Global error control in adaptive Nordsieck methods (Q2904815)

This paper is concerned with the numerical solution of initial value problems (IVP) for systems of first order differential equations by means of variable step size Nordsieck methods. In the proposed methods, for a given approximation \( X_k\) to the Nordsieck vector of the solution \( x = x(t)\) of the IVP at the grid point \( t_k\) defined by the \((r+1)\)-dimensional vector \( X(t_k) \equiv \left( x(t_k), h_k x'(t_k), (h_k^2 /2!) x''(t_k), \ldots , (h^r /r!) x^{(r)}(t_k) \right)^T \), the methods compute a new approximation \( X_{k+1}\) to \( X(t_{k+1})\) with \( t_{k+1}= t_k + h_{k+1}\) by means of consistent or quasi-consistent Nordsieck formulas that include a global error control algorithm that is used to select the variable stepsize. In the case of quasi-consistent Nordsieck formulas the author considers optimally stable Nordsieck methods in the sense that the eigenvalues of the transition matrix satisfies the strict root condition and a modification is proposed to avoid the order reduction that appears on non uniform grids.NEWLINENEWLINEIt is claimed that, with some restrictions between the step size ratio between consecutive steps, the proposed strategy to select the variable step size based on some estimates of local and global errors leads to adaptive numerical integrators that allow the user to obtain numerical solutions with a prescribed tolerance TOL given by the user in contrast with the situation of standard adaptive integrators based on local error estimates that do not guarantee the size of the global error.NEWLINENEWLINEAfter some implementation details, the paper ends presenting the results of numerical experiments with two test problems integrated by the proposed Nordsieck methods with orders 3, 4, 5 and 6, showing that the global error agrees with the prescribed TOL particularly for small TOL and low order methods.