61. Experience With The Solution Of A Finite Difference Discretization On Sparse Gri Translate this page independent of the meshsize. Keywords. finite differences, Sparse Grids, Cascadic Iteration, BiCGStab Kurzfassung. Vor kurzem haben wir http://www.bi.fraunhofer.de/publications/report/0098/

GMD Report No. 98 Pieter W. Hemker, Frauke Sprengel Experience with the Solution of a Finite Difference Discretization on Sparse Grids May 2000, 16 pages, fee 5.00 EUR REP-SCAI-2000-98 Abstract In a recent paper [10], we described and analyzed a finite difference discretization on adaptive sparse grids in three space dimensions. In this paper, we show how the discrete equations can be efficiently solved in an iterative process. Several alternatives have been studied before in Sprengel [16], where multigrid algorithms were used. Here, we report on our experience with BiCGStab iteration. It appears that, applied to the hierarchical representation and combined with Nested Iteration in a cascadic algorithm, BiCGStab shows fast convergence, although the convergence rate is not truly independent of the meshsize. Keywords Finite Differences, Sparse Grids, Cascadic Iteration, BiCGStab Kurzfassung Schlagworte This document is not available online. The purpose of the GMD Report is the dissemination of scientific work for scientific use. The commercial distribution of this document is prohibited, as is any modification of its content. ISSN: 1435-2702

62. Finite Difference Gradients of the nonlinear element functions, the derivatives will be approximated by finite differences whenever the FINITEDIFFERENCE-GRADIENTS keyword is specified. http://www.numerical.rl.ac.uk/lancelot/spec/node8.html

Next: The Second Derivative Approximations Up: Keywords Previous: Checking Derivatives Finite Difference Gradients keyword: FINITE-DIFFERENCE-GRADIENTS The user is assumed to have provided analytic expressions for the values of the nonlinear element functions in the SEIF file. The specification of analytic first derivatives is optional, but recommended whenever possible. If the user is unable to provide analytic first derivatives of the nonlinear element functions, the derivatives will be approximated by finite differences whenever the FINITE-DIFFERENCE-GRADIENTS keyword is specified. The user should note, however, that the accuracy of the solution obtained by LANCELOT may be compromised if finite-difference approximations are used.

63. Finite Differences Finite difference methods. The method of finite differences relies upon making finite approximations to the conventional derivative of a continuous function. http://www.tc.cornell.edu/~slantz/SPUR/SPUR94/Reports/David_html/DAVID_findiff.h

Finite difference methods The method of finite differences relies upon making finite approximations to the conventional derivative of a continuous function. Recall that a regular derivative can be defined as: A finite derivative, on the other hand can defined several ways, the simplest of which is: or Where xi is the ith element of the matrix x representing the continuous function x. [[Delta]] is the grid spacing, or the distance between each point in x. More complicated expressions can also be derived from similar principles, as shown by the definition below: Where f is a two-dimensional function, represented as a two dimensional matrix. It is assumed in this example that the grid spacing in both directions is the same, i.e. [[Delta]]x=[[Delta]]y=[[Delta]]. This is typically a desirable property for modeling physical systems, as it simplifies most calculations. In our model, we made this assumption. Back to paper

64. Finite Differences Of Arbitrary Order finite differences of Arbitrary Order http://www.omatrix.com/manual/diff.htm

65. Search ASCE'S Civil Engineering Database SacramentoSan Joaquin Delta 1993 Automatic Calibration of 3D finite differences Model Parameters Application to the Finnsjon Site (Sweden) 1993 Breakpoint http://www.pubs.asce.org/WWWsrchkwx.cgi?Finite difference method

66. A Finite Difference Calculation Of Impurity Migration In Semiconductors By The K of the simulation methods 3. 7 The implicit form of the finite differences method (IFDM...... Cs(0) = Cs(d) = Cs* the penetration profiles are given by http://www.ttp.net/3-908450-83-7/89.htm

Defects and Diffusion in Semiconductors First part of the paper. Find the complete, well formatted version here: www.scientific.net A Finite Difference Calculation of Impurity Migration in Semiconductors by the Kick-Out Mechanism E D C B A 0,2 0,4 0,6 0,8 1,0 0,0 x/d Fig. 11. Distribution profiles of self-interstitial atoms in a dislocation-free crystal ( d = 0cm-2) using small values of hm (hm = 10-6cm/s) for various diffusion times; A: 5min ; B: 1min; C: 2min; D: 6min; E : 20min ; F : 40min ; G : 60min ; H : 240min; I: 480min. b- Analyses of the penetration profiles. To determine the domain of validity corresponding to limiting-case solutions, we represent in Fig. 12 the variation y = erf ( Ln(C s C sm ) ) with reduced depth xr. In this representation we note at times t < 120min a good agreement between the numerical profiles and the analytical solution (12) corresponding to a regime controlled by interstitial atom migration. At times, 120min < t < 120min. The obtained value of the effective solute diffusivity from the fit of Eq. 12 to numerical data Deff = 1.10 x 10-7cm2.s-1 is close to the reduced interstitial diffusivity used in the simulation, Di* = Ci*/Cs* = 1.25 x 10-7cm2.s-1. After the intermediate regime, the concentration ~ C(d / 2) will be This is an extract of www.scientific.net

