In this talk, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and achieve high-order accuracy in time. To accomplish this, our method first incorporates the energy dissipation law into the target gradient flow equation by introducing a time-dependent spectral renormalization (TDSR) factor. Then, the coupled equations are discretized using the spectral approximation in space and the exponential time differencing (ETD) in time. Finally, the resulting fully discrete nonlinear system is decoupled and solved using the Picard iteration at each time step. Furthermore, we introduce an extra enforcing term into the system for updating the TDSR factor, which greatly relaxes the time step size restriction of the proposed method and enhances its computational efficiency. Extensive numerical tests with various gradient flows are presented to demonstrate the accuracy and effectiveness of our method as well as its high efficiency when combined with an adaptive time-stepping strategy for long-term simulations.
专家简介:鞠立力教授1995年毕业于武汉大学数学系获数学学士学位，1998年在中国科学院计算数学与科学工程计算研究所获得计算数学硕士学位，2002年在美国爱荷华州立大学获得应用数学博士学位。2002-2004年在美国明尼苏达大学数学与应用研究所从事博士后研究。随后进入美国南卡罗莱纳大学工作，历任数学系助理教授（2004-2008），副教授（2008-2012），和教授（2013-现在）。主要从事偏微分方程数值方法与分析，非局部模型与计算，深度学习算法，计算机视觉，高性能科学计算，及其在材料与地球科学中的应用等方面的研究工作。至今已发表科研论文150多篇，Google学术引用约5800多次。自2006年起主持了十多项由美国国家科学基金会和能源部资助的科研项目。2012至2017年担任SIAM J. Numer. Anal.的副主编，目前担任J. Sci. Comput.，Numer. Methods PDEs等相关学术期刊的副主编。与合作者关于合金微结构演化在“神威·太湖之光”超级计算机上的相场模拟工作入围2016年国际高性能计算应用领域“戈登·贝尔”奖提名。
