Efficient Numerical Integration Methods Overview
Explore various numerical integration techniques including Richardson extrapolation, Bulirsch-Stoer method, Rosenbrock method, and more. Understand the caveats, approaches, and error control mechanisms for efficient computation. Dive into practical examples and step-by-step processes for implementing these methods in numerical analysis.
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CSE291: Numerical Methods Lecture Note 5.1 Numerical Integration Prof. Chung-Kuan Cheng 1
Numerical Integration: Outline Richardson extrapolation (Bulirsch-Stoer) Rosenbrock method (Runge Kutta) Predictor-corrector method Matrix exponential 2
Bulirsch-Stoer Method Caveats: Nonsmoothed function: RK Contain singular points: RK Very smooth and right-hand sides expensive to compute: Predictor-corrector 4
Bulirsch-Stoer Method Approach: Modified midpoint method Extrapolation Step size control 5
Bulirsch-Stoer Method: midpoint method Given ??/?? = ?(?,?),? and ?, set = ?/? ?0= ?(?) ?1= ?0+ ? ?,?0 ??+1= ?? 1+ 2 ?(? + ? ,??) for ? = 1,2,..,? 1 ?(? + ?): ?? = 1/2[?? + ?? 1 + ?(? + ?,??)] Error: ?? ?(? + ?) = ?=1?? 2?[1,2,3] Example Sequence: n=2,4,6,8,10,12,14, (Deuflhard) 6
Bulirsch-Stoer Method: Extrapolation ?00 ?10 ?20 ?11 ?21 ?22 ??0=?? ??,?+1= ???+ (????? 1,?)/[(??/?? ?)2 1], ? = 0,1, ,? 1 Solution: ??? Error: |??? ??,? 1| Errk: ?2?+1 7
Bulirsch-Stoer Method: Step size Control 1 ?2 ???? 2?+1 Stepzie ??= ??1 Complexity ?0= ?0+ 1 ??+1= ?? + ??+1 Work per unit step ??= ??/?? Strategy minimize ??([4].17.3.3) For ?(? + ?) ??+ (?? ??/2)/3, we use 1.5 derivative evaluations per step h. For Runge-Kutta, it takes 4 evaluations. 8
Rosenbrock Methods ODE: ??/?? = ?(?) Step size: Process: ?(?0+ ) = ?0+ i=1,s???? 1 ? ? ??= ?(?0+ j=1,i-1?????) + ? j=1,i-1?????, ? = 1, ,? Runge-Kutta: ? = ???= 0 for all ??. 9
Rung-Kutta Method (4th order) For 4th order RK method, we evaluate the derivatives four times: once at the initial points, twice at trial midpoints, and once at a trial endpoint. The final solution is calculated from the 4 derivatives. ?1= ?(??,??) ?2= ?(?? + 0.5 ,?? + 0.5?1) ?3= ?(?? + 0.5 ,?? + 0.5?2) ?4= ?(??+ ,?? + ?3) ??+1 = ?? + 1/6?1+ 1/3?2+ 1/3?3+ 1/6?4+ ?( 5) 10
Predictor-Corrector Methods: Admas-Bashforth ODE: ??/?? = ?(?) Predictor ??+1= ??+ /12(23?? 1 16?? 1+ 5?? 2) + ?( 4) Corrector ??+1= ??+ /12(5??+1+ 8?? ?? 1) + ?( 4) 11
References 1. J.A. Gaunt, The deferred approach to the limit, II- interpenetrating lattices, Trans. Roy, Soc., Lond. 226, 350-361, 1927 2. R. De Vogelaere, On a paper of Gaunt concerned with the start of numerical solutions of differential equations, Z. Angew. Math. Phys, 151-156, 1957 3. W.B. Gragg, On extrapolation algorithms for ordinary initial value problems, J. of SIAM, 384-403, 1965 4. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical recipes, 3rd Edition, 2007 12
