A new class of explicit two-step fourth order methods for \(y''=f(t,y)\) with extended intervals of periodicity (Q1069277)

The author introduces a new class of explicit two-step fourth order methods for the initial value problem: \(y''=f(t,y)\), \(y(t_ 0)=y_ 0\), \(y'(t_ 0)=y'\!_ 0\). The method is essentially a modification of Numerov's method: \[ y_{n+1}=2y_ n-y_{n-1}+h^ 2(\bar f_{n+1}+10f_ n+f_{n-1})/12, \] based on \(m+2\) evaluations of f. He shows a result on the stability when applied to \(y''=-\lambda^ 2y\), \(\lambda >0\), possessing an interval of periodicity of length nearly \(2((m+1)(m+3))^{1/2}\). Similarly, he mentions a class of explicit second order methods with \(m+1\) evaluations of f possessing an interval of length nearly \(2(m+1)\).