Efficient error control in numerical integration of ordinary differential equations and optimal interpolating variable-stepsize peer methods

Automatic global error control of numerical schemes is examined. A new approach to this problem is presented. Namely, the problem is reformulated so that the global error is controlled by the numerical method itself rather than by the user. This makes it possible to find numerical solutions satisfying various accuracy requirements in a single run, which so far was considered unrealistic. On the other hand, the asymptotic equality of local and global errors, which is the basic condition of the new method for efficiently controlling the global error, leads to the concept of double quasi-consistency. This requirement cannot be satisfied within the classical families of numerical methods. However, the recently proposed peer methods include schemes with this property. There exist computational procedures based on these methods and polynomial interpolation of fairly high degree that find the numerical solution in a single run. If the integration stepsize is sufficiently small, the error of this solution does not exceed the prescribed tolerance. The theoretical conclusions of this paper are supported by the numerical results obtained for test problems with known solutions.