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We propose a practical implementation of high-order fully implicitRunge-Kutta(IRK) methods in a multiple precision floating-point environment.Although implementations based on IRK methods in an IEEE754 double precisionenvironment have been reported as RADAU5 developed by Hairer and SPARK3developed by Jay, they support only 3-stage IRK families. More stages andhigher-order IRK formulas must be adopted in order to decrease truncationerrors, which become relatively larger than round-off errors in a multipleprecision environment. We show that SPARK3 type reduction based on theso-called W-transformation is more effective than the RADAU5 type one forreduction in computational time of inner iteration of a high-order IRK process,and that the mixed precision iterative refinement method is very efficient in amultiple precision floating-point environment. Finally, we show that ourimplementation based on high-order IRK methods with embedded formulas canderive precise numerical solutions of some ordinary differential equations.

Practical Implementation of High-Order Multiple Precision Fully Implicit Runge-Kutta Methods with Step Size Control Using Embedded Formula