Runge-Kutta pairs of orders 8(7) with extended stability regions for addressing linear inhomogeneous systems

The non-stiff Initial Value Problem is a wider subject classified in Mathematics. Here, we consider an interesting subclass. Namely, the Linear Inhomogeneous system that shares constant coefficients. Runge-Kutta pairs of high orders are chosen in order to achieve stringent accuracies when solving these systems numerically. It is theoretically interesting to equip these methods with large stability intervals for addressing the problems at hand. Thus, at first, we present an explicit algorithm for deriving the coefficients of such pairs of orders eight and seven. Then we adjust this in an optimization precess for extending the stability region and simultaneously keep the principal truncation error as low as possible. The resulting pair outperforms other standard pairs in a series of relevant problems.