Course Category：Professional Basic Course
Prerequisites：Advanced Mathematics (Mathematical Analysis and Linear Algebra), Equations of Mathematical Physics, Computational Methods, Programming.
Aims & Requirements：
This course is a speciality foundation course for Doctor and Master degree candidates majoring in Mathematics-related specialities. It can also be taken as an elective course for Master degree candidates majoring in Physics, Mechanics, Chemistry, and Engineering Science. The scope of the course is the algorithms, and their convergence and stability analysis for initial and boundary value problems of ordinary differential equations (ODEs), and difference methods for partial differential equations (PDEs) of parabolic, hyperbolic, and elliptic types. Participants of this course are expected to command the basic techniques of numerically solving differential equations, to lay the foundation for further study of scientific computing.
Chapter 1 Numerical methods for initial and boundary value problems of ODEs
Euler methods; Runge-Kutta methods; linear multi-step methods; stability, convergence, and error estimates; numerical methods for boundary value problems of ODEs.
Chapter 2 Difference methods for equations of parabolic type
Foundations of constructing difference schemes; explicit and implicit difference schemes; stability and convergence of difference schemes; difference methods for higher dimensional equations of parabolic type; alternating direction implicit difference methods.
Chapter 3 Difference methods for equations of hyperbolic type
Methods of characteristic line for 1-dimensional equation of hyperbolic type; difference methods for the first order linear equations of hyperbolic type; hyperbolic conservation law equations and conservational difference schemes; difference methods for the second order wave equations.
Chapter 4 Difference methods for equations of elliptic type
Difference methods for the boundary value problems of the first kind of Poisson equations; finite volume methods for Poisson equations; Convergence and error estimate for difference methods; iterative solution of difference equations of elliptic type; multi-mesh methods.
YU Dehao, TANG Huazhong, Numerical Solution of Differential Equations, Science Press, Beijing, 2002.
ZHANG Wensheng, Finite Difference Methods for Partial Differential Equations in Scientific Computing, Science Press, Beijing, 2006.
 J. W. Thomas. Numerical Partial Differential Equations: Finite Difference Methods. Springer-Verlag New York Inc. 1995.
 HU Jianwei, TANG Huaimin, Numerical Methods for Differential Equations, Science Press, Beijing, 1999.
Author：Weiying Zheng (Academy of Mathematics and Systems Science)