21. Finite Differences Finite difference methods. The method of finite differences relies upon making finiteapproximations 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

22. Finite Differences On A Helix Examples of simple 2d finite differences on a helix. The function,(4). is an autocorrelation function. It is symmetrical about the http://sep.stanford.edu/sep/jon/optical/paper_html/node5.html

Next: Matrix view of the Up: Multidimensional recursive filters via Previous: Examples of simple 2-d Finite differences on a helix The function is an autocorrelation function. It is symmetrical about the ``4'' and its Fourier transform is positive for all frequencies. Digging out an old textbook Claerbout (1976) , we discover how to compute a causal wavelet with this autocorrelation. I used the ``Kolmogoroff spectral-factorization method'' to find this wavelet Wind the signal around a vertical-axis helix to see its two-dimensional shape This 2-D filter is is the negative of the finite-difference representation of the negative of the Laplacian operator, generally denoted .Now wind the signal around the same helix to see its two-dimensional shape In the 2-D representation ( ) we see the coefficients diminishing rapidly away from maximum value 1.791. My claim is that the 2-D autocorrelation of ( ) is ( ). You verified this idea at the beginning of this paper where the numbers were all ones. You can check it again in a few moments if you drop the small values, say 0.2 and smaller. Since the autocorrelation of is is a second derivative, the operator

23. Interpolation By Finite Differences Interpolation by finite differences. When the known. The differenceare then called finite differences of first order. Furthermore http://www.geocities.com/RainForest/Vines/2977/gauss/formulae/interpolation.html

Interpolation by Finite Differences When the interpolation points are equally spaces, the values of the interpolation polynomial can be expressed in terms of finite differences. Suppose that for , where h is the grid size, the values of the function f(x) are known. The difference are then called finite differences of first order. Furthermore, we define the differences of order k + 1 inductively by . If we use the shift operator E defined by , the difference operator is represented as . Sometimes the backwards difference is , the operator is called the forward difference. The central difference which is defined by is also used. Table 1 shows the relations between the finite differences and the differentiation operator D defined by Table 2, in which each entry after the second column is the or we can express each entry of table 2 in terms of or . For example, in the second column, is equal to and . If f(x) is a polynomial of degree k, then f(x) is a polynomial of degree is a constant, and is zero. Therefore, looking at the difference table, we can find the degree of an interpolation polynomial that can satisfactory approximate f(x) . It should also be noted that, if the computation of each entry of the table is carried out with a finite number of significant figures, the error in the values

24. FINITE DIFFERENCES ON A HELIX Causality in twodimensions finite differences ON A HELIX. The function,(8). is an autocorrelation function. It is symmetrical about http://sepwww.stanford.edu/sep/prof/gee/hlx/paper_html/node8.html

Next: Matrix view of the Up: The helical coordinate Previous: Causality in two-dimensions FINITE DIFFERENCES ON A HELIX The function is an autocorrelation function. It is symmetrical about the ``4'' and its Fourier transform is positive for all frequencies. Digging out our old textbooks we discover how to compute a causal wavelet with this autocorrelation. I used the ``Kolmogoroff spectral-factorization method'' to find this wavelet Now for the magic: Wind the signal around a vertical-axis helix to see its two-dimensional shape This 2-D filter is the negative of the finite-difference representation of the Laplacian operator, generally denoted .Now wind the signal around the same helix to see its two-dimensional shape In the representation ( ) we see the coefficients diminishing rapidly away from maximum value 1.791. My claim is that the autocorrelation of ( ) is ( ). You verified this idea earlier when the numbers were all ones. You can check it again in a few moments if you drop the small values, say 0.2 and smaller. Since the autocorrelation of is is a second derivative, the operator

25. FINITE DIFFERENCES ON A HELIX FEATURES OF 1D THAT finite differences ON A HELIX. The function,(7). is an autocorrelation function. It is symmetrical about the http://sepwww.stanford.edu/sep/jon/helix/paper_html/node6.html

