《科学计算选讲》教学大纲 课程基本信息( Course Information) 课程代码 学时 学分 MA430 Credit hours) 32 (Course code (Credits 课程名称 科学计算选讲 Course Name) Selected Topics in Scientific Computing 课程属性 (Course Type) Selective(专业方向选修课B组) 开课学期 Department of Mathematics fall semester (School) (Term) 先修课程 (Prerequisite Numerical analysis, Functional Analys course 授课教师 应文俊 Wen jun Y (Instructors 本课程为科学计算和计算数学方向选讲课程,注重算法的分析与实现及 应用,面向高年级本科生,要求学生已修《数值分析》、《泛函分析》或相 关课程,同时有应用高级程序设计语言设计和实现数值算法的基础。本课程 希望学生能够应用课上所授科学计算方法求解一些来自工程和科学研究的有 课程简介代表性的问题,为进一步的深造打下一定的基础。本课程的内容包括但不局 (Do限于求解有限差分方程组的快速傳里叶变换方法,求解有限元方程组的几何 3050字多重网格方法,求解离散边界积分方程的快速多极算法,以及一些求解双曲 型偏微分方程的TⅧ格式和若干求解椭圆型和双曲型偏微分方程的笛卡尔直 角网格法,如嵌入边界法和嵌入界面法。本课程内容还将包括求解椭圆型偏 微分方程的自适应有限元方法和求解反应扩散方程及描述不可压流体运动的 Navier- Stokes方程的算子分裂方法等。 This course is intended for senior undergraduate students. The content includes the fast Fourier transform method for elliptic pdes with the finite difference method, the multigrid method for elliptic PDEs with the finite element method, the fast multipole method for elliptic pdes with the boundary integral method, the Tvd scheme for 课程简介 hyperbolic PDEs with the finite volume method, the operator splitting (Description) methods for PDEs involving multiple operators such as the reaction 300-500 diffusion equation and the Navier-Stokes equations, etc.. This course will also cover some adaptive finite element methods and multilevel methods for discrete equations on locally refined grids as well as some structured grid me thods such as the immersed boundary me thod, immersed interface method and the kernel free boundary integral method. A prerequisite for the course is an introductory course on numerical
《科学计算选讲》教学大纲 课程基本信息（Course Information） 课程代码 （Course Code） MA430 学时 （Credit Hours） 32 学分 （Credits） 2 课程名称 （Course Name） 科学计算选讲 Selected Topics in Scientific Computing 课程属性 (Course Type) Selective（专业方向选修课 B 组） 开课院系 （School） Department of Mathematics 开课学期 （Term） fall semester 先修课程 （Prerequisite course） Numerical analysis, Functional Analysis 授课教师 （Instructors） 应文俊 Wenjun Ying 课程简介 （Description） 300-500 字 本课程为科学计算和计算数学方向选讲课程，注重算法的分析与实现及 应用，面向高年级本科生，要求学生已修《数值分析》、《泛函分析》或相 关课程，同时有应用高级程序设计语言设计和实现数值算法的基础。本课程 希望学生能够应用课上所授科学计算方法求解一些来自工程和科学研究的有 代表性的问题，为进一步的深造打下一定的基础。本课程的内容包括但不局 限于求解有限差分方程组的快速傅里叶变换方法，求解有限元方程组的几何 多重网格方法，求解离散边界积分方程的快速多极算法，以及一些求解双曲 型偏微分方程的 TVD 格式和若干求解椭圆型和双曲型偏微分方程的笛卡尔直 角网格法，如嵌入边界法和嵌入界面法。本课程内容还将包括求解椭圆型偏 微分方程的自适应有限元方法和求解反应扩散方程及描述不可压流体运动的 Navier-Stokes 方程的算子分裂方法等。 课程简介 （Description） 300-500 字 This course is intended for senior undergraduate students. The content includes the fast Fourier transform method for elliptic PDEs with the finite difference method, the multigrid method for elliptic PDEs with the finite element method, the fast multipole method for elliptic PDEs with the boundary integral method, the TVD scheme for hyperbolic PDEs with the finite volume method, the operator splitting methods for PDEs involving multiple operators such as the reaction diffusion equation and the Navier-Stokes equations, etc.. This course will also cover some adaptive finite element methods and multilevel methods for discrete equations on locally refined grids as well as some structured grid methods such as the immersed boundary method, immersed interface method and the kernel free boundary integral method. A prerequisite for the course is an introductory course on numerical
analysis and a course on partial differential equations and functional analysis or their equivalents. The students for this course should be experienced with at least one computer programming language such as C, C++ or MatLab for numerical experiments 课程教学大纲( course syllabus) After completing the course, students should be able to 1. Solve some elliptic PDEs with the finite difference method 2. Apply the fast Fourier transform method to solve finite difference equations 3. Solve some elliptic PDEs with the finite element method *学习目标 4. Apply the geometric multigrid method to solve finite element equations 5. Solve some elliptic PDEs with the boundary integral method Outcomes) 6. Solve some parabolic PDEs with the operator splitting technique 7. Solve some hyperbolic PDes with some typical TVD schemes 8. Understand the fast multipole method 9. Understand some typical Cartesian methods 10. Understand some adaptive finite element methods 教学内容 教学方式作业及要求基本要求 查方式 Credit Teaching Intended Assessment hours methodology learning methods outcomes blackboard Finite difference ng homework method for linear blackboard problems by diffusion riting plus method homework equation in 1D computer computer lectured demonstration implementation *教学内容进度| Finite difference blackboard 安排及要求 writing homework understand reaction -diffusion blackboard problems by the gradIng (Class Schedule equation and hand and method homework techniques demonstration implementation LOD and ADI riting homework understand methods for blackboard problems by equation in 2D lectured demonstration implementation Finite difference blackboard understand ng homework
