On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs

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Truong Nguyen-Ba Abdulrahman Alzahrani Thierry Giordano Remi Vaillancourt

Abstract

New optimal, contractivity-preserving (CP), \(d\)-derivative, 2- and 3-step, predictor-corrector,  Hermite-Birkhoff-Obrechkoff series methods, denoted by \(HBO(d,k,p)\), \(k=2,3\), with  nonnegative coefficients are constructed  for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\).  The upper bounds \(p_u\) of order \(p\) of \(HBO(d,k,p)\), \(k=2,3\) methods are approximately 1.4 and 1.6 times the number  of derivatives \(d\), respectively.  Their stability regions have generally  a good shape and grow with decreasing \(p-d\).  Two selected CP HBO methods: 9-derivative 2-step HBO of order 13, denoted by HBO(9,2,13),  which has maximum order 13 based on the CP conditions, and  8-derivative 3-step HBO of order 14, denoted by HBO(8,3,14), compare well  with Adams-Cowell of order 13 in PECE mode, denoted by AC(13),  in solving standard N-body problems over an interval of 1000 periods  on the basis of the relative error of energy as a function of the CPU time.  They also compare well with AC(13) in solving standard N-body problems on the basis of the growth of relative error of energy and 10000 periods of integration.  The coefficients of selected HBO methods are listed in the  appendix.

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How to Cite
NGUYEN-BA, Truong et al. On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs. Journal of Modern Methods in Numerical Mathematics, [S.l.], v. 8, n. 1-2, p. 17-39, jan. 2017. ISSN 2090-4770. Available at: <http://m-sciences.com/index.php?journal=jmmnm&page=article&op=view&path%5B%5D=1130>. Date accessed: 15 dec. 2017. doi: https://doi.org/10.20454/jmmnm.2017.1130.
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