67. Finite Difference Programs electrical conductivity programs can be operated quite similarly to the finite element electrical conductivity programs, with some small differences given below http://ciks.cbt.nist.gov/garbocz/manual/node22.html

Next: Exact solutions for Up: Actual program operation Previous: Eigenstrain programs Finite difference programs The finite difference electrical conductivity programs can be operated quite similarly to the finite element electrical conductivity programs, with some small differences given below. (1) Since the finite difference programs only handle diagonal conductivity tensors, the local phase conductivity variable, sigma , is not a full tensor but only has space for the diagonal elements. (2) The subroutine DEMBX is not exited until the gtest criterion is met. The total number of conjugate gradient cycles possible within DEMBX is given by ncgsteps , a parameter that is set by the user. Intermediate results are printed out every ncheck cycles, from within DEMBX. (3) The main variables, like the voltage vector u , are dimensioned ns2 = (nx + 2) x (ny + 2) x (nz + 2) , rather than ns = nx x ny x nz like in the finite element programs. Also, these dimensions occur explicitly only in the main program, so only have to be changed there. There are other differences, but these are not in sections that must be changed by the user. Sections 2 and 3 of this manual and the comments in the actual programs should be consulted if the user wishes to know more about these differences.

68. A Vertical Finite-Difference Scheme For Hydrostatic And Nonhydrostatic Equations Vertical finitedifferences schemes that preserve the important integral properties of the continuum equations are derived for the basic hydrodynamic and http://ams.allenpress.com/amsonline/?request=get-abstract&issn=1520-0493&volume=

69. Finite Difference From MathWorld finite Difference from MathWorld The finite difference is the discrete analog of the derivative. The finite forward difference of a function f_p is defined as \Delta f_p\equiv f_{p+1}f_p, and http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/FiniteDifference.h

70. Bessel's Finite Difference Formula -- From MathWorld Bessel s finite Difference Formula. 3940, 1967. Eric W. Weisstein. Bessel s finite Difference Formula. From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/BesselsFiniteDifferenceFormula.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Applied Mathematics Numerical Methods Finite Differences Bessel's Finite Difference Formula An interpolation formula also sometimes known as for where is the central difference and where are the coefficients from Gauss's backward formula and Gauss's forward formula and and are the coefficients from Everett's formula . The s also satisfy for Everett's Formula search Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972. Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 90-91, 1990. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987. Whittaker, E. T. and Robinson, G. "The Newton-Bessel Formula." §24 in

71. FDTD.org Home finiteDifference Time-Domain Literature Database. Citations with most recent comments Numerical Modeling of Microstrip Circuits http://www.fdtd.org/

home about search theses ... feedback Certificate changed; details here Please log in! Finite-Difference Time-Domain Literature Database Citations with most recent comments: Numerical Modeling of Microstrip Circuits and Antennas , D. M. Sheen, Cambridge, MA, Massachusetts Institute of Technology, 1991. (bibkey=sheen1991m) Comment Page (30 comments to date, last on 2004-06-01 at 07:11) GTEM Fields FDTD Modeling , T. E. Harrington, IEEE Int. EMC Symp., Austin, TX, 614-619, August, 1997. (bibkey=harrington1997c) Comment Page (2 comments to date, last on 2004-03-21 at 04:23) Application of the Three-Dimensional Finite-Difference Time-Domain Method to the Analysis of Planar Microstrip Circuits , D. M. Sheen, S. M. Ali, M. D. Abouzahra, and J. A. Kong, IEEE Transactions on Microwave Theory and Techniques , vol. 38, no. 7, 849-857, July, 1990. (bibkey=sheen1990a) Comment Page (5 comments to date, last on 2004-03-18 at 18:36) Analyses of Self-Resonant Bent Antennas , M. Ali, University of Victoria, 1997. (bibkey=ali1997m) Comment Page (2 comments to date, last on 2004-03-07 at 10:47)

72. Untitled 164, Spring 2001Due DateName(s)Honors Project 6 finite DierencesIntroductionIn this project we view a sequence http://www.mapleapps.com/powertools/calcProjects/calc2honors/HP7-finiteDifferenc

