## 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.

### MSC:

 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
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### References:

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