Superstable single-step methods for second-order initial-value problems
This paper is concerned with the numerical solution of initial value problems for general second-order differential equations. Two implicit one step methods of orders 4 and 6 which are superstable (except for isolated values of the parameters) in the sense of Chawla are proposed. Here the term superstability means that when the method is applied to the test equation \(y''+2\alpha y'+\beta \quad 2y=0\) with \(\alpha\),\(\beta\geq 0\), \(\alpha +\beta >0\), we get a linear difference equation whose characteristic polynomial \(\rho (x)=A_ 1x\quad 2+A_ 2x+A_ 3\) \((A_ i=A_ i(h\alpha,h\beta))\) satisfies: i) If \(\rho (x)=0\), then \(| x| <1\) for all \(h\alpha >0\), \(h\beta >0\). ii) If \(\alpha =0\), the roots of \(\rho\) are complex conjugate and with modulus one. iii) If \(\beta =0\) the two roots of \(\rho\) (x) are such that \(x_ 1=1\) and \(| x_ 2| <1\).
scientific article; zbMATH DE number 4066249
scientific article