Global error estimation with Runge-Kutta triples
For the system: \(y'(x)=f[y(x)],\quad y(x_ 0)=y_ 0,\) the Zadunaisky technique for global error estimation of an embedded Runge-Kutta formula consists on calculating the error obtained with the same formula in a neighbouring problem with known solution; for example: \(y'_ h(x)=f_ h[y_ h(x)]=f[y_ h(x)]+d_ h(x),\quad y_ h(x_ 0)=y(x_ 0)=P(x_ 0);\quad d_ h(x)=P'(x)-f[P(x)]\) whose solution is \(y_ h(x)=P(x)\), [see: the first and the fourth author, IMA J. Numer. Anal. 5, 481-497 (1985; Zbl 0606.65048)]. The novelty of the present paper is that P(x) is formed using a triple Runge-Kutta formula [see: same authors, Comput. Math. Appl., Part A 12, 1007-1017 (1986; Zbl 0618.65059)] which allows the application of the estimation process at each integration step and not after a block of m steps. Some formulae, numerical tests and references are included.
scientific article; zbMATH DE number 4119521
scientific article