Stepsize selection for tolerance proportionality in explicit Runge-Kutta codes (Q1373127)

An adaptive stepsize formula is introduced for application in adaptive explicit Runge-Kutta approximations to the solution of an ordinary differential equation. Suppose that at the \(n\)th step of the approximation the error is known and has the form \[ e_n= h^p_n\psi(t_{n- 1},y_n)+ O(h^{p+ 1}_n),\tag{i} \] \(h_n= t_n- t_{n-1}\), \(n= 0,1,2,\dots, N\), \(t_N= T\), where \(y_n\) is the approximated value. Given a tolerance \(\delta\), if \(\psi\neq 0\), \(h_n\) is chosen such that \(\text{est}_n= |e_n|\leq\delta\) and the step \(h_{n+1}\) is produced by the formula \[ h_{n+ 1}=\theta(\delta/\text{est}_n)^{1/p}h_n,\quad 0<\theta< 1.\tag{ii} \] The authors present examples which show that (ii) becomes ineffective if \(\psi= 0\) for some \(t_n\). They introduce a modified formula replacing \(\text{est}_n\) in (ii) by \(\text{estmax}_n\), where \(\text{estmax}_n= \max(\text{est}_n, \text{EST}_n)\), \(\text{EST}_n= h^p_n\min(A_n, B)\), \(A_n= k {1\over t_n}\sum^n_{i= 1} (\text{est}_i/ h^{p-1}_i)\), \(B\), \(k\) depending on the coefficients of the Runge-Kutta algorithm. The modified procedure allows \(\psi\) to become zero and is proved to lead to globally accurate results. Results of numerical tests are presented.