Speaker: Prof. Haijun Yu (于海军教授）

Time: 16:00-17:00, 02 December 2024 (Monday) (Beijing time)

Venue: A103,Lijiao Building, BNU

Abstract

We propose two efficient energetic spectral-element methods in time for marching nonlinear　gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit　nonlinear method and one semi-implicit linear method. Different from other spectral methods in　time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit　method employs an energetic variational Galerkin form that can maintain the mass conservation　and energy dissipation property of the continuous dynamical system. Another advantage of this　method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term　to get the semi-implicit method. The semi-implicit method does not have superconvergence, but　can be improved by a few Picard-like iterations to recover the superconvergence of the implicit　method. Numerical experiments verify that the method using Legendre elements of degree three　outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order　exponential time difference Runge-Kutta method, which were known to have best performances　in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply　the method to a conservative Allen–Cahn equation, in which the conservation of discrete total　mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

About Prof. Yu

于海军，中国科学院数学与系统科学研究院研究员, 博士生导师，中国科学院大学岗位教授。分别于2002年,2007年获得北京大学学士学位和博士学位。曾先后在美国普林斯顿大学和普渡大学从事博士后研究。主要研究方向为高精度数值方法. 研究内容涉及复杂流体建模和计算，高维偏微分方程稀疏网格方法，相变路径计算方法，以及微分方程计算深度学习方法等. 作为课题负责人先后承担过自然科学基金委青年、面上、重大研究计划、国际合作等科研项目。