Stability Analysis and Performance Evaluation of Additive Mixed-Precision Runge-Kutta Methods

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed. These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations. In this work, we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors, and demonstrate their performance in terms of accuracy and efficiency. We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta (MP-ARK) methods. The convergence, accuracy, and runtime of these methods are explored. We show that for a given level of accuracy, suitably chosen MP-ARK methods may provide significant reductions in runtime.