Cheap global error estimation in some Runge-Kutta pairs

IMA Journal of Numerical Analysis, 33(1) (2013), 136-163
http://dx.doi.org/10.1093/imanum/drr060

The global error estimation and control problem is studied in embedded Runge–Kutta methods applied to ordinary differential equations (ODEs). We consider both nonstiff equations and stiff ones. Such problems arise in many areas of application and their accurate numerical solution is an important issue of applied mathematics. Usually, global error estimation and control techniques implemented in ODE solvers considerably extend execution time and they are rarely used in practical computations, especially, in implicit methods applied to stiff ODEs. In this paper we show that useful information can be obtained via simple summation of local error estimates calculated cheaply in the course of integration by some embedded Runge–Kutta pairs. Thus, since the embedded method local error estimation is a standard option of any practical solver for differential equations, our technique allows the global error estimation to be implemented naturally and without extra computations. Numerical tests presented here confirm this conclusion in practice. A comparison with explicit and implicit ODE solvers implemented with only local error control in MATLAB is also conducted.