同的空间方向呈现不同的性态特征或不同变化规律的物理问题时,用非等距网格离散的四阶紧致格式的多重网格算法和二阶中心差分格式的多重网格算法都比等距网格离散得高效。同时,在非等距网格下下,部分半粗化多重网格算法比完全粗化多重网格算法具有更高的计算效率。针对不同的松弛算子对误差残量的磨光效果比较研究表明,线松弛算子是最高效的。而且,非等距网格离散的高精度紧致格式的多重网格算法对于对流扩散问题中大网格雷诺数情形也是收敛的。     

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.     

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996; and J. Comput. Phys. 133(2), 1997) and Ditkowski (Ph.D. thesis, Tel Aviv University, 1997), for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the Stefan problem. Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher than 2nd-order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies, that the numerical solution remains consistent and valid for a long integration time. A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2nd and a 4th-order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method. This research was supported by the Israel Science Foundation (grant No. 1362/04).     

TVD Fluxes for the High-Order ADER Schemes

In this paper we propose to use a TVD flux, instead of a first-order monotone flux, as the building block for designing very high-order methods; we implement the idea in the context of ADER schemes via a new flux expansion. Systematic assessment of the new schemes shows substantial gains in accuracy; these are particularly evident for problems involving long time evolution     

Application of Compact Schemes to Large Eddy Simulation of Turbulent Jets

We present 3-D large eddy simulation (LES) results for a turbulent Mach 0.9 isothermal round jet at a Reynolds number of 100,000 (based on jet nozzle exit conditions and nozzle diameter). Our LES code is part of a Computational Aeroacoustics (CAA) methodology that couples surface integral acoustics techniques such as Kirchhoff's method and the Ffowcs Williams– Hawkings method with LES for the far field noise estimation of turbulent jets. The LES code employs high-order accurate compact differencing together with implicit spatial filtering and state-of-the-art non-reflecting boundary conditions. A localized dynamic Smagorinsky subgrid-scale (SGS) model is used for representing the effects of the unresolved scales on the resolved scales. A computational grid consisting of 12 million points was used in the present simulation. Mean flow results obtained in our simulation are found to be in very good agreement with the available experimental data of jets at similar flow conditions. Furthermore, the near field data provided by the LES is coupled with the Ffowcs Williams–Hawkings method to compute the far field noise. Far field aeroacoustics results are also presented and comparisons are made with experimental measurements of jets at similar flow conditions. The aeroacoustics results are encouraging and suggest further investigation of the effects of inflow conditions on the jet acoustic field.     

Finite Volume Formulation of Compact Upwind and Central Schemes with Artificial Selective Damping

The present paper describes the use of compact upwind and compact central schemes in a Finite Volume formulation with an extension towards arbitrary meshes. The different schemes are analyzed and tested on several numerical experiments. A new formulation of artificial selective damping that is applicable on non-uniform Cartesian meshes is presented. Results are shown for a 1D advection equation, a 2D rotating Gaussian pulse and a subsonic inviscid vortical flow on uniform and non-uniform meshes and for a non-linear acoustic pulse.     

On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws

We discuss a new fifth-order, semi-discrete, central-upwind scheme for solving one-dimensional systems of conservation laws. This scheme combines a fifth-order WENO reconstruction, a semi-discrete central-upwind numerical flux, and a strong stability preserving Runge–Kutta method. We test our method with various examples, and give particular attention to the evolution of the total variation of the approximations.     

Compact high-order schemes for the Euler equations

An implicit approximate factorization (AF) algorithm is constructed, which has the following characteristics.

Bounded Error Schemes for the Wave Equation on Complex Domains

This paper considers the application of the method of boundary penalty terms (SAT) to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell’s equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g., the staggered Yee scheme)—we achieve a decrease of two orders of magnitude in the level of the L₂-error.     

Staggered Finite Difference Schemes for Conservation Laws

In this work, we introduce new finite-difference shock-capturing central schemes on staggered grids. Staggered schemes may have better resolution of the corresponding unstaggered schemes of the same order. They are based on high-order nonoscillatory reconstruction (ENO or WENO), and a suitable ODE solver for the computation of the integral of the flux. Although they suffer from a more severe stability restriction, they do not require a numerical flux function. A comparison of the new schemes with high-order finite volume (on staggered and unstaggered grids) and high order unstaggered finite difference methods is reported.     

