Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a na\"ive mixed-precision implementation of any Runge-Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed-precision RKC schemes that are instead unaffected by this limiting behaviour. We present three mixed-precision schemes: a first- and a second-order RKC method, and a first-order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first- and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method.

翻译：混合精密算法结合了低精度和高精度计算法,以便从降低精度的绩效增益中获益,同时又不牺牲准确性。在这项工作中,我们设计了混精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精精