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Numerical Error (numerical + error)
Selected AbstractsNumerical errors of the volume-of-fluid interface tracking algorithmINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2002Gregor, erne Abstract One of the important limitations of the interface tracking algorithms is that they can be used only as long as the local computational grid density allows surface tracking. In a dispersed flow, where the dimensions of the particular fluid parts are comparable or smaller than the grid spacing, several numerical and reconstruction errors become considerable. In this paper the analysis of the interface tracking errors is performed for the volume-of-fluid method with the least squares volume of fluid interface reconstruction algorithm. A few simple two-fluid benchmarks are proposed for the investigation of the interface tracking grid dependence. The expression based on the gradient of the volume fraction variable is introduced for the estimation of the reconstruction correctness and can be used for the activation of an adaptive mesh refinement algorithm. Copyright © 2002 John Wiley & Sons, Ltd. [source] Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Linearized and non-linear acoustic/viscous splitting techniques for low Mach number flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2003Mohammad Farshchi Abstract Computation of the acoustic disturbances generated by unsteady low-speed flow fields including vortices and shear layers is considered. The equations governing the generation and propagation of acoustic fluctuations are derived from a two-step acoustic/viscous splitting technique. An optimized high order dispersion,relation,preserving scheme is used for the solution of the acoustic field. The acoustic field generated by a corotating vortex pair is obtained using the above technique. The computed sound field is compared with the existing analytic solution. Results are in good agreement with the analytic solution except near the centre of the vortices where the acoustic pressure becomes singular. The governing equations for acoustic fluctuations are then linearized and solved for the same model problem. The difference between non-linear and linearized solutions falls below the numerical error of the simulation. However, a considerable saving in CPU time usage is achieved in solving the linearized equations. The results indicate that the linearized acoustic/viscous splitting technique for the simulation of acoustic fluctuations generation and propagation by low Mach number flow fields seems to be very promising for three-dimensional problems involving complex geometries. Copyright © 2003 John Wiley & Sons, Ltd. [source] Valence and extra-valence orbitals in main group and transition metal bondingJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 1 2007C. R. Landis Abstract We address the issue first raised by Maseras and Morokuma with regard to the questionable treatment of empty p-orbitals in the algorithm for natural atomic/bond orbitals (NAOs, NBOs) and associated natural population analysis. We quantify this issue in terms of the numerical error (root-mean-square density deviation) resulting from the two alternative treatments of empty p-sets, leading to distinct NAOs, atomic charges, and idealized Lewis structural representations. Computational application of this criterion to a broad spectrum of main group and transition group species (employing both single- and multi-structure resonance models) reveals the interesting general pattern of (i) relatively insignificant differences for normal-valent species, where a single resonance structure is usually adequate, but (ii) clear superiority of the standard NAO algorithm for hypervalent species, where multi-resonance character is pronounced. These comparisons show how the divisive issue of "valence shell expansion" in transition metal bonding is deeply linked to competing conceptual models of hypervalency (viz., "p-orbital participation" in skeletal hybridization vs. 3c/4e resonance character). The results provide a quantitative measure of superiority both for the standard NAO evaluation of atomic charges as well as the general 3c/4e (A: B-C , A-B :C resonance) picture of main- and transition-group hypervalency. © 2006 Wiley Periodicals, Inc. J Comput Chem, 2007 [source] Numerical instabilities in the computation of pseudopotential matrix elementsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 2 2006Christoph van Wüllen Abstract Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion. © 2005 Wiley Periodicals, Inc., J Comput Chem 27: 135,141 2006 [source] ADI-FDTD method perturbed by the second order cross derivative termsMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 7 2008Ki-Bok Kong Abstract A two-step FDTD method as a compromise of conditional stability and reduced splitting error is formulated and its numerical stability is investigated. It is the perturbed form to the ADI-FDTD method by the addition of second order cross derivative term. It is validated from the comparison of numerical anisotropy and numerical error over the ADI-FDTD that numerical performances can be improved by controlling the perturbed term within the stable region of the cross derivative term. © Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1822,1826, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23479 [source] Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous mediaNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2010Florin A. Radu Abstract In this article, we analyze an Euler implicit-mixed finite element scheme for a porous media solute transport model. The transporting flux is not assumed given, but obtained by solving numerically the Richards equation, a model for subsurface fluid flow. We prove the convergence of the scheme by estimating the error in terms of the discretization parameters. In doing so we take into account the numerical error occurring in the approximation of the fluid flow. The article, is concluded by numerical experiments, which are in good agreement with the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] Radial basis collocation method and quasi-Newton iteration for nonlinear elliptic problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2008H.Y. Hu Abstract This work presents a radial basis collocation method combined with the quasi-Newton iteration method for solving semilinear elliptic partial differential equations. The main result in this study is that there exists an exponential convergence rate in the radial basis collocation discretization and a superlinear convergence rate in the quasi-Newton iteration of the nonlinear partial differential equations. In this work, the numerical error associated with the employed quadrature rule is considered. It is shown that the errors in Sobolev norms for linear elliptic partial differential equations using radial basis collocation method are bounded by the truncation error of the RBF. The combined errors due to radial basis approximation, quadrature rules, and quasi-Newton and Newton iterations are also presented. This result can be extended to finite element or finite difference method combined with any iteration methods discussed in this work. The numerical example demonstrates a good agreement between numerical results and analytical predictions. The numerical results also show that although the convergence rate of order 1.62 of the quasi-Newton iteration scheme is slightly slower than rate of order 2 in the Newton iteration scheme, the former is more stable and less sensitive to the initial guess. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] Simulating Seepage into Mine Shafts and Tunnels with MODFLOWGROUND WATER, Issue 3 2010Jacob Zaidel In cases when an equivalent porous medium assumption is suitable for simulating groundwater flow in bedrock aquifers, estimation of seepage into underground mine workings (UMWs) can be achieved by specifying MODFLOW drain nodes at the contact between water bearing rock and dewatered mine openings. However, this approach results in significant numerical problems when applied to simulate seepage into an extensive network of UMWs, which often exist at the mine sites. Numerical simulations conducted for individual UMWs, such as a vertical shaft or a horizontal drift, showed that accurate prediction of seepage rates can be achieved by either applying grid spacing that is much finer than the diameter/width of the simulated openings (explicit modeling) or using coarser grid with cell sizes exceeding the characteristic width of shafts or drifts by a factor of 3. Theoretical insight into this phenomenon is presented, based on the so-called well-index theory. It is demonstrated that applying this theory allows to minimize numerical errors associated with MODFLOW simulation of seepage into UMWs on a relatively coarse Cartesian grid. Presented examples include simulated steady-state groundwater flow from homogeneous, heterogeneous, and/or anisotropic rock into a vertical shaft, a horizontal drift/cross-cut, a ramp, two parallel drifts, and a combined system of a vertical shaft connected to a horizontal drift. [source] Numerical error patterns for a scheme with hermite interpolation for 1 + 1 linear wave equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2004Zuojin Zhu Abstract Numerical error patterns were presented when the fourth-order scheme based on Hermite interpolation was used to solve the 1 + 1 linear wave equation. Since most non-linear equations for real systems can be converted into linear forms by using proper transformations, this study certainly pertains its practical significance. The analytical solution was obtained under inhomogeneous initial and boundary conditions. It was found that not only the Hurst index of an error train at a given position but also its spatial distribution is dependent on the ratio of temporal to spatial intervals. The solution process with the fourth-order scheme based on Hermite interpolation diverges as the ratio is greater than unity. The results show that regular error pattern and smaller maxima of absolute values of numerical errors can be obtained when the ratio is set as unity; while chaotic phenomena for the numerical error propagation process can appear when the ratio is less than unity. It was found that it is better to choose the ratio as unity for the numerical solution of 1 + 1 linear wave equation with the scheme; while other selections for the ratio in the scheme can bring about chaotic patterns for the numerical errors. Copyright © 2004 John Wiley & Sons, Ltd. [source] Conjugate filter approach for shock capturing,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 2 2003Yun Gu Abstract This paper introduces a new scheme for the numerical computation involving shock waves. The essence of the scheme is to adaptively implement a conjugate low-pass filter to effectively remove the accumulated numerical errors produced by a set of high-pass filters. The advantages of using such an adaptive algorithm are its controllable accuracy, relatively low cost and easy implementation. Numerical examples in one and two space dimensions are presented to illustrate the proposed scheme. Copyright © 2003 John Wiley & Sons, Ltd. [source] Guaranteed-quality triangular mesh generation for domains with curved boundariesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2002Charles Boivin Guaranteed-quality unstructured meshing algorithms facilitate the development of automatic meshing tools. However, these algorithms require domains discretized using a set of linear segments, leading to numerical errors in domains with curved boundaries. We introduce an extension of Ruppert's Delaunay refinement algorithm to two-dimensional domains with curved boundaries and prove that the same quality bounds apply with curved boundaries as with straight boundaries. We provide implementation details for two-dimensional boundary patches such as lines, circular arcs, cubic parametric curves, and interpolated splines. We present guaranteed-quality triangular meshes generated with curved boundaries, and propose solutions to some problems associated with the use of curved boundaries. Copyright © 2002 John Wiley & Sons, Ltd. [source] TLM for diffusion: consistent first time step.INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 3 2002Two-dimensional case Abstract In initializing a transmission line matrix (TLM) diffusion model it is necessary to consider both initial concentration (temperature) and initial flow. As usual, only one of them is given; an auxiliary formula is necessary to calculate the distribution for the first time step. It has been shown that the standard formula may introduce additional numerical errors (International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 1993; 6:135; International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 1993; 6:161) and that these errors can persist over many time steps. In this paper, we show how an initial modification to the normal TLM algorithm can remove such errors, and we demonstrate the method by applying it to a two-dimensional TLM heat diffusion model for a copper plate. Copyright © 2002 John Wiley & Sons, Ltd. [source] Role of the one-body Jastrow factor in the transcorrelated self-consistent field equationINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 7 2006Naoto Umezawa Abstract The one-body Jastrow factor has been introduced into the transcorrelated variational Monte Carlo (TC-VMC) method. The principal role of the one-body Jastrow factor in the Jastrow,Slater-type wave function is to prevent an unfavorable effect of the two-body Jastrow factor that alters the charge density. In the TC-VMC method, since the one-body orbitals are optimized by the transcorrelated self-consistent field (TC-SCF) equations, which take into account the electron,electron correlation interactions originating from the two-body Jastrow factor, the unfavorable effect of altering charge density can be avoided without introducing the one-body Jastrow factor. However, it is found that it is still better to incorporate a one-body Jastrow factor into the TC-VMC method for the practical effect of reducing numerical errors caused by the Monte Carlo sampling and the re-weighting calculations in solving the TC-SCF equations. Moreover, since the one-body Jastrow function adopted in the present work is constructed from the two-body Jastrow factor without increasing any variational parameter, the computational cost is not significantly increased. The preferable effect of the use of the one-body Jastrow factor in the TC-VMC calculation is demonstrated for atoms. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] Modeling And Solving An Engine Intake Manifold With Turbo Charger For Predictive ControlASIAN JOURNAL OF CONTROL, Issue 3 2006Long Xie ABSTRACT In this paper, we build the intake manifold model of an engine with a turbo charger and develop a high speed calculation algorithm for model-based predictive control in real time. The model is built according to the analysis of its thermodynamic and hydrodynamic characteristics and the sampled experiment data. The model equations are presented as a set of differential equations with condition selection (bifurcation) on the right hand side. The switching surface is divided into two parts, sliding and crossing. The sliding mode on the switching surface is analyzed in detail, and a calculation algorithm is proposed to remove illegal crossing caused by the numerical errors on this surface. Also, the control formula and the condition guiding the bifurcation between these two parts are demonstrated. Using this method, we can solve this model over the entire region of input throttle angles, the stability is greatly increased, and the calculation time is greatly reduced for real time control systems. [source] | |||||||||||||