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Finite-difference Scheme (finite-difference + scheme)
Selected AbstractsMultilayer Analytic Element Modeling of Radial Collector WellsGROUND WATER, Issue 6 2005Mark Bakker A new multilayer approach is presented for the modeling of ground water flow to radial collector wells. The approach allows for the inclusion of all aspects of the unique boundary condition along the lateral arms of a collector well, including skin effect and internal friction losses due to flow in the arms. The hydraulic conductivity may differ between horizontal layers within the aquifer, and vertical anisotropy can be taken into account. The approach is based on the multilayer analytic element method, such that regional flow and local three-dimensional detail may be simulated simultaneously and accurately within one regional model. Horizontal flow inside a layer is computed analytically, while vertical flow is approximated with a standard finite-difference scheme. Results obtained with the proposed approach compare well to results obtained with three-dimensional analytic element solutions for flow in unconfined aquifers. The presented approach may be applied to predict the yield of a collector well in a regional setting and to compute the origin and residence time, and thus the quality, of water pumped by the collector well. As an example, the addition of three lateral arms to a collector well that already has three laterals is investigated. The new arms are added at an elevation of 2 m above the existing laterals. The yield increase of the collector well is computed as a function of the lengths of the three new arms. [source] A new numerical approach for solving high-order non-linear ordinary differential equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2003Songping Zhu Abstract There have been many numerical solution approaches to ordinary differential equations in the literature. However, very few are effective in solving non-linear ordinary differential equations (ODEs), particularly when they are of order higher than one. With modern symbolic calculation packages, such as Maple and Mathematica, being readily available to researchers, we shall present a new numerical method in this paper. Based on the repeated use of a symbolic calculation package and a second-order finite-difference scheme, our method is particularly suitable for solving high-order non-linear differential equations arising from initial-value problems. One important feature of our approach is that if the highest-order derivative in an ODE can be written explicitly in terms of all the other terms of lower orders, our method requires no iterations at all. On the other hand, if the highest-order derivative in an ODE cannot be written explicitly in terms of all the other lower-order terms, iterations are only required before the actual time marching begins. Copyright © 2003 John Wiley & Sons, Ltd. [source] Solving singularly perturbed Riccati equation with MathematicaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 6 2003M. D. Mikhailov Abstract An exponentially fitted finite-difference scheme of order one for singularly perturbed Riccati equation has been presented and tested on three problems in this journal (Selvakumar K. Commun. Numer. Meth. Engng 1997; 13: 1,12). This note demonstrates the superiority of Mathematica in solving the same problems. Copyright © 2003 John Wiley & Sons, Ltd [source] A higher-order predictor,corrector scheme for two-dimensional advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Chuanjian Man Abstract A higher-order accurate numerical scheme is developed to solve the two-dimensional advection,diffusion equation in a staggered-grid system. The first-order spatial derivatives are approximated by the fourth-order accurate finite-difference scheme, thus all truncation errors are kept to a smaller order of magnitude than those of the diffusion terms. Therefore, there is no need to add an artificial diffusion term to balance the unwanted numerical diffusion. For the time derivative, the fourth-order accurate Adams,Bashforth predictor,corrector method is applied. The stability analysis of the proposed scheme is carried out using the Von Neumann method. It is shown that the proposed algorithm has good stability. This method also shows much less spurious oscillations than current lower-order accurate numerical schemes. As a result, the proposed numerical scheme can provide more accurate results for long-time simulations. The proposed numerical scheme is validated against available analytical and numerical solutions for one- and two-dimensional transport problems. One- and two-dimensional numerical examples are presented in this paper to demonstrate the accuracy and conservative properties of the proposed algorithm by comparing with other numerical schemes. The proposed method is demonstrated to be a useful and accurate modelling tool for a wide range of transport problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2005K. S. Erduran Abstract A hybrid scheme composed of finite-volume and finite-difference methods is introduced for the solution of the Boussinesq equations. While the finite-volume method with a Riemann solver is applied to the conservative part of the equations, the higher-order Boussinesq terms are discretized using the finite-difference scheme. Fourth-order accuracy in space for the finite-volume solution is achieved using the MUSCL-TVD scheme. Within this, four limiters have been tested, of which van-Leer limiter is found to be the most suitable. The Adams,Basforth third-order predictor and Adams,Moulton fourth-order corrector methods are used to obtain fourth-order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ,HYWAVE', based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi-chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd. [source] On the validity of the perturbation approach for the flow inside weakly modulated channelsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002H. Zhou The equations governing the flow of a viscous fluid in a two-dimensional channel with weakly modulated walls have been solved using a perturbation approach, coupled to a variable-step finite-difference scheme. The solution is assumed to be a superposition of a mean and perturbed field. The perturbation results were compared to similar results from a classical finite-volume approach to quantify the error. The influence of the wall geometry and flow Reynolds number have extensively been investigated. It was found that an explicit relation exists between the critical Reynolds number, at which the wall flow separates, and the dimensionless amplitude and wavelength of the wall modulation. Comparison of the flow shows that the perturbation method requires much less computational effort without sacrificing accuracy. The differences in predicted flow is kept well around the order of the square of the dimensionless amplitude, the order to which the regular perturbation expansion of the flow variables is carried out. Copyright © 2002 John Wiley & Sons, Ltd. [source] Toward accurate hybrid prediction techniques for cavity flow noise applicationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2009W. De Roeck Abstract A large variety of hybrid computational aeroacoustics (CAA) approaches exist differing from each other in the way the source region is modeled, in the way the equations are used to compute the propagation of acoustic waves in a non-quiescent medium, and in the way the coupling between source and acoustic propagation regions is made. This paper makes a comparison between some commonly used numerical methods for aeroacoustic applications. The aerodynamically generated tonal noise by a flow over a 2D rectangular cavity is investigated. Two different cavities are studied. In the first cavity (L/D=4, M=0.5), the sound field is dominated by the cavity wake mode and its higher harmonics, originating from a periodical vortex shedding at the cavity leading edge. In the second cavity (L/D=2, M=0.6), shear-layer modes, due to flow-acoustic interaction phenomena, generate the major components in the noise spectrum. Source domain modeling is carried out using a second-order finite-volume large eddy simulation. Propagation equations, taking into account convection and refraction effects, are solved using high-order finite-difference schemes for the linearized Euler equations and the acoustic perturbation equations. Both schemes are compared with each other for various coupling methods between source region and acoustic region. Conventional acoustic analogies and Kirchhoff methods are rewritten for the various propagation equations and used to obtain near-field acoustic results. The accuracy of the various coupling methods in identifying the noise-generating mechanisms is evaluated. In this way, this paper provides more insight into the practical use of various hybrid CAA techniques to predict the aerodynamically generated sound field by a flow over rectangular cavities. Copyright © 2009 John Wiley & Sons, Ltd. [source] Stable high-order finite-difference methods based on non-uniform grid point distributionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2008Miguel Hermanns Abstract It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundström theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q,N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q=N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q On the influence of numerical schemes and subgrid,stress models on large eddy simulation of turbulent flow past a square cylinderINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2002A. Nakayama Abstract Influence of finite difference schemes and subgrid-stress models on the large eddy simulation calculation of turbulent flow around a bluff body of square cylinder at a laboratory Reynolds number, has been examined. It is found that the type and the order of accuracy of finite-difference schemes and the subgrid-stress model for satisfactory results are dependent on each other, and the grid resolution and the Reynolds number. Using computational grids manageable by workstation-level computers, with which the near-wall region of the separating boundary layer cannot be resolved, central-difference schemes of realistic orders of accuracy, either fully conservative or non-conservative, suffer stability problems. The upwind-biased schemes of third order and the Smagorinsky eddy-viscosity subgrid model can give reasonable results resolving much of the energy-containing turbulent eddies in the boundary layers and in the wake and representing the subgrid stresses in most parts of the flow. Noticeable improvements can be obtained by either using higher order difference schemes, increasing the grid resolution and/or by implementing a dynamic subgrid stress model, but each at a cost of increased computational time. For further improvements, the very small-scale eddies near the upstream corners and in the laminar sublayers need to be resolved but would require a substantially larger number of grid points that are out of the range of easily accessible computers. Copyright © 2002 John Wiley & Sons, Ltd. [source] Highly efficient finite-difference schemes for structures of nonrectangular cross-sectionMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 6 2002D. Z. Djurdjevic Abstract A novel finite-difference beam-propagation method (FD-BPM) algorithm for the analysis of photonics structures with sloped sides in the transverse plane is presented. By introducing nonorthogonal coordinate systems in the transverse plane, the staircasing effects inherent when one is modeling with a rectangular mesh are eliminated, and accuracy is substantially improved. The algorithm is tested in the analysis of sloped rib waveguides. The results obtained confirm the accuracy and high efficiency of the new algorithm. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 33: 401,407, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10335 [source] The Poisson equation with local nonregular similaritiesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2001Alexander Yakhot Abstract Moffatt and Duffy [1] have shown that the solution to the Poisson equation, defined on rectangular domains, includes a local similarity term of the form: r2log(r)cos(2,). The latter means that the second (and higher) derivative of the solution with respect to r is singular at r = 0. Standard high-order numerical schemes require the existence of high-order derivatives of the solution. Thus, for the case considered by Moffatt and Duffy, the high-order finite-difference schemes loose their high-order convergence due to the nonregularity at r = 0. In this article, a simple method is outlined to regain the high-order accuracy of these schemes, without the need of any modification in the scheme's algorithm. This is a significant consideration when one wants to use a given finite-difference computer code for problems with local nonregular similarity solutions. Numerical examples using the modified scheme in conjunction with a sixth-order finite difference approximation are provided. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:336,346, 2001 [source] On the finite-differences schemes for the numerical solution of two dimensional Schrödinger equationNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2002Murat Suba Abstract In this study three different finite-differences schemes are presented for numerical solution of two-dimensional Schrödinger equation. The finite difference schemes developed for this purpose are based on the (1, 5) fully explicit scheme, and the (5, 5) Noye-Hayman fully implicit technique, and the (3, 3) Peaceman and Rachford alternating direction implicit (ADI) formula. These schemes are second order accurate. The results of numerical experiments are presented, and CPU times needed for this problem are reported. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 752,758, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10029. [source]
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