Mixed precision and mixed accuracy explicit two-derivative Runge--Kutta methods

TL;DR

This paper develops a mixed-precision framework for explicit two-derivative Runge--Kutta methods to time-evolve PDEs, extending Grant2022's perturbation theory to the $\dot{F}$ term and deriving order conditions that bound $O(\\varepsilon \\Delta t^m)$ perturbations. It designs and analyzes third–sixth order explicit MP-TDRK schemes that compute the second derivative in lower precision, achieving global error $E = O(\\Delta t^p) + O(\\varepsilon \\Delta t^m)$ under appropriate conditions. Numerical experiments on linear and nonlinear PDEs (e.g., linear advection and inviscid Burgers' equation) confirm the predicted convergence behavior and illustrate how perturbation magnitude $\\varepsilon$ influences stability regions and allowable time steps. The results highlight the potential of mixed-precision TDRKs to balance efficiency and accuracy in PDE time integration, with a roadmap for further optimization and extension to broader linear methods.

Abstract

Mixed precision Runge--Kutta methods have been recently developed and used for the time-evolution of partial differential equations. Two-derivative Runge--Kutta schemes may offer enhanced stability and accuracy properties compared to classical one-derivative methods, making them attractive in a wide variety of problems. However, their computational cost can be significant, motivating the use of a mixed-precision paradigm that employs different floating-point precisions for different function evaluations to balance efficiency and accuracy. To ensure that the perturbations introduced by the low precision computations do not destroy the accuracy of the solution, we need to understand how these perturbation errors propagate. We extend the numerical analysis mixed precision framework previously developed for Runge--Kutta methods to characterize the propagation of the perturbation errors arising from mixed precision computations in explicit and implicit two-derivative Runge--Kutta methods. We use this framework for analyzing the order of the perturbation errors, and for designing new methods that are less sensitive to the effect of the low precision computations. Numerical experiments on linear and nonlinear representative PDEs, demonstrate that appropriately designed mixed-precision two-derivative Runge--Kutta methods achieve the predicted accuracy.

Figures (12)

• Figure 1: Linear stability regions for the MP-TDRK methods with error $O(\Delta t^p)$ and perturbation error $O(\varepsilon \Delta t^m)$. Third order in blue, fourth order in red, fifth order in cyan, sixth order in magenta. Solid lines are perturbation error $m=1$, dashed lines are $m=2$, and dotted line $m=3$. Left: Third order methods. Center: Fourth order methods. Right: All methods with perturbation order $m=1$.
• Figure 2: Perturbed Linear stability regions for the third order MP-TDRK methods. Top: $\varepsilon = 0.1$; Bottom: $\varepsilon = 0.5$. Left: the MP-TDRK2s3p1e method \ref{['MP2s3p1e']}. Middle: MP-TDRK2s3p2e method \ref{['MP2s3p2e']}. Right: MP-TDRK3s3p3e method \ref{['MP3s3p3e']}.
• Figure 3: Perturbed Linear stability regions for the fourth order MP-TDRK methods. Left: two-stage fourth-order method \ref{['MP2s4p1e']}. Right: three-stage fourth-order method \ref{['MP3s4p2e']}. Top: $\varepsilon = 0.1$; Bottom: $\varepsilon = 0.5$.
• Figure 4: Perturbed Linear stability regions for the fifth order MP-TDRK method \ref{['Tsai3s5p1e']} with $\varepsilon = 0.1$ (left) and $\varepsilon = 0.5$ (right);
• Figure 5: Perturbed Linear stability regions for the sixth order MP-TDRK method \ref{['MP4s6p1e']} with $\varepsilon = 0.1$ (left) and $\varepsilon = 0.5$ (right);

Theorems & Definitions (2)

• Remark 2.1
• Remark 4.1