Runge–Kutta Embedded Methods of Orders 8(7) for Use in Quadruple Precision Computations

Abstract

High algebraic order Runge–Kutta embedded methods are commonly used when stringent tolerances are demanded. Traditionally, various criteria are satisfied while constructing these methods for application in double precision arithmetic. Firstly we try to keep the magnitude of the coefficients low, otherwise we may experience loss of accuracy; however, when working in quadruple precision we may admit larger coefficients. Then we are able to construct embedded methods of orders eight and seven (i.e., pairs of methods) with even smaller truncation errors. A new derived pair, as expected, is performing better than state-of-the-art pairs in a set of relevant problems.