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Truncation Error (truncation + error)
Selected AbstractsQuasi optimal finite difference method for Helmholtz problem on unstructured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010Daniel T. Fernandes Abstract A quasi optimal finite difference method (QOFD) is proposed for the Helmholtz problem. The stencils' coefficients are obtained numerically by minimizing a least-squares functional of the local truncation error for plane wave solutions in any direction. In one dimension this approach leads to a nodally exact scheme, with no truncation error, for uniform or non-uniform meshes. In two dimensions, when applied to a uniform cartesian grid, a 9-point sixth-order scheme is derived with the same truncation error of the quasi-stabilized finite element method (QSFEM) introduced by Babu,ka et al. (Comp. Meth. Appl. Mech. Eng. 1995; 128:325,359). Similarly, a 27-point sixth-order stencil is derived in three dimensions. The QOFD formulation, proposed here, is naturally applied on uniform, non-uniform and unstructured meshes in any dimension. Numerical results are presented showing optimal rates of convergence and reduced pollution effects for large values of the wave number. Copyright © 2009 John Wiley & Sons, Ltd. [source] Insights on a sign-preserving numerical method for the advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2009E. SousaArticle first published online: 17 DEC 200 Abstract In this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one-dimensional unsteady advection,diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first-order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second-order terms of the truncation error and the other considers additionally the third-order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd. [source] Incorporating spatially variable bottom stress and Coriolis force into 2D, a posteriori, unstructured mesh generation for shallow water modelsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009D. Michael Parrish Abstract An enhanced version of our localized truncation error analysis with complex derivatives (LTEA,CD ) a posteriori approach to computing target element sizes for tidal, shallow water flow, LTEA+CD , is applied to the Western North Atlantic Tidal model domain. The LTEA + CD method utilizes localized truncation error estimates of the shallow water momentum equations and builds upon LTEA and LTEA,CD-based techniques by including: (1) velocity fields from a nonlinear simulation with complete constituent forcing; (2) spatially variable bottom stress; and (3) Coriolis force. Use of complex derivatives in this case results in a simple truncation error expression, and the ability to compute localized truncation errors using difference equations that employ only seven to eight computational points. The compact difference molecules allow the computation of truncation error estimates and target element sizes throughout the domain, including along the boundary; this fact, along with inclusion of locally variable bottom stress and Coriolis force, constitute significant advancements beyond the capabilities of LTEA. The goal of LTEA + CD is to drive the truncation error to a more uniform, domain-wide value by adjusting element sizes (we apply LTEA + CD by re-meshing the entire domain, not by moving nodes). We find that LTEA + CD can produce a mesh that is comprised of fewer nodes and elements than an initial high-resolution mesh while performing as well as the initial mesh when considering the resynthesized tidal signals (elevations). Copyright © 2008 John Wiley & Sons, Ltd. [source] A posteriori pointwise error estimation for compressible fluid flows using adjoint parameters and Lagrange remainderINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005A. K. Alekseev Abstract The pointwise error of a finite-difference calculation of supersonic flow is discussed. The local truncation error is determined by a Taylor series with the remainder being in a Lagrange form. The contribution of the local truncation error to the total pointwise approximation error is estimated via adjoint parameters. It is demonstrated by numerical tests that the results of the numerical calculation of gasdynamics parameter at an observation point may be refined and an error bound may be estimated. The results of numerical tests for the case of parabolized Navier,Stokes are presented as an illustration of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd. [source] The use of classical Lax,Friedrichs Riemann solvers with discontinuous Galerkin methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3-4 2002W. J. Rider Abstract While conducting a von Neumann stability analysis of discontinuous Galerkin methods we discovered that the classic Lax,Friedrichs Riemann solver is unstable for all time-step sizes. We describe a simple modification of the Riemann solver's dissipation returns the method to stability. Furthermore, the method has a smaller truncation error than the corresponding method with an upwind flux for the RK2-DG(1) method. These results are verified upon testing. Copyright © 2002 John Wiley & Sons, Ltd. [source] Towards entropy detection of anomalous mass and momentum exchange in incompressible fluid flowINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2002G. F. Naterer An entropy-based approach is presented for assessment of computational accuracy in incompressible flow problems. It is shown that computational