analysis and a course on partial differential equations and functional analysis or their equivalents. The students for this course should be experienced with at least one computer programming language such as C, C++ or MatLab for numerical experiments. 课程教学大纲（course syllabus） *学习目标 (Learning Outcomes) After completing the course, students should be able to: 1．Solve some elliptic PDEs with the finite difference method 2．Apply the fast Fourier transform method to solve finite difference equations 3．Solve some elliptic PDEs with the finite element method 4．Apply the geometric multigrid method to solve finite element equations 5．Solve some elliptic PDEs with the boundary integral method 6．Solve some parabolic PDEs with the operator splitting technique 7．Solve some hyperbolic PDEs with some typical TVD schemes 8．Understand the fast multipole method 9．Understand some typical Cartesian methods 10．Understand some adaptive finite element methods *教学内容、进度 安排及要求 (Class Schedule & Requirements) 教学内容 topics 学时 Credit hours 教学方式 Teaching methodology 作业及要求 tasks 基本要求 Intended learning outcomes 考查方式 Assessment methods Finite difference method for linear diffusion equation in 1D 2 blackboard writing blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework Finite difference for reaction-diffusion equation and operator splitting techniques 2 blackboard writing blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework LOD and ADI methods for linear diffusion equation in 2D 2 blackboard writing blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework Finite difference method for elliptic PDEs 3 blackboard writing blackboard solve homework problems by understand the method grading homework
demonstration implementation Fast Fourier riting homework understand transform blackboard problems by grading methods for hand and method homework finite difference writing plus computer lectured equations demonstration implementation Finite element ing plt the method fo 3 computer hand and method homework elliptic Pdes demonstration comp lectured implementation Multigrid method homework ng plu home quations demonstrationcomputer lectured implementati A posteriori error blackboard homework understand te and adaptive fin element method demonstration lectured implementation solve blackboard problems by the integral method3 riting hand and method homework for elliptic PDES computer lectured implementation Fast multipole blackboard homework understand method for writing plus problems by mute hand and demonstration lectured equations mplementation a few structured understand blackboard he methods fo homework elliptic PDEs lectured computer
writing plus computer demonstration hand and computer implementation lectured Fast Fourier transform methods for finite difference equations 2 blackboard writing blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework Finite element method for elliptic PDES 3 blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework Multigrid method for finite element equations 2 blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework A posteriori error estimate and adaptive finite element method 2 blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework Boundary integral method for elliptic PDES 3 blackboard writing solve homework problems by hand and computer implementation understand the method lectured grading homework Fast multipole method for boundary integral equations 2 blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework A few structured (Cartesian) grid methods for elliptic PDEs 2 blackboard writing solve homework problems by hand and computer understand the method lectured grading homework
mplementation solve blackboard homework understan Kernel-free writing plus problems by grading boundary method homework ntegral method demonstration computer lectured mplementation solve blackboard homework understand Finite volume writing plus problems by he method fo hand and method homework hyperbolic PDEs demonstration lectured implementation solve blackboard Total variation writing plus problems by grading diminishing computer hand and schemes demonstration lectured mplementation 考核方式 (Assessment seven homework assignment and one semester project methods and Grading 1. Finite difference schemes for PDEs, John Striwerda, SIAM 2004 2. Numerical solution of PDEs, Morton and Mayers, Cambridge 2005 教材或参考资料 3. Finite elements: theory, fast solvers and applications in solid mechanics, Dietrich Braess, Cambridge 2007. Other Reading 4. Iterative methods for sparse linear systems, Yousef Saad, SIAM 2003. Materials) 5. Selected topics in finite element methods, Z. Chen and H. Wu Beijing, 2010 6. The numerical solution of integral equations of the second kind, K. E. Atkinson, Cambridge 1997. 备注 (Notes)
implementation Kernel-free boundary integral method 2 blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework Finite volume method for hyperbolic PDEs 3 blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework Total variation diminishing schemes 2 blackboard writing plus computer demonstration solve homework problems by hand and computer implementation understand the method lectured grading homework 考核方式 (Assessment methods and Grading) seven homework assignment and one semester project 教材或参考资料 (Textbooks & Other Reading Materials) 1. Finite difference schemes for PDEs, John Striwerda, SIAM 2004. 2. Numerical solution of PDEs, Morton and Mayers, Cambridge 2005. 3. Finite elements: theory, fast solvers and applications in solid mechanics, Dietrich Braess, Cambridge 2007. 4. Iterative methods for sparse linear systems, Yousef Saad, SIAM 2003. 5. Selected topics in finite element methods, Z. Chen and H. Wu, Beijing, 2010. 6. The numerical solution of integral equations of the second kind, K. E. Atkinson, Cambridge 1997. 备注 （Notes）
