Stabilization techniques offer significant advantages in developing highly stable algorithms for gradient flows; however, the considerable exponential decay of solutions, damping of high frequencies, and lagging effects arising from different integrations of stabilization terms continues to pose major challenges. (1) Considering the exponential decay of solutions in stabilized integrating factor Runge--Kutta (sIFRK) schemes, we propose recursive approximations of exponential functions related to stabilization terms to remove the decay effects, and develop up-to-fourth-order unconditionally maximum principle preserving parametric IFRK (pIFRK) schemes. (2) To further remove the exponential damping of high frequencies and preserve equilibria, we develop an exponential-free Runge--Kutta (EFRK) framework by incorporating stiff linear operators in the recursive approximations. A unified strategy is proposed to prove the original energy dissipation property of EFRK schemes. (3) Finally, considering a class of stabilized single-step schemes, we develop a novel strategy to characterize the time delay introduced by stabilization and eliminate the lagging phenomenon through a relaxation technique.

张弘, 国防科技大学副教授。2012年毕业于浙江大学数学系, 2014年获国防科学技术大学硕士学位, 2018年获荷兰乌特勒支大学数学博士学位。主要从事偏微分方程保结构算法、自适应方法的研究。工作发表在SIAM NUMER. ANAL., SIAM J. SCI. COMPUT., J. COMPUT. PHYS.等期刊上, 入选湖南省青年A类项目, 荷尖人才, 国防科技大学高层次卓青人才。主持国家自然科学基金面上、青年项目、军委科技委项目等。