entropy can serve as an effective parameter in detecting erroneous or anomalous predictions of mass and momentum transport in the flow field. In the present paper, the fluid flow equations and second law of thermodynamics are discretized by a Galerkin finite-element method with linear, isoparametric triangular elements. It is shown that a weighted entropy residual is closely related to truncation error; this relationship is examined in an application problem involving incompressible flow through a converging channel. In particular, regions exhibiting anomalous flow behaviour, such as under-predicted velocities, appear together with analogous trends in the weighted entropy residual. It is anticipated that entropy-based error detection can provide important steps towards improved accuracy in computational fluid flow. Copyright © 2002 John Wiley & Sons, Ltd. [source] A Petrov,Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2001Seung-Buhm Woo Abstract A new finite element method is presented to solve one-dimensional depth-integrated equations for fully non-linear and weakly dispersive waves. For spatial integration, the Petrov,Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C2 -continuity. For the time integration an implicit predictor,corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth-order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd. [source] Distributed Gaussian discrete variable representationINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 1 2005Hasan Karabulut Abstract A discrete variable representation (DVR) made from distributed Gaussians gn(x) = e, (n = ,,, ,, ,) and its infinite grid limit is described. The infinite grid limit of the distributed Gaussian DVR (DGDVR) reduces to the sinc function DVR of Colbert and Miller in the limit c , 0. The numerical performance of both finite and infinite grid DGDVRs and the sinc function DVR is compared. If a small number of quadrature points are taken, the finite grid DGDVR performs much better than both infinite grid DGDVR and sinc function DVR. The infinite grid DVRs lose accuracy due to the truncation error. In contrast, the sinc function DVR is found to be superior to both finite and infinite grid DGDVRs if enough grid points are taken to eliminate the truncation error. In particular, the accuracy of DGDVRs does not get better than some limit when the distance between Gaussians d goes to zero with fixed c, whereas the accuracy of the sinc function DVR improves very quickly as d becomes smaller, and the results are exact in the limit d , 0. An analysis of the performance of distributed basis functions to represent a given function is presented in a recent publication. With this analysis, we explain why the sinc function DVR performs better than the infinite grid DGDVR. The analysis also traces the inability of Gaussians to yield exact results in the limit d , 0 to the incompleteness of this basis in this limit. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source] Hybrid Framework for Managing Uncertainty in Life Cycle InventoriesJOURNAL OF INDUSTRIAL ECOLOGY, Issue 6 2009Eric D. Williams Summary Life cycle assessment (LCA) is increasingly being used to inform decisions related to environmental technologies and polices, such as carbon footprinting and labeling, national emission inventories, and appliance standards. However, LCA studies of the same product or service often yield very different results, affecting the perception of LCA as a reliable decision tool. This does not imply that LCA is intrinsically unreliable; we argue instead that future development of LCA requires that much more attention be paid to assessing and managing uncertainties. In this article we review past efforts to manage uncertainty and propose a hybrid approach combining process and economic input,output (I-O) approaches to uncertainty analysis of life cycle inventories (LCI). Different categories of uncertainty are sometimes not tractable to analysis within a given model framework but can be estimated from another perspective. For instance, cutoff or truncation error induced by some processes not being included in a bottom-up process model can be estimated via a top-down approach such as the economic I-O model. A categorization of uncertainty types is presented (data, cutoff, aggregation, temporal, geographic) with a quantitative discussion of methods for evaluation, particularly for assessing temporal uncertainty. A long-term vision for LCI is proposed in which hybrid methods are employed to quantitatively estimate different uncertainty types, which are then reduced through an iterative refinement of the hybrid LCI method. [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] Computationally Efficient Algorithm For Frequency-Weighted Optimal H, Model ReductionASIAN JOURNAL OF CONTROL, Issue 3 2003Fen Wu ABSTRACT In this paper, a frequency-weighted optimal H, model reduction problem for linear time-invariant (LTI) systems is considered. The objective of this class of model reduction problems is to minimize H, norm of the frequency-weighted truncation error between a given LTI system and its lower order approximation. A necessary and sufficient solvability condition is derived in terms of LMIs with one extra coupling rank constraint, which generally leads to a non-convex feasibility problem. Moreover, it has been shown that the reduced-order model is stable when both stable input and output weights are included, and its state-space data are given explicitly by