课程大纲
COURSE SYLLABUS
1.
课程代码/名称
Course Code/Title
MAT7088 偏微分方程数值解
Numerical Solutions to Partial Differential Equations
2.
课程性质
Compulsory/Elective
选修 Elective
3.
课程学分/学时
Course Credit/Hours
3/48
4.
授课语言
Teaching Language
英文 English
5.
授课教师 Instructor(s)
Alexander Kurganovm, 讲席教授
Alexander Kurganov, Chair Professor;
6.
是否面向本科生开放
Open to undergraduates
or not
是 Yes
7.
先修要求
Pre-requisites
（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates,
please indicate the difference.）
1.MA201a 常微分方程 A,
2.MA303 偏微分方程,
3.MA325 偏微分方程数值解 或者 MAT8028 科学计算
1.MA201a Ordinary Differential Equations,
2.MA303 Partial Differential Equations，
3.MA325 Numerical Solutions to Partial Differential Equations OR MAT8028
Scientific Computing
8.
教学目标 Course Objectives
（ 如 面 向 本 科 生 开 放 ， 请 注 明 区 分 内 容 。 If the course is open to undergraduates, please indicate the
difference.）
教给学生时间独立和演化的微分方程的经典的和传统的数值方法。介绍它们的算法，分析，优缺点
和相关软件。这会帮助学生在今后的科研工作中，运用他们学到的技能，设计自己的算法，解决非
标准的微分方程问题。
To teach the students both classical and modern numerical methods for both time-independent and
evolutionary differential equations. To introduce both numerical algorithms and analysis of their properties
as well as the reasons behind success and failure of numerical methods and software. This will help the
students to apply the studied numerical techniques and to successfully design their own solution approach
for any nonstandard problems they may encounter in their research work.
9.
教学方法 Teaching Methods
（ 如 面 向 本 科 生 开 放 ， 请 注 明 区 分 内 容 。 If the course is open to undergraduates, please indicate the
difference.）
理论与编程并重，并辅以前沿课题应用
Teaching in both theory and programming, including applications to cutting edge problems
10.
教学内容 Course Contents
（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the
difference.）
Part I. Numerical Methods for Time-Independent Differential Equations
Some Model Problems
Finite-Difference Approximations
Steady States and Boundary Value Problems
- simple finite-difference methods
- local truncation error, global error
- stability, consistence, convergence
- high-order methods
- compact finite-difference schemes
- fast Poisson solver
Part II. Numerical Methods for Evolutionary Differential Equations
Numerical Methods for ODEs
- initial value problems
- linear multistep methods
- Runge-Kutta methods
- stability and stiffness
- local truncation error, global error
- stability, consistence and convergence
- adaptive error control
Finite-Difference Methods
- local truncation error and order of accuracy
- semi-discretization (method of lines): boundary conditions, stability and
convergence
- fully-discrete schemes: general linear stability and convergence
- Lax equivalence theorem
- CFL condition
- Lax-Friedrichs scheme
- Lax-Wendroff scheme
- modified equation
- explicit and implicit schemes
- operator splitting methods
Stability for Constant Coefficient Problems
- Fourier analysis for scalar equations and for systems
- eigenvalue analysis
Variable Coefficient and Nonlinear Problems
- freezing coefficients and dissipativity
- schemes for one-dimensional hyperbolic systems
- nonlinear stability and energy methods
Dispersion and Dissipation
- dispersion relation, phase velocity, group velocity
- the wave equation
- the KdV equation
- Lagrangian methods
Numerical Methods for Initial-Boundary Value Problems
- parabolic problems
- hyperbolic problems
- infinite or large domains and artificial boundaries
Several Space Variables and Dimensional Splitting Methods
- Alternating Direction Implicit scheme
- Locally One-Dimensional (LOD) scheme
Finite-Volume Methods
- finite-volume piecewise polynomial approximations
- one-dimensional scalar conservation laws
- first-order Godunov and Lax-Friedrichs schemes
- total variation and nonlinear stability
- higher-order schemes
- systems of conservation laws
- multidimensional problems
- central and upwind methods
Collocation and spectral methods
11.
课程考核 Course Assessment
作业（30%）+期中（30%）+期末考试（40%）
Assignment (30%) + Mid-term exam(30%) + final-term exam (40%)
12.
教材及其它参考资料 Textbook and Supplementary Readings
参考教材 Textbook：
1. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-
Dependent Problems, by Randall J. LeVeque, SIAM, 2007.
2. Numerical Methods for Evolutionary Differential Equations, by Uri M. Ascher, SIAM Computational
Science and Engineering, 2008
3. Numerical Solution of Partial Differential Equations: An Introduction, 2nd edition, by K. W. Morton,
D. F. Mayers, 2005.
4. Finite-Volume Methods for Hyperbolic Problems, by Randall J. LeVeque, Cambridge Texts in Applied
Mathematics, 2002.
其他参考资料 Supplementary Readings：
1. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, by L. N.
Trefethen, Cornell University, 1996.
2. Numerical Methods for Conservation Laws, by Randall J. LeVeque, Birkhauser-Verlag, 1990.