High Accuracy Compact Schemes and Gibbs' Phenomenon

Compact difference schemes have been investigated for their ability to capture discontinuities. A new proposed scheme (Sengupta, Ganerwal and De (2003). J. Comp. Phys. 192(2), 677.) is compared with another from the literature Zhong (1998). J. Comp. Phys. 144, 622 that was developed for hypersonic transitional flows for their property related to spectral resolution and numerical stability. Solution of the linear convection equation is obtained that requires capturing discontinuities. We have also studied the performance of the new scheme in capturing discontinuous solution for the Burgers equation. A very simple but an effective method is proposed here in early diagnosis for evanescent discontinuities. At the discontinuity, we switch to a third order one-sided stencil, thereby retaining the high accuracy of solution. This produces solution with vastly reduced Gibbs' phenomenon of the solution. The essential causes behind Gibbs' phenomenon is also explained.     

Compact finite volume methods for the diffusion equation

We describe an approach to treating initial-boundary-value problems by finite volume methods in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, we are able to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl=0 and curl grad=0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second-order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and ADI methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results.     

一种基于多相带通结构的信道化滤波器算法

针对信道化滤波器要求运算速度快、消耗资源多、难以实时处理的突出问题,从多相滤波器,信道化滤波器的结构、原理和运算效率分析出发,推导了一种基于多相带通结构的信道化滤波器算法模型。这种算法将现有多相结构信道化滤波器模型中的低通设计改为带通设计,实现了复数乘法运算全部集中在带通滤波环节当中,并采用协调分级DFT算法的实现方案,大幅度节省了硬件资源,提高了运算效率,实现了信道化滤波器在通用FP—GA和DSP芯片中的实时处理,硬件仿真结果验证了算法模型的正确性和有效性。     

Cahn-Hilliard方程的紧致指数时间差分法研究

我们将一种快速稳定的紧致指数时间差分法应用于求解Cahn-Hilliard方程,并采用多步逼近法和龙格库塔方法有效地解决了方程中的非线性项带来的稳定问题。通过与经典的半隐式欧拉方法对比,分别对不同体自由能模型和不同扩散迁移率下的相场方程求解进行收敛性测试,验证了算法的正确性和高效性。最后我们用提出的方法对Flory-Huggins模型的粗化率进行研究,得到了与理论预测值一致的结果。     

A Note on Two Classical Enhancement Filters and Their Associated PDE's

We establish in 2D, the P.D.E. associated with a classical image enhancement filter, the Kramer operator and compare it with another classical shock filter, the Osher-Rudin filter. We show that each one corresponds to a non-flat mathematical morphology operator conditioned by a the sign of an edge detector. In the case of the Kramer operator, the equation is conditioned by the Canny edge detector while in the case of the original Rudin-Osher filter, the equation is conditioned by the sign of the Laplacian.     

High Order Fluctuation Schemes on Triangular Meshes

We develop a new class of schemes for the numerical solution of first-order steady conservation laws. The schemes are of the residual distribution, or fluctuation-splitting type. These schemes have mostly been developed in the context of triangular or tetrahedral elements whose degrees of freedom are their nodal values. We work here with more general elements that allow high-order accuracy. We introduce, for an arbitrary number of degrees of freedom, a simple mapping from a low-order monotone scheme to a monotone scheme that is as accurate as the degrees of freedom will allow. Proofs of consistency, convergence and accuracy are presented, and numerical examples from second, third and fourth-order schemes.     

积分清零与积分梳状滤波器在抽取滤波中的应用研究

抽取滤波器的设计是全数字软件接收机中的关键技术,选择合适的抽取滤波器可以使得效率和资源达到最佳的平衡.积分清零和积分梳状滤波器是两种实现简单、滤波性能较好的数字抽取滤波器.这两种滤波器在中频数字接收机中都有重要的应用.从原理上分析和研究了这两种滤波器频域响应,总结了积分梳状滤波器的设计方法.提出了两种滤波器结合应用的方法,并建立模型验证在性能方面的提高.在实际应用中,说明了两者在本质上相同但在结构上的差异,因此在应用中加以区分.通过两种滤波器在相干解调电路和同步电路中的仿真,进一步说明两种滤波器的设计和实现准则.     

Residual Distribution Schemes on Quadrilateral Meshes

We propose an investigation of the residual distribution schemes for the numerical approximation of two-dimensional hyperbolic systems of conservation laws on general quadrilateral meshes. In comparison to the use of triangular cells, usual basic features are recovered, an extension of the upwinding concept is given, and a Lax–Wendroff type theorem is adapted for consistency. We show how to retrieve many variants of standard first and second-order accurate schemes. They are proven to satisfy this theorem. An important part of this paper is devoted to the validation of these schemes by various numerical tests for scalar equations and the Euler equations system for compressible fluid dynamics on non Cartesian grids. In particular, second-order accuracy is reached by an adaptation of the Linearity preserving property to quadrangle meshes. We discuss several choices as well as the convergence of iterative method to steady state. We also provide examples of schemes that are not constructed from an upwinding principle