Next: Matrix view of the Up: Multidimensional recursive filters via Previous: FEATURES OF 1-D THAT FINITE DIFFERENCES ON A HELIX The function is an autocorrelation function. It is symmetrical about the ``4'' and its Fourier transform is positive for all frequencies. Digging out an old textbook Claerbout (1976) , we discover how to compute a causal wavelet with this autocorrelation. I used the ``Kolmogoroff spectral-factorization method'' to find this wavelet Wind the signal around a vertical-axis helix to see its two-dimensional shape This 2-D filter is is the negative of the finite-difference representation of the negative of the Laplacian operator, generally denoted .Now wind the signal around the same helix to see its two-dimensional shape In the 2-D representation ( ) we see the coefficients diminishing rapidly away from maximum value 1.791. My claim is that the 2-D autocorrelation of ( ) is ( ). You verified this idea at the beginning of this paper where the numbers were all ones. You can check it again in a few moments if you drop the small values, say 0.2 and smaller. Since the autocorrelation of is is a second derivative, the operator

26. The Finite Differences Method The finite differences Method. If a finite differences scheme needs informationof the n row to compute the n+1 row, it is called one step scheme. http://www.ii.uam.es/~jlara/investigacion/ecomm/pdes/FDM.html

The Finite Differences Method The method consists in replacing each derivative in the equation by a discretization (usually truncated Taylor series). There are a lot of schemes, depending on the chosen discretization for each derivative. After the discretization, we can obtain explicit schemes - if there's no need to solve a system of equations, just to walk the grid nodes - or implicit if we have to solve a system of equations for each row of the grid. Domain discretization have to be accomplished by means of quadrilaterals parallel to the X and Y axis. Usually quadrilaterals are of equal size. If a finite differences scheme needs information of the n row to compute the n+1 row, it is called one step scheme. Those that need information about several rows, are called multi-step. A multi-step scheme using m steps needs the solution values in the first (m-1) levels, or they must be calculated using other method. If the approximate solution that a method obtain converge to the true equation solution when the mesh spacing tends to zero, the scheme is convergent. A scheme is stable if the generated errors by the computation, such as the round or the truncation ones, vanish when the computation advances in the mesh. A scheme is consistent if the local truncation errors obtainded when discretizing the Taylor series tend to zero when h k and the elemental time interval tend to zero. The discretization error is a combination of the truncation error in the equation and the errors in the initial and boundary conditions.

27. A Limited Tutorial On Using Finite Differences In Soil Physics Problems Written A Limited Tutorial on Using finite differences in Soil Physics Problems Writtenby Donald L. Baker Reviewed by H. Don Scott Adapted with permission from www http://www.aquarien.com/sptutor/findifa8/

Tutorial Home Click here to start Table of Contents Introduction PPT Slide Finite Difference Approximations of Derivatives PPT Slide ... PPT Slide Author: Donald L. Baker Download presentation source

28. Derivatives Defined Using Finite Differences Derivatives Defined Using finite differences. The class defines derivativesusing finite differences and the ``mapping method''. http://www.llnl.gov/CASC/Overture/henshaw/documentation/opHTML/node41.html

Next: Conservative Difference Approximations Up: Class MappedGridOperators Previous: Example 1: Differentiation of Derivatives Defined Using Finite Differences The class defines derivatives using finite differences and the ``mapping method''. Simply put, each derivative is written, using the chain use, in terms of derivatives on the unit square (or cube). The derivatives on the unit square are discretized using standard central finite differences. Each MappedGrid M , consists of a set of grid points, One or two extra lines of fictitious points are added for convenience in discretizing to second or fourth-order. Boundaries of the computational domain will coincide with the boundaries of the unit cubes, or The derivatives are discretized with second or fourth-order accurate central differences applied to the equations written in the unit cube coordinates, as will now be outlined. Define the shift operator in the coordinate direction m by and the difference operators Let D r m D r m r n D x m and D x m x n denote second-order accurate derivatives with respect to r and x . The derivatives with respect to r are the standard centred difference approximations. For example

