Adaptive step size methods and error control

Instead of evaluating an error by solving the IVP twice, by using two different fixed step sizes, e.g. h and A/2, the adaptive methods rely on error estimations from embedded pairs of different orders. As the solution proceeds, two solutions with different orders can be calculated in parallel and compared at each step to ensure that the accuracy is within 

Another method that uses fourth- and fifth-order embedded pairs is the Dormand-Prince method. The Dormand-Prince method is more accurate than the Runge-Kutta Fehlberg method and it is used by the MATLAB ode45 solver. Both methods have in common that the difference between the fourth- and fifth-order accurate solutions is calculated to determine the error, and to adapt the step size. The error estimate, e +, for the step is 

Note that ki is not nsed except for calculating ki, and so on. The definitions are 

Consequently, the error estimate in the Dormand-Prince method is given by 

Given this estimate, s +, and a specified target tolerance for the discretization error, the adaptive method will decrease the step size to comply with the tolerance. If the accuracy is met unecessarily well, the step size can be increased to reduce the computational effort. Furthermore, the step size can be controlled or bounded by a specified minimum step size, hmm, and maximum step size, hmax, to improve stability. 

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