A note on the stability of rational Runge-Kutta methods
Rational Runge-Kutta methods are studied for solution of a system of ordinary differential equations \(U'=f(U(t)),\) \(U(0)=u_ 0,\) \(f: R^ n\to R^ n.\) This method is called \(A_ 0\)-bounded if and only if there exists a constant \(\tau >0\) such that for every dimension \(n\geq 1\) \(| \bar u_{k+1}-u_{k+1}| \leq \tau | \bar u_ k-u_ k|\) holds where \(\{\) \(\bar u_ k\}\) and \(\{u_ k\}\) denote any two numerical sequences generated by applying the method to the problem \(U'=AU(t),\quad U(0)=u_ 0,\quad A=diag(\lambda_ 1,...,\lambda_ n)\) with the same (arbitrary choosen) step \(h>0,\quad \lambda_ i\leq 0\) and with different (arbitrary choosen) \(\bar u_ 0\), \(u_ 0\in R^ n.\) It is proved that there exists no rational Runge-Kutta method which is \(A_ 0\)-bounded.
scientific article; zbMATH DE number 4117759