On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

The contractivity-preserving 2- and 3-step predictor-corrector series methods for ODEs (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt, On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs, J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17--39. doi:10.20454/jmmnm.2017.1130) are expanded into new optimal, contractivity-preserving (CP), d-derivative, k-step, predictor-corrector, Hermite- Birkhoff--Obrechkoff series methods, denoted by HBO(d,k,p), k=4,5,6,7, with nonnegative coefficients for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The main reason for considering this class of formulae is to obtain a set of methods which have larger regions of stability and generally higher upper bound \(p_u\) of order \(p\) of HBO(d,k,p) for a given d. Their stability regions have generally a good shape and grow generally with decreasing \(p-d\). A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14) which has maximum order 14 based on the CP conditions compares satisfactorily 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. HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error, relative energy error and 10000 periods of integration. The coefficients of HBO(6,4,14) are listed in the appendix.

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

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.