讲座主题：High-order complex-resolvent IMEX Runge--Kutta methods for semilinear parabolic equations

专家姓名：张弘

工作单位：国防科技大学

讲座时间：2026年10月20日 10:30-11:30

讲座地点：腾讯会议：202-293-760

主办单位：烟台大学数学与信息科学学院

内容摘要：

We construct high-order complex-resolvent IMEX Runge--Kutta (CIMEXRK) methods for semilinear parabolic equations by composing first-order IMEX Euler maps. Each stage combines an explicit forward-Euler evaluation of the nonlinearity with a backward-Euler resolvent and can be implemented with two solution-sized work vectors. A sequential composition with positive real increments has a second-order barrier. We remove this obstruction by using complex increments with positive real parts and determine an operator sector in which every resolvent factor is contractive. The complete additive Runge--Kutta order conditions are derived up to order four. Algebraic elimination shows that three stages cannot attain order three and that four, five, or six stages cannot attain order four. For nonnegative self-adjoint generators, we prove discrete parabolic smoothing, the nonsmooth-data estimate, and a weakly singular convolution estimate for nonlinearities. These estimates yield stability and order-$p$ convergence with constants independent of the spatial cutoff under the stated regularity assumptions. Numerical experiments for the Allen--Cahn,Cahn--Hilliard,phase-field-crystal,and two-dimensional Navier--Stokes equations examine the initial-layer error, positive-time order recovery, work--precision behavior, and stability for large time steps.

主讲人介绍：

张弘，国防科技大学数学系副教授，硕士生导师。本科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。在SIAM Numer. Anal., SIAM J. Sci. Comput., J. Comput. Phys.等期刊发表论文60余篇，入选湖南省青年杰青，湖湘青年英才，国防科技大学高层次创新人才。主持国家自然科学基金面上、青年项目、军委科技委项目等。