AI 中文总结
研究混合精度龙格 - 库塔方法中稳定校正的性能，通过谱半离散化两个非线性偏微分方程，在多种精度配对下比较未校正、显式校正和稳定变体的DIRK方法，表明稳定校正提高精度且节省运行时。
AI 中文摘要
混合精度龙格 - 库塔方法通过低精度评估对角隐式龙格 - 库塔（DIRK）格式中的昂贵隐式求解来降低成本，同时保持较大时间步长下格式的精度。低精度扰动造成的精度损失可通过廉价的显式校正恢复，但这些校正对稳定性有不利影响。最近提出的稳定校正通过对校正步骤应用稳定矩阵来解决此问题，但其运行时成本此前未被量化。在这项工作中，我们对这些稳定校正的运行时性能进行了数值研究。使用两个非线性偏微分方程（无粘伯格斯方程和多孔介质方程）的谱半离散化，我们在半精度、单精度、双精度和四倍精度配对下，比较了未校正的混合精度DIRK方法与显式校正和稳定变体，针对二阶到四阶SDIRK方法。我们报告了收敛性、运行时和加速比，并表明稳定校正提高了混合精度格式的精度，同时保留了大量运行时节省。所有实验均在配备Julia版本1.11.4的英特尔至强铂金8480 + CPU上进行。
英文摘要
Mixed precision Runge--Kutta methods reduce the cost of the expensive implicit solves in diagonally implicit Runge--Kutta (DIRK) schemes by evaluating them in low precision, while retaining the accuracy of the scheme for larger time steps. The accuracy lost to the low-precision perturbation can be recovered through inexpensive explicit corrections; however, these corrections have an adverse impact on stability. Recently proposed stabilized corrections remedy this by applying a stabilization matrix to the correction step, but their runtime cost has not previously been quantified. In this work, we present a numerical study of the runtime performance of these stabilized corrections. Using spectral semi-discretizations of two nonlinear partial differential equations, the inviscid Burgers' equation and the porous medium equation, we compare uncorrected mixed precision DIRK methods against explicitly corrected and stabilized variants across half, single, double, and quadruple precision pairings, for SDIRK methods of orders two through four. We report convergence, runtime, and speedups, and show that the stabilized corrections improve the accuracy of the mixed precision schemes while preserving substantial runtime savings. All experiments were performed on an Intel Xeon Platinum 8480+ CPU with Julia version 1.11.4.
