Global error estimation with adaptive explicit Runge-Kutta methods

Users of locally-adaptive software for initial value ordinary differential equations are likely to be concerned with global errors. At the cost of extra computation, global error estimation is possible. Zadunaisky's method and 'solving for the error estimate' are two techniques that have been successfully incorporated into Runge-Kutta algorithms. The standard error analysis for these techniques, however, does not take account of the stepsize selection mechanism. In this paper, some new results are presented which, under suitable assumptions show that these techniques are asymptotically valid when used with an adaptive, variable stepsize algorithm—the global error estimate reproduces the leading term of the global error in the limit as the error tolerance tends to zero. The analysis is also applied to Richardson extrapolation (step halving). Numerical results are provided for the technique of solving for the error estimate with several Runge-Kutta methods of Donnand, Lockyer, McGorrigan and Prince.