Local Discontinuous Galerkin Method Coupled with   Implicit Integration Factor Methods for Reaction-Diffusion Systems

Xijun Yu, Na An, Chaobao Huang, Maochang Duan

Laboratory of Computational Physics

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Abstract: In this take, on two-dimension unstructured triangular meshes, an efficient fully -discrete local discontinuous Galerkin (LDG) method is presented for solving the nonlinear reaction-diffusion systems, which are often used as mathematical models for many biological, physical and chemical sciences. In order to overcome LDG method for the strict time-step restriction (=O() of explicit schemes for stability, the implicit integration factor (IIF) method based on Krylov subspace approximation is introduced into the temporal discretization, in which the time-step can be taken as =O(). The method can derive the numerical approximations not only for solutions but also for their gradients at the same time comparing other methods for the numerical solutions only. Moreover, the method can compute element by element and avoid solving a global system of nonlinear algebraic equations as the standard implicit schemes do, so that the method can reduce the computational cost greatly. Numerical experiments for the reaction-diffusion equations with exact solutions and the morphogenesis systems from developmental biology are conducted to illustrate the efficient, accurate and advantages of the LDG method coupled with the Krylov IIF time discretization.

Keywords: Nonlinear Reaction-diffusion Systems; Local Discontinuous Galerkin (LDG) Method Krylov Implicit Integration Factor Methods; Numerical Simulation

局部间断Galerkin方法结合隐式积分因子

方法求解非线性反应扩散方程组

蔚喜军，安娜，黄朝宝，段茂昌

北京应用物理与计算数学研究所，计算物理实验室，北京，100088

摘要：非线性反应扩散方程组常常被用作许多物理和生物科学的数学模型。本文在二维非结构三角形网格上给出了一种有限地求解非线性反应扩散方程组的全离散局部间断Galerkin方法。为了克服显式时间离散稳定性要求时间步长=O()，基于Krylov子空间的隐式积分因子（IIF）方法（时间步长 (=O()）被引入。相比其它方法，LDG结合Krylov IIF方法不仅能够得到近似解，同时能够得到解的梯度。方法能够逐单元计算，避免求解整体非线性代数方程组，极大地节约计算花费。数值实验模拟了具有精确解的反应扩散方程组和生物生态发展模型，展示了LDG结合Krylov IIF 方法有效性和精度。

关键词：非线性反应扩散方程组；局部间断Galerkin方法；Krylov隐式积分因子方法；数值模拟

报告时间： 10:00- 11:00, June 28, (Wednesday), 2017

报告地点： Room 316, the 3^rd floor of the environmental building, Renmin University of China

Curriculum Vitae of Prof. YU

Xijun Yu (蔚喜军), Doctor and Professor, graduated in 1993 from Institute of Computational Mathematics and Scientific and Engineering Computing of Chinese Academy of Science. Now Yu is vice-director of Laboratory of Computational Physics (LCP) of Institute of Applied Physics and Computational Mathematics (IAPCM) and Doctoral supervisor of IAPCM, University of Science and Technology of China，Northeast Normal University，member of the editorial board in Chinese Journal of Computational Physics and ACTA Aerodynamica Sinica and Journal of Mathematical Research and Applications (JMRA).

Yu has long been engaged in study on the numerical methods of fluid dynamics, especially on the structured and unstructured grid, the high resolution finite volume and finite element method, including numerical mesh generation and adaptive method, multimedia fluid and interface treatment. In the study of finite element numerical method of fluid dynamics, he has achieved the great scientific research achievements. More than 90 papers were published in journals at home and abroad, such as the J Comput Math, J Comput Phys, J Comput Applied Math, SIAM Multiscal Model Simul, Int J Numer Meth Fluids, Comput Meth Appl Mech Engrg, Computer Phys Commun.etc.

蔚喜军, 博士和研究员，1993年中国科学院计算中心获得博士学位，1995年北京应用物理与计算数学研究所博士后流动站出站。现任北京应用物理与计算数学研究所究员，计算物理国防科技重点实验室副主任，博士生导师，中国科技大学、东北师范大学兼职博士生导师。《计算物理》、《空气动力学学报》和《Journal of Mathematical Research and Applications (JMRA)》杂志编委。

蔚喜军研究员长期从事流体力学数值方法研究，特别是结构和非结构网格上高分辨率有限体积和有限元方法，包括数值网格生成与自适应方法，多介质流体力学界面处理及数值模拟等。在流体力学有限元数值方法研究方面，取得国内一流原创性科研成果。在J Comput Math、J Comput Phys、J. Comp Applied Math、SIAM Multiscal Model Simul、Int J Numer Meth Fluids、Comput Meth Appl Mech Engrg、Computer Phys. Commun.等国内外期刊发表论文90多篇，合作撰写信息与计算科学丛书《多介质流体动力学计算方法》（科学出版社）1本。

联系方式：电话 10-59872510， 61935157，Email: yuxj@iapcm.ac.cn