73. The FDTD BibTeX Database The FDTD BibTeX Database John Schneider and Kurt Shlager at the School of Electrical Engineering and Computer Science at Washington State University provide the finiteDifference Time-Domain (FDTD http://rdre1.inktomi.com/click?u=http://www.fdtd.org/&y=0224348FE147C970&

74. Finite Difference finite Difference. Note If your WWW browser cannot display special symbols, like ² or 2 , then click here for the alternative finite Difference page. http://www.jimloy.com/algebra/finite.htm

Return to my Mathematics pages Go to my home page Finite Difference , then click here for the alternative Finite Difference page Let's say that you have some unknown function of x, y=f(x), which gives these values: x=0, y=5 x=1, y=0 x=2, y=1 x=3, y=20 x=4, y=69 x=5, y=160 x=6, y=305 And you would like to know which function fits those values. One possibility is an n-degree polynomial: y=ax +bx +cx +dx +ex +fx+g, for example. You could actually plug the above x and y values into this equation. Then you would have seven linear equations (like 1=64a+32b+16c+8d+4e+2f+g) with seven unknowns. And there are a few fairly easy ways to solve them, to get a, b, c... A valuable short-cut is called the Finite Difference method. We take the numbers in the table, and find their differences (between consecutive elements), then we find the differences between the differences, etc: x y diff1 diff2 diff3 diff4 5 -5 1 6 1 12 2 1 18 19 12 3 20 30 49 12 4 69 42 91 12 5 160 54 145 6 305 It can be shown that for an n-degree polynomial, the nth difference is constant (and the (n+1)th difference is 0). So our function is

75. Finite Difference finite Difference. Note This is the alternative finite Difference page, for WWW browser which cannot display special symbols. In http://www.jimloy.com/algebra/finitez.htm

Return to my Mathematics pages Go to my home page Finite Difference Note: This is the alternative Finite Difference page, for WWW browser which cannot display special symbols. In particular, I am using ^2 for "squared" in this version. Please let me know if this is working for you or not, by sending me email Return to the primary Finite Difference page Let's say that you have some unknown function of x, y=f(x), which gives these values: x=0, y=5 x=1, y=0 x=2, y=1 x=3, y=20 x=4, y=69 x=5, y=160 x=6, y=305 And you would like to know which function fits those values. One possibility is an n-degree polynomial: y=ax^6+bx^5+cx^4+dx^3+ex^2+fx+g, for example. You could actually plug the above x and y values into this equation. Then you would have seven linear equations (like 1=64a+32b+16c+8d+4e+2f+g) with seven unknowns. And there are a few fairly easy ways to solve them, to get a, b, c... A valuable short-cut is called the Finite Difference method. We take the numbers in the table, and find their differences (between consecutive elements), then we find the differences between the differences, etc: x y diff1 diff2 diff3 diff4 5 -5 1 6 1 12 2 1 18 19 12 3 20 30 49 12 4 69 42 91 12 5 160 54 145 6 305 It can be shown that for an n-degree polynomial, the nth difference is constant (and the (n+1)th difference is 0). So our function is

76. 9.3 Finite-difference Derivatives 9.3 finitedifference derivatives. finite-difference formulas make it possible to use arithmetic operations to determine derivatives. http://www.asp.ucar.edu/colloquium/1992/notes/part1/node72.html

Next: 9.4 Interpolation and extrapolation Up: 9. Numerical Methods Previous: 9.2 The Taylor series 9.3 Finite-difference derivatives Finite-difference formulas make it possible to use arithmetic operations to determine derivatives. They are based on the Taylor series truncated at various orders in the expansion. The first-order expansion, of course, just gives the form or where indicates that the error in this approximation is of order . If an index is used to indicate consecutive values at intervals , so that if f i f x ), then is called the first forward difference and is the first backward difference formula for the derivative. Similar expressions can be found for higher-order derivatives. For example, the second derivative is The corresponding backward-difference formula is The preceding formulas were obtained by taking only the first term in the Taylor series containing the desired derivative. More accurate formulas can be obtained by retaining more terms. For example, or Another improvement is to use central differences The centered-difference formula is more accurate than the forward or backward-difference formulas, but still involves only two terms. Table 9 .1 lists some finite-difference formulas for derivatives.

