Contractivity-preserving explicit multistep Hermite–Obrechkoff series differential equation solvers

Abstract

New optimal, contractivity-preserving (CP), explicit, d-derivative, k-step Hermite–Obrechkoff series methods of order p up to $p=20$, denoted by CP HO(d, k, p), with nonnegative coefficients are constructed. These methods are used to solve nonstiff first-order initial value problems $y'=f(t,y)$, $y(t_0)=y_0$. The upper bound $p_u$ of order p of HO(d, k, p) can reach, approximately, as high as 2.4 times the number of derivatives d. The stability regions of HO(d, k, p) have generally a good shape and grow with decreasing $p-d$. We, first, note that three selected CP HO methods: 4-derivative 7-step HO of order 13, denoted by HO(4, 7, 13), 5-derivative 6-step HO of order 13, denoted by HO(5, 6, 13), and 9-derivative 2-step HO of order 13, denoted by CMDAHO(13) compare favorably with Adams–Cowell of order 13, 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. Next, the three HO methods compare positively with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error and relative energy error over 10, 000 periods of integration. Finally, these three methods compare also well with P-stable methods of Cash and Franco et al. on some quasi periodic, second-order linear and nonlinear problems. The coefficients of selected HO methods are listed in the appendix.

Appendix: Coefficients of two selected HO methods

Table 20 of the appendix lists the two selected CP HO methods: HO(4, 7, 13) and HO(5, 6, 13) with their $c(\text {HO}(d,k,p))$ and $x_{\text {min}}$ (of the unscaled stability intervals $( x_{\text {min}}, 0)$).

Table 20 $c(\text {HO}(d,k,p))$, $x_{\text {min}}$ and coefficients of the integration formulae of HO(4, 7, 13) and HO(5, 6, 13)