COMPUTATIONAL METHODS MCEN 3030
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This 4 page Class Notes was uploaded by Deron Schmitt on Thursday October 29, 2015. The Class Notes belongs to MCEN 3030 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/232015/mcen-3030-university-of-colorado-at-boulder in Mechanical Engineering at University of Colorado at Boulder.
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Date Created: 10/29/15
Chapter 4 Numerical Differentiation We would like to calculate the derivative of a smooth function de ned on a discrete set of grid points 101171 1 i 7 1N1 Assume that the data are the exact values of the function at the data points7 and we need the derivative only at the data points We will look into the construction of numerical approximations of the derivative called nite differences There are two approaches to such constructions using interpolation formulas7 or Taylor series approximations 41 Finite Differences from Interpolation From linear piecewise Lagrange interpolation we have 7 1H1 7 z I z 7 11 I fz 7 fl 7 1 m1 Tl 7 xi fem Differentiating7 we nd that the derivative is a constant and the same at the two ends of the interval fi1 7 f i 7 fi 1 f 1 1 I39727A39 z HiE7V f 1 hi hi fu f 11 hi hi fz17 where A and V are forward and backward difference operators Note that the derivative is discontinuous at the end points From quadratic piecewise Lagrange interpolation we have I MAXI 1H1 Ii 7 Ii711i 7 n1 f Ii fltzgt I W Ii mm 1H 7 IiIi71 7 n1 I 7 zi1z 7 11 i n1 7 Ii711i1 7 m f 1 Differentiating and evaluating the result at the midpoint of the interval we obtain hi 1 1 ill1 17 174r 777 i iA 161 hi71hi71 hi f 1 lthi71 higt f hi 01271 hif1 For equally spaced intervals hi1 hi h7 so fi1 7 f 13971 1 i f a 2h 2hltAvm This is the central difference formula7 and is the average of the forward and backward formulasi If we differentiate twice and evaluate the result at 11 we obtain a constant in the interval 1121 S I S 1 2 2 2 H 39 i7 7i i7 f 11 hi71hi71 hif 1 hi71hif T hi hi71 hif1 19 20 CHAPTER 4 NUMERICAL DIFFERENTIATION which for equally spaced intervals reduces to f39iii2f39f391 1 1 f zi 1 h I PAw vm Since f is constant it is called a forward central or backward formula depending whether it is evaluated at 1 Ii or 1141 respectively The use of cubic splines to derive nite difference approximations has received little attention It requires the solution of the tridiagonal system hi4 hi hi hi i 7 i i 7 ii Tlf 120 3 f 10gf Ii1 fx 121 101 fr I 1 13971 For uniform intervals we obtain 1 2 1 f397 i 2f f39 g 1 flggfll Note that the solution of this system gives us the approximation for the second derivative and the effect of the spline is to distribute the previous result over the central point and its neighbors with weights 16 23 and 16 Once this tridiagonal system is solved the appropriate can be used in the rst derivative approximation obtained by differentiating the spline approximation for and evaluating the result at min 42 Finite Differences from Taylor Series Finite difference formulas can be easily derived from Taylor series expansionsi For example to obtain an approximation for the derivative of at the point Ii we use 2 z 7 z f1i1 n1 xiv10 J HIi n Rearrangement leads to Hm e mwn i 2 When the grid points are uniformly spaced the above formula can be recast in the following form fi 10141 fi 00 This formula is referred to as the rst order forward difference The exponent of h in 0h is the order of accuracy of the method With a rst order scheme if we re ne the mesh size by a factor of 2 the error called the truncation error is reduced by approximately a factor of 2 Similarly 7 fi fiil fi TJFOUL is called the rst order backward difference formulai Higher order more accurate schemes can be derived by Taylor series of the function f at different points about the point min For example the widely used central difference formula can be obtained from subtraction of the two Taylor series expansions 7 2 hisH fz17fzhfi2fi6fi quot39 41 7 7h EvilsH 12141 fi2fi 612 4 2 43 DIFFERENCE OPERATORS 21 This leads to i i i7 12 f12hf 1 igfiHJFHquot 43 This is of course a second order formula In general we can obtain higher accuracy if we include more points Here is a fourth order formula 16122 v 816121 8fi1 fi2 Z 0 h4 f1 12 lt gt The main difficulty with higher order formulas occurs near the boundaries of the domain They require the functional values at points outside the domain which are not available Near the boundaries one usually resorts to lower order formulas Similar formulas can be derived for second or higher order derivatives For example the second order central difference formula for the second derivative is f 1 2f39fgt1 f 7 h I 0W and is obtained by adding formulas 41 and 42 43 Difference Operators In order to develop approximations to differential equations we will occasionally be using the following operators h The shift operator h 7 The forward difference operator 7 7 h The backward difference operator 7 7 The central difference operator 7 The average operator f Differential operator where h is the difference interval For linking the difference operators with the differential operator we consider Taylor s formu a 1 Huh fzhf r h2f ru ln operator notations we can write 1 Efz1hD hD2 39fI The series in brackets is the expression for the exponential and hence we have formally E ehD This relation is very important since it can be used by symbolic programs such as Maple or Mathe matica to analyze the accuracy of nite difference schemes For example consider the nite difference operator de ned by Eq 43 Applying the shift operator to we can write fi1 e and 121 E e thi Now substituting these expressions into central difference approximation for the derivative we obtain fl 44 fi1 fi71 ethiJre thi ehD 67 2h 2h 2h 22 CHAPTER 4 NUMERICAL DIFFERENTIATION Now using the Taylor series expansion for ehD and e hD for small values of h we obtain 1 1 1 ehD 1hD h2D2gh3D3mjhnDnm TL 1 1 71 e hD 17hD h2D27Eh3D3m hnDnim TL Substituting these expressions to Eq 44 we obtain fi1 fiil 7 1 2 3 2h 7 D6hD 1 fr 415 Reorganizing left and right hand sides of this equation would result in Eq 43
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