29. Citations: Calculus Of Finite Differences - Jordan (ResearchIndex) C. Jordan, Calculus of finite differences, 3rd ed. New York Chelsea, 1979. C.Jordan, The calculus of finite differences, 2nd ed., Chelsea, 1947. http://citeseer.nj.nec.com/context/265690/0

13 citations found. Retrieving documents... C. Jordan, Calculus of finite differences , 3rd edition, Chelsea Publ. Comp., New York, 1965. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: The asymptotic behavior of the Stirling numbers of the first kind - Wilf (1 citation) (Correct) ....numbers of the first kind are notoriously difficult to compute. Unlike the numbers of the second kind, the closed formulas for whose summands are the familiar factorials, binomial coefficients, etc. involve two summation signs. The first asymptotic formula for them was given by Jordan , who showed, for k fixed, that 1 (log n fl) 8) Moser and Wyman [3] gave a comprehensive study of the asymptotics of for large n. Further, their study was not restricted to fixed k, but indeed covered the entire range of 1 k n by breaking it into three subranges and dealing .... C. Jordan, The calculus of finite differences , 2nd ed., Chelsea, 1947. On The Distribution Of The Sum Of - Non-Identically Distributed Uniform (Correct) ....n 2p n X j=1 (2m j 1) j n Gamma2k Gamma1 Theta sign 2p n X j=1 (2m j 1) j n Y j=1 j : Substituting (3.8) into (3.7) and interchanging the order of summation completes the proof of Theorem 2. Remark . The equations (3.4) 3.5) 3. 6) can also be obtained using Jordan s (1979 , x74, p.

30. Solution Of The Diffusion Equation By Finite Differences Solution of the Diffusion Equation by finite differences. The basic ideaof the finite differences method of solving PDEs is to replace http://www.math.princeton.edu/~jmoehlis/APC591/tutorials/tutorial5/node3.html

Next: Numerical Solution of the Up: APC591 Tutorial 5: Numerical Previous: The Diffusion Equation Solution of the Diffusion Equation by Finite Differences The basic idea of the finite differences method of solving PDEs is to replace spatial and time derivatives by suitable approximations, then to numerically solve the resulting difference equations. Specifically, instead of solving for with and continuous, we solve for , where define the grid shown in Figure Figure 1: Grid for our finite difference approximations. The point labelled corresponds to , etc. Derivatives of are approximated in terms of the values of at grid points. For example, we know that This derivative evaluated at the grid point can be approximated in many different ways, the simplest being the following: Forward Difference: Backward Difference: Central Difference: The second derivative at the grid point may be approximated by using Instead of using approximations for in terms of the values of at as for the forward difference, or at the points as for the backward difference, let's imagine instead that we evaluate it at the (fictitious) points

31. Enhanced Cell-Centered Finite Differences For Elliptic Equations On General Geom 404425 © 1998 Society for Industrial and Applied Mathematics. Enhanced Cell-Centeredfinite differences for Elliptic Equations on General Geometry. http://epubs.siam.org/sam-bin/dbq/article/26454

SIAM Journal on Scientific Computing Volume 19, Number 2 pp. 404-425 Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry Todd Arobogast, Clint N. Dawson, Philip T. Keenan, Mary F. Wheeler, Ivan Yotov Abstract. Key words. mixed finite element, finite difference, elliptic differential equation, tensor coefficient, error estimates, logically rectangular grid, unstructured mesh, hierarchical mesh AMS Subject Classifications PII Retrieve PostScript document ( 26454.ps : 2258514 bytes) Retrieve GNU Compressed PostScript document ( ... : 111812 bytes) For additional information contact service@siam.org

32. I V /i -Cycle Multigrid For Cell-Centered Finite Differences 552564 © 1999 Society for Industrial and Applied Mathematics. V-CycleMultigrid for Cell-Centered finite differences. Do Y. Kwak. Abstract. http://epubs.siam.org/sam-bin/dbq/article/32731