the solution of the feasibility problem. An efficient model reduction scheme based on cone complementarity algorithm (CCA) is proposed to solve the non-convex conditions involving rank constraint. [source] Incorporating spatially variable bottom stress and Coriolis force into 2D, a posteriori, unstructured mesh generation for shallow water modelsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009D. Michael Parrish Abstract An enhanced version of our localized truncation error analysis with complex derivatives (LTEA,CD ) a posteriori approach to computing target element sizes for tidal, shallow water flow, LTEA+CD , is applied to the Western North Atlantic Tidal model domain. The LTEA + CD method utilizes localized truncation error estimates of the shallow water momentum equations and builds upon LTEA and LTEA,CD-based techniques by including: (1) velocity fields from a nonlinear simulation with complete constituent forcing; (2) spatially variable bottom stress; and (3) Coriolis force. Use of complex derivatives in this case results in a simple truncation error expression, and the ability to compute localized truncation errors using difference equations that employ only seven to eight computational points. The compact difference molecules allow the computation of truncation error estimates and target element sizes throughout the domain, including along the boundary; this fact, along with inclusion of locally variable bottom stress and Coriolis force, constitute significant advancements beyond the capabilities of LTEA. The goal of LTEA + CD is to drive the truncation error to a more uniform, domain-wide value by adjusting element sizes (we apply LTEA + CD by re-meshing the entire domain, not by moving nodes). We find that LTEA + CD can produce a mesh that is comprised of fewer nodes and elements than an initial high-resolution mesh while performing as well as the initial mesh when considering the resynthesized tidal signals (elevations). Copyright © 2008 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] A Crank,Nicholson-based unconditionally stable time-domain algorithm for 2D and 3D problemsMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 2 2007Xin Xie Abstract It has been shown that both ADI-FDTD and CN-FDTD are unconditionally stable. While the ADI is a second-order approximation, CN is only in the first order. However, analytical expressions reveal that the CN-FDTD has much smaller truncation errors and is more accurate than the ADI-FDTD. Nonetheless, it is more difficult to implement the CN than the ADI, especially for 3D problems. In this paper, we present an unconditionally stable time-domain method, CNRG-TD, which is based upon the Crank,Nicholson scheme and implemented with the Ritz,Galerkin procedure. We provide a physically meaningful stability proof, without resorting to tedious symbolic derivations. Numerical examples of the new method demonstrate high precision and high efficiency. In a 2D capacitance problem, we have enlarged the time step, ,t, 400 times of the CFL limit, yet preserved good accuracy. In the 3D antenna case, we use the time step, ,t, 7.6 times larger that that of the ADI-FDTD i.e., more than 38 times of the CFL limit, with excellent agreement of the benchmark solution. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 261,265, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22101 [source] Systematic intensity errors caused by spectral truncation: origin and remedyACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2001A. T. H. Lenstra The wavelength dispersion of graphite(002)-monochromated X-ray beams has been determined for a Cu, a Mo and an Rh tube. The observed values for ,,, were 0.03, 0.14 and 0.16, respectively. The severe reduction in monochromaticity as a function of wavelength is determined by the absorption coefficient , of the monochromator. ,(monochromator) varies with ,3. For an Si monochromator with its much larger absorption coefficient, ,,, values of 0.03 were found, regardless of the X-ray tube. This value matches a beam divergence defined by the size of the focus and of the crystal. This holds as long as the monochromator acts as a mirror, i.e.,(monochromator) is large. In addition to monochromaticity, homogeneity of the X-ray beam is also an important factor. For this aspect the mosaicity of the monochromator is vital. In cases like Si, in which mosaicity is practically absent, the reflected X-ray beam shows an intensity distribution equal to the mass projection of the filament on the anode. Smearing by mosaicity generates a homogeneous beam. This makes a graphite monochromator attractive in spite of its poor performance as a monochromator for , < 1,Å. This choice means that scan-angle-induced spectral truncation errors are here to stay. These systematic intensity errors can be taken into account after measurement by a software correction based on the real beam spectrum and the applied measuring mode. A spectral modeling routine is proposed, which is applied on the graphite-monochromated Mo K, beam. Both elements in that spectrum, i.e. characteristic ,1 and ,2 emission lines and the Bremsstrahlung, were analyzed using the 6318 reflection of Al2O3 (s = 1.2,Å,1). The spectral information obtained was used to calculate the truncation errors for intensities measured in an ,2, scan mode. The results underline the correctness of previous work on the structure of NiSO4·6H2O [Rousseau, Maes & Lenstra (2000). Acta Cryst. A56, 300,307]. [source] | |||||||||||||