Numerical Methods for Ordinary Dierential Equations Episode 9 ppsx

... −b s 280 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 38 Algebraic Properties of Runge–Kutta Methods 380 Motivation For any specific N-dimensional initial value problem, Runge–Kutta methods can ... acceptable for many problems. In each of the values of ξ for which there is a single underline, the method is A(α)-stable with α ≥ 1.55 ≈ 89 ◦ . 272 NUMERICAL METHOD...

Ngày tải lên: 13/08/2014, 05:21

... b 1 b 2       =0 192 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS The first methods of this order, derived by Kutta ( 190 1), have a different structure. One of these, as corrected by Nystr¨om ( 192 5), ... manner: 0 1 4 1 4 1 4 1 8 1 8 1 2 00 1 2 3 4 3 16 − 3 8 3 8 9 16 1 − 3 7 8 7 6 7 − 12 7 8 7 7 90 0 16 45 2 15 16 45 7 90 . 166 NUMERICAL METHODS FOR...

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... byAxelsson ( 196 9, 197 2), Butcher ( 197 7), Ehle ( 197 3), Ehle and Picel ( 197 5), Watts and Shampine ( 197 2) and Wright ( 197 0). 354 Order stars We have identified some members of the Pad´e table for the ... P k−1 (v) is √ 2k −1  2k−2 k−1  , with a similar formula for the leading coefficient of P k (u). 240 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS The fact that...

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... Third order methods 95 234 Introduction to order conditions 95 235 Fourth order methods 98 236 Higher orders 99 237 Implicit Runge–Kutta methods 99 238 Stability characteristics 100 2 39 Numerical ... Multistep–multistage–multiderivative methods 90 225 Implicit methods 91 226 Local error estimates 91 23 Runge–Kutta Methods 93 230 Historical introduction 93 2...

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... these assumptions, the equations of motion become 38 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 13 Difference Equation Problems 130 Introduction to difference equations While differential equations ... k<1, such that, for any η, ζ ∈ M, ρ(φ(η),φ(ζ)) ≤ kρ(η, ζ). Then there exists a unique ξ ∈ M such that φ(ξ)=ξ. 46 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL...

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... 1130400.0252×10 −10 4.4125 40 256178 .98 89 10 −10 4.1 893 80 61150.2626×10 −10 4. 090 4 160 1 494 9.6176×10 −10 4.0442 320 3 696 . 596 7×10 −10 4.0218 640 91 9.1362×10 −10 4.0108 1280 2 29. 16 29 10 −10 4.0054 2560 57.2134×10 −10 4.0026 5120 ... 0.014 790 21 −0.04016858 3 .96 76 3 .96 88 64 0.00372781 −0.01012 098 4.0417 4.0001 128 0.00 092 233 −0.00253020 4.0361 4.0041 256 0.00022852 −0....

Ngày tải lên: 13/08/2014, 05:21

... −0.0015 495 7 2.1 899 7 .97 97 64 −0.002153 39 −0.000 194 19 3.2451 8.1221 128 −0.00066358 −0.00002 391 3.6551 8.1017 256 −0.00018155 −0.00000 295 3.8351 8.0620 512 −0.00004734 −0.00000037 3 .91 94 8.03 39 1024 ... − 1 2 2 3 2 − 1 2 5 12 3 23 12 − 4 3 5 12 − 3 8 4 55 24 − 59 24 37 24 − 3 8 251 720 5 190 1 720 − 1387 360 1 09 30 − 637 360 251 720 − 95 288 6 4277 1440 − 2641 480 499...

Ngày tải lên: 13/08/2014, 05:21

... labelled by Numerical Methods for Ordinary Differential Equations, Second Edition. J. C. Butcher © 2008 John Wiley & Sons, Ltd. ISBN: 97 8-0-470-72335-7 158 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... is              000 − 5 29 3375 390 4 3375 4232 3375 1472 3375 1 89 92 00 152 25 − 127 25 − 4 19 100 − 1118 575 3375 5152 25 168 0 10 19 96 − 1 552 3...

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... − 1 2 3 4 − 93 22 0 2 492 1 1408 − 100 59 704 735 1408 735 704 17 19 86547313055 10 295 610642 0 − 96 707067 2867062 1552 695 1 598 57 197 88 69 2 794 9088 81711267 − 452648800 245133801 2701 895 68 46 798 2711 1 98 765 00 − 9 83 1071 1600 − 11 75 64 225 390 963 2257600 188 3315 00 1 593 7553 294 3 20800 197 97 5 576 2275 2476 099 293 48800 2 39 − 142 198 9 00 2412 7553...

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... requirements. Numerical Methods for Ordinary Differential Equations, Second Edition. J. C. Butcher © 2008 John Wiley & Sons, Ltd. ISBN: 97 8-0-470-72335-7 332 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL ... necessary. For the fourth order case, we need to choose the coefficients of the method so that α 1 =1, α 2 = 1 2 , α 4 = 1 6 , α 8 = 1 24 , 308 NUMERICAL METHODS...

Ngày tải lên: 13/08/2014, 05:21