-Cycle Multigrid for Cell-Centered Finite Differences SIAM Journal on Scientific Computing Volume 21, Number 2 pp. 552-564 V -Cycle Multigrid for Cell-Centered Finite Differences Do Y. Kwak Abstract. -Cycle Multigrid for Cell-Centered Finite Differences: SIAM Journal on Scientific Computing Vol. 21, Iss. 2 We introduce and analyze a V -cycle multigrid algorithm for cell-centered finite difference methods applied to second-order elliptic boundary value problems. Unlike conventional cell-centered multigrid algorithms that use the natural injection operator for prolongation, we use a new prolongation operator whose energy norm we prove is bounded by 1 in the constant coefficient case and 1+ Ch in the nonconstant case. We are thus able to use general finite element multigrid theory to conclude that the V -cycle either converges well or serves as a reasonably good preconditioner, respectively. While our theory does not establish optimal performance, our numerical experiments do show that the resulting algorithm converges much faster than the conventional schemes. In fact, these results show that the energy norm convergence factor is small and remains bounded uniformly in the finest mesh size, while that of the conventional algorithm grows. Key words.

33. NuTec Sciences - Energy - Finite Differences FINITE DIFFERENCE 2D/3D Full Acoustic Prestack Finite Difference DepthMigration NuTec Energy Services Inc. of Stafford, Texas proudly http://www.nutecsciences.com/finite.html

FINITE DIFFERENCE 2D/3D Full Acoustic Prestack Finite Difference Depth Migration NuTec Energy Services Inc. of Stafford, Texas proudly announces a worldwide exclusive agreement with Sandia National Laboratories to offer software sales of SALVO, a 2D/3D Prestack shot-based Finite Difference Solution to the Full Acoustic Wave Equation with Migrated Gathers*, capable of producing structural and amplitude representations of seismic prospects with accuracy never before attainable. The ability to accurately obtain the structure and amplitude responses in areas of complex subsurface geology, such as salt domes in the gulf of Mexico and thrusts in mountainous regions, is a key to reducing present day risks and costs in oil and gas exploration. Up to now imaging these structures correctly has been technically difficult and computationally expensive. In 1995 Sandia National Laboratories was selected by the Department of Energy Advanced Computational Technologies Initiative program to accelerate the development of practical finite difference depth migration algorithms, the theoretically best approach to solving complex seismic problems. After more than three years of work the product is now ready for commercialization.

34. MATH 922 - INTRO TO FINITE DIFFERENCES MATH 922 INTRO TO finite differences. The following *.m files wereused to create the plots shown on the first day of classes http://www.math.sfu.ca/~mkropins/apma930/lectures_old/intro.html

MATH 922 - INTRO TO FINITE DIFFERENCES The following *.m files were used to create the plots shown on the first day of classes: The main driver file is finitediff.m The function is defined in func.m , and its derivatives are in funcp.m funcpp.m funcppp.m funcpppp.m The difference operators are found in dp.m dm.m , and d0.m MATH 992 Home Dr. Kropinski class list

35. By Finite Differences Translate this page by finite differences. with. requires solvesof direct problem per step in optimization. http://www.numa.uni-linz.ac.at/Staff/haase/my-articles/opti99/tsld034.htm

by finite differences with requires solves of direct problem per step in optimization Vorherige Folie Nächste Folie Zurück zur ersten Folie Graphik-Version anzeigen

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

37. Quadric Finite Differences Quadric finite differences. November 18, 1994. Derivation of finite differencealgorithms for classification of a regular ray grid against a quadric surface. http://www.mae.cornell.edu/cpa/RCE/RCE_UG/QFD.html

