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

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