Multiple order double output Runge–Kutta Fehlberg formulae: strategies for efficient application

When attempting to solve initial value problems numerically, a scientist may choose from several well-known codes, e.g. RKSUITE, RK45 and GERK, due to Shampine et al., Dverk (Verner) or diverse codes due to Dormand and Prince. In any case, the stepsize control is done with respect to a strategy introduced by Fehlberg, i.e., the local error is estimated by the help of the two approximations resulting from each step. on this strategy, we have designed a new code named EMDRKF. Here, special Runge–Kutta formulae yielding approximations of multiple orders and in two different knots are implemented. They are called Multiple Order Double Output Runge–Kutta Fehlberg formulae (MODORKF formulae). We have implemented them applying three different strategies, i.e. Fehlberg-, Multiple order- and Multiple order double output strategy. Their benefit is demonstrated by some numerical results.