Quadric Finite Differences November 18, 1994 Derivation of finite difference algorithms for classification of a regular ray grid against a quadric surface. E. Eugene Hartquist HTML version by Rich Marisa This document is currently being formatted. We wish to verify ray-grid vs. quadric halfspace classification calculations by Michael Tate (Duke University's Electrical Engineering Department Technical Report number TR86-11 and at the same time pull together several other peoples notations so as to be able track the calculation's incranation through several documents with the specific aim of being able to interpret implementations of the calculations in RCE hardware and RCE simulation software. Material presented here deals with "what is" rather than "what might be" and covers the ground from concepts to the minutia of the parameters available for computing ray representations, RR, in parallel machines via finite differences. We generally write a quadric surface as a matrix of ten coefficients, Q, such that when pre- and post-multipiled by a point in Cartisian co-ordiantes we get the usual quadratic expression1. Qsurf := Transpose[P].Q.P

38. PDE's Finite Differences Graduate Course PDE's finite differences Spring 2002, UCSB There will not be lecture on June4 6 but office hours will be held instead from 2 to 5pm on these days. http://www.math.ucsb.edu/~hdc/teaching/Math206C/

PDE's FINITE DIFFERENCES Spring 2002, UCSB SYLLABUS HOMEWORK FINAL PROJECTS NOTES ... COMMENTS? Instructor: Hector D. Ceniceros hdc@math.ucsb.edu SH 6710, 893-3462 Class: TR 2:00-3:15 BRDA 2015 Office hours: TU 3:30-5PM, TH 4:30-6PM Course description: This graduate course will focus on the study of finite differences for approximating solutions to partial differential equations (PDE's). The course will cover consistency, stability, and convergence of finite difference schemes as well as dissipation and dispersion. An introduction to spectral methods will also be given. Several aplications in science and engineering will be discussed.

39. Comparison Of Path-Summation And Finite Differences Waveforms Comparison of PathSummation and finite differences waveforms for acomplicated, smooth, 2D model (top left). Three snapshots show http://alomax.free.fr/abstracts/PathFdiff.html

Comparison of Path-Summation and Finite Differences waveforms for a complicated, smooth, 2D model (top left). Three snapshots show the finite differences wavefield from a point source (green dot). The waveforms from 33 observation points (green line of dots) are shown at the bottom of the figure. There is a good agreement in amplitude and phase between the Path-Summation (thick red traces) and Finite Differences (thin black traces) waveforms for all of the principal signals. This agreement includes diffracted signals (i.e. the secondary arrival at time ~450sec and distance ~1000km, ), which cannot be modelled with geometrical ray theory. The noise in the coda of the Path-Summation waveforms indicates that the Monte-Carlo summation over paths has not yet fully converged. Each Path-Summation waveform takes about 1/1000th of the time of the entire finite differences calculation, though the finite difference calculation produces the full response throughout the grid. (The set of 33 Path-Summation waveforms requires about 1/30th the time of the finite differences calculation.) The relative difference in calculation times between the two methods may be much larger in 3D.

40. 4.2 Residuals And Finite Differences 4.2 Residuals and finite differences. The Falling Stone. A stone isdropped from a high bridge into the water. A camera takes a photo http://www.casio.edu.shriro.com.au/pages/tasks/4_2.htm

[Previous Task] [Task Index] [Next Task] 4.2 Residuals and finite differences The Falling Stone A stone is dropped from a high bridge into the water. A camera takes a photo every 0.4 seconds showing how far the stone has fallen. The distances are shown in the table below for the first two seconds. Time Elapsed, t (seconds) Distance Fallen, d (metres) Question 1. (a) Enter the data and draw a scatter plot with time as the independent variable. (b) Is a straight line model appropriate? Check by finding and drawing on your scatterplot the straight line regression and state the value of r, the correlation coefficient. What does this suggest? (c) As an additional check on how reasonable the linear model is, we should look at the residuals , the differences between the observed values and the fitted values (found from the regression equation) - if the plot of these residuals is randomly scattered about zero (so no pattern appears), then the model is a good fit. (i) Enter the fitted values in List 3 as shown in the following graphic. (ii) Calculate the residuals (ie. List 2 - List 3), place them in List 4 and then draw a scatterplot of the residuals against the time elapsed. Explain why this plot suggests that the linear model is inadequate.

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