77. Risk Management & Trading System Software OEX options. Fair values are calculated with finite Difference Model that takes account of the entire Local Volatilities Surface. http://www.intermarkit.com/research/default.asp

INTERMARK'S RESEARCH AND WORKING PAPERS Java FX Exotics Options Calculator FX Exotic Options Calculator enables you to calculate Fair Values and "Greeks" of European and American vanilla options, European single and double barrier options and Binary Range "Bet" options. Equity Options: To price an equity option, enter dividend yield in the foreign rate field. Metal Options: To price a metal option, enter lease rate in the foreign rate field. This calculator uses the same pricing models as FOCUS and Advanced Exotics Toolkit. EGAR’s Model for pricing Weather HDD and CDD swaps and options This model uses the selection of an analytical distribution function using open statistical methods of distribution function feed and then calculates the price of the derivative using Monte-Carlo method, flexibly accounting for the nuances of the statistical distribution. Download Word document Using our Local Volatility model to calculate OEX Volatilities and Fair Values A well-known limitation of the Black-Scholes model is that each Implied Volatility represents a single option price. Different Implied Volatilities for options on the same asset creates the Volatility "Smile". Market practice is to calculate single implied volatility for pricing and risk management needs. This single volatility is either interpolated from the smile or calculated as an average volatility. The Local Volatilities Surface allows traders to use multiple volatilities for pricing and risk management needs. Local Volatilities Surface can be used to forecast underlying asset prices at future dates.

78. ENVI2200: Dynamical Systems - Lecture 6 6.2 Recap finitedifference approximations Centred-difference approximation, df (4). where Dx = x j+1 -x j . 6.3 An example of a finite-difference approximation. http://www.env.leeds.ac.uk/envi2200/lecture6/lecture6.html

6. Numerical solutions and Ordinary Differential Equations 6.1 Recap definition of a derivative [df/ dx] is the slope of the tangent to the curve y = f(x) at the point x. [df/ dx] can be approximated by [( D f)/( D x)] i.e. how much f changes per D x change in x. In fact df dx = lim D x D f D x is the definition of the derivative. 6.2 Recap finite-difference approximations Centred-difference approximation df dx f(x j+1 )-f(x j-1 x j+1 -x j-1 Forward-difference approximation df dx f(x j+1 )-f(x j x j+1 -x j Backward-difference approximation df dx f(x j )-f(x j-1 x j -x j-1 On a regular grid these expressions are slightly simpler e.g. for a forward-difference approximation, df dx f j+1 -f j D x where D x = x j+1 -x j 6.3 An example of a finite-difference approximation Consider the very simple example f(x) = sin(2x) We can, of course, immediately write down the derivative: df dx Note: the accuracy of a finite-difference approximation to [df/ dx] is determined by both the scheme we choose and the resolution (i.e. the size of D x).

79. Finite Difference Equations Nondimensionalization. finite Difference Equations. Advection advected quantity. The finite difference form of the advection term is http://www.es.ucsc.edu/~rop/pgman/node4.html

Next: Model Basics Up: Planetary-Geostrophic Ocean Model: Previous: Nondimensionalization Finite Difference Equations Advection is effected using centered differencing in flux form, except at the upper boundary where an upwind scheme is implemented. Time-stepping uses a second order Runge-Kutta algorithm. In the discussion following, if salinity is not explicitly mentioned its treatment is the same as that of temperature. Let the domain be covered by a three-dimensional grid, denoting increments in the x y -, and z -directions respectively, and let temperature, T , salinity S , density, , and pressure, p , be defined on that grid. Let and be the respective grid increments. If and are the number of grid points in the x and y directions, the lateral boundaries are considered to lie at and , namely one half a grid interval in from the edge. Define the following variables: and similarly for p and Define also the difference operators: where is an arbitrary field. Then the horizontal velocities are defined on a staggered grid, obtained from pressure on as follows: and where , with

80. A Finite Difference Method A finite Difference Method. This involves a finitedifference scheme which requires the velocity gradient to be known or calculated at the boundary. http://www.ph.ed.ac.uk/~jmb/thesis/node84.html

Next: Dirichlet Boundary Conditions Up: Higher-Order Boundary Conditions Previous: Higher-Order Boundary Conditions A Finite Difference Method A second-order boundary condition has been proposed by Skordos [ ]. This involves a finite-difference scheme which requires the velocity gradient to be known or calculated at the boundary. If the velocity gradient is known at the boundary the boundary conditions produce good accuracy, however this method can only be applied if the velocity distribution is already known. When the velocity gradient is not know it can be calculated using first- or second-order asymmetric differences. Both methods produce good accuracy but neither are as accurate as the exact gradient method [ ]. As expected the second-order scheme is more accurate than the first-order, however both of the difference methods are unstable at high values of the `computational Mach number', , the second-order method being less stable than the first-order. This method has the disadvantage that it rejects the simplicity of the lattice Boltzmann method at the boundaries and instead requires a finite-difference approach. James Buick Tue Mar 17 17:29:36 GMT 1998 [ Please note that the University of Edinburgh is not responsible for the content of these WWW pages. For queries please contact

Page 4     61-80 of 90    Back | 1  | 2  | 3  | 4  | 5  | Next 20