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by: Camryn Rogahn
Camryn Rogahn
GPA 3.61


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Class Notes
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This 17 page Class Notes was uploaded by Camryn Rogahn on Tuesday October 20, 2015. The Class Notes belongs to MATH542 at San Diego State University taught by P.Blomgren in Fall. Since its upload, it has received 25 views. For similar materials see /class/225268/math542-san-diego-state-university in Mathematics (M) at San Diego State University.

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Date Created: 10/20/15
riii e t e e M I 1 H M tin s i Peter Blomgren blomgren peter gmail com Department of Mathematics and Statistics Dynamicai Systems Grou Computationai Sciences Research Center San Diego State University San Diego CA 9218277720 PelerBlom i A m rrCol cto Nun er L Outline 0 Variable Order Predictor Corrector Methods 0 Introduction and Model Problem 0 Sample Code 0 Numerical Results 0 Plots 0 Comments 9 Looking Forward 0 Future Menu 0 Final Projects Format Example Variable Order LMM Scheme Introduction and Model Problem Variable Order PredictoryCorrecl Ni 39 L Closing 0th the Initial Value Problem To finish up the discussion on numerical methods for the initial value problem as we know it so far we will develop a variable order predictor corrector method For simplicity we will look at the following problem y f ff7yf where YO 07 ft7 t 07 t lt Tstan 1 This way we will not have to worry about generating starting values the forcing turns on after a while Peter Blomgren Example Variable Order LMM Schemes ds m 39 I 5 Look F0 Variable Order PredictoryCorreclor Mellio Ni 3916 Rest Introduction and Model Problem Our first forcing function For our first example we choose the following forcing function 0 t 6 01 ftyt 710m sin2met1 2 t e 1 8 710m sin27rte t 172e t 82 t 6 812 Peter Blomgren Example Variable Order LMM Schemes Variable Order PredictorrCorreclor Methods Numer uIE L Sample Code Fun The Code 397 Variable order Predictoricltgtrrector method 7 in PEC2E mode 397 clear all 7 The rightihand side of y t ftyt function f fty f tgt1 710y s1n2p1t exp71t1t1 8 tgt8exp7t78t78 to endfunction 397 Coefficients for AdamsiBashforth Predictors 397 AB 1 o o o o o 3 r1 0 0 0 02 23 716 5 0 0 012 55 759 37 79 o 024 1901 72774 2616 71274 251 0720 4277 77923 9982 77298 2877 74751440 Example Variable Order LMM Scheme Variable Order PredictorrCorreclor Methods 39 I Y N Ier uIE un L Sample Code Fun The Code 397 Error Coefficients for AdamsiBashforth Predictors ABE 12 512 924 251720 4751440 1908760480 397 Coefficients for AdamsiMoulton Correctors 397 251 646 264 106 19 0720 475 1427 798 482 173 27 1440 397 Error Coefficients for AdamsiMoulton Correctors AME 12 112 124 19720 271440 86360480 7 Currently we cannot go beyond sixth order MALURDER 6 m Example Variable Order LMM Scheme quot 39 5 Sample Code Variable Order PredictoryCorreclor Methods M 1939 L In 30 39 The Code IIIVI 397 Step size h 112 397 Starting order order 1 397 Number of corrections 397 Tolerance TUL 0 00007 397 Tmax 12 tv 0hTmax zerossize tv Variable Order LMM Schen Variable Order PredictorrCorreclor Methods I N uIE urn er L m i w Sample Code Fun The Code ix 2 while ix lt 1engthtv 397 Extract the time 397 t tvixil 397 Extract the AdamsiBashforth and AdamsiMoulton Coefficients ab ABorder1order am AMorder1order 397 Predictor Step iquot P 397 fJJistory fvix71 1 ixiorderD ypred y1xr1 h sumabunstory 397 EvaluateCoIrect ECloop 397 fJJistory fvix 1 ixiorder1 fJJistory1 ftypred for u 1mu ycorr yix71 h t sumamfJnistory finistory1 ftycorr Example Variable Order LMM Scheme Variable Order PredictorrCorreclor Methods Numer uIE L m t w Sample Code Fun The Code 7 Evaluate iquot E 397 Leval finistory1 397 Milne s Error Estimate LTEest AMEorder ABEorder AME order ycorriypred Errv ix t if abs LTEest lt TUL YiX yecorr if order gt 1 0 AMEorder ABEorderAMEorder Cpml AMEorder1 ABEorder1AMEorder1 if absLTEest Cpml Cp h lt 05TUL rd r 139 or er 7 o e 7 fprintfquotDecreasing order to 397d at time 397fnquot ordert en end m Example Variable Order LMM Scheme munquot v14 Variable Order PredictorrCorreclor Methods M 1939 m 2 In LN Sample Code The Code er 1 if order lt MALURDER fprintfquot1ncreasing order to 39Zd at time 397fnquot ordert e se fprintfquotCannot increase order beyond 397dnquot MAX RDER fprintfquotProgram failed at time fnnnquot t fprintfquotTry a smaller stepisizenquot errorquotExecution stoppedquot 397 Dump data to files for plotting 397 D tv y save ascii varsolndat D D tv OV save ascii varorderdat D D tv Errv save ascii varerrordat D D tv fv save ascii varfdat D m Variable Order LMM Schen 0 05 Peter Blom Solution Variable Order F39rodixarrC 39r cto Mnlho Numerical Resul39s L Fun o Estimated Error 6e05 4e05 ZeOS 2e05 4e05 6e05 Peter Blom Example Variable Order LMM Schen Orderp Peter Blom Peter Blom Variable Order F39I39e 39 Commems Comments The example is carefully cooked in order to trigger order changes up to 6 but not break The forcing function is somewhat synthetic to put it kindly We have completely ignored the stability analysis But since there is no strange blow up everything seems to be OK Implementing a variable order BDF scheme as described in lecture 12 may be a good final project Peter Blomgren Example Variable Order LMM Schemes 7 1517 Fu Lu re Menu 1 Looking Forward Looking ahead Boundary Value Problems Next we will open up a some new cans of worms o A discussion of hybrid methods combining some of the ideas introduced so far 0 A look at Boundary Value Problems BVP for ODEs Example Variable Order LMM Scheme Final Projec 7 Format Final Project Presentations Last Weeks of Class 15 20 min ute presentations 1 Background describe the physical problem and equations 2 Approach describe the numerical scheme selected and analysis performed to get a feel for the parameters eg stability analysis eigenvalues of the system if applicable to get a working step size h 3 Results document how well your approach solves the problem also report on approaches that went wrong 4 Conclusions comment on possible improvements which may be beyond what is reasonable for a small class project Do not hesitate to use office hours and email to get help Ask early and ask often 7 Hand in a draft of 1 and 2 by 12 34pm Friday 4202009 m Peter Blon ren Variable Order LMM Schemes 7 1717


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