An explicit sixth-order method with phase-lag of order eight for \(y''=f(t,y)\). (English) Zbl 0614.65084

Recently, R. Thomas [BIT 24, 225-238 (1984; Zbl 0569.65052)] gave a two-step sixth-order method with phase-lag (1/101 \(478.67)H^ 8\) for the numerical integration of periodic initial value problems: \(y''=f(t,y)\), \(y(t_ 0)=y_ 0\), \(y'(t_ 0)=y_ 0'\). However, Thomas’ method is implicit, it possesses an interval of periodicity of size 2.77 and for nonlinear problems her method requires \(6I+1\) function evaluations for I modified Newton iterations for the solution of the resulting implicit equations at each step. In the present paper we present a new two-step sixth-order method which also has phase-lag of order eight but with a smaller constant given by (1/3 628 800)H\({}^ 8\). In contrast with Thomas’ method our method is explicit, possesses a larger interval of periodicity of size 4.63 and it is decidedly more economical since it involves only six function evaluations per step. Numerical experiments confirm the superiority of our present method over Thomas’ method.


65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C25 Periodic solutions to ordinary differential equations


Zbl 0569.65052
Full Text: DOI


[1] Cash, J. R., High-order \(P\)-stable formulae for the numerical integration of periodic initial value problems, Numer. Math., 37, 355-370 (1981) · Zbl 0488.65029
[2] Chawla, M. M., A sixth order tridiagonal finite difference method for non-linear two-point boundary value problems, BIT, 17, 128-133 (1977) · Zbl 0361.65077
[3] Chawla, M. M., Numerov made explicit has better stability, BIT, 24, 117-118 (1984) · Zbl 0568.65042
[4] Chawla, M. M.; Rao, P. S., A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems II. Explicit method, J. Comput. Appl. Math., 15, 329-337 (1986) · Zbl 0598.65054
[5] Gladwell, I.; Thomas, R. M., Damping and phase analysis for some methods for solving second order ordinary differential equations, Internat. J. Numer. Meth. Engng., 19, 493-503 (1983) · Zbl 0513.65053
[6] Thomas, R. M., Phase properties of high order, almost P-stable formulae, BIT, 24, 225-238 (1984) · Zbl 0569.65052
[7] van Dooren, R., Stabilization of Cowell’s classical finite difference methods for numerical integration, J. Comput. Phys., 16, 186-192 (1974) · Zbl 0294.65042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.