Course Introduction

• Prerequisite：Numerical analysis, differential equations
• Recommended for： graduate students
• Introduction：

This course addresses students who are interested in numerical methods for partial differential equations. After introducing basic numerical approximation of polynomial interpolation and Fourier approximation, we particularly focus on fundamentals of finite difference methods and important concepts such as stability, convergence, and error analysis. Other methods such as finite volume, finite element and spectral methods will also be introduced briefly. Partial differential equations will be solved in this course include Poisson equation, heat equation, wave equation, convection-diffusion problems, Maxwell equation and Navier-Stokes equations etc.

• Syllabus：

1. Lagrange and Fourier approximation
2. Numerical differentiation and numerical integration
3. Finite difference method for initial value problems (hyperbolic equations)
4. Finite difference method for boundary value problems (elliptic equations)
5. Finite difference method for initial boundary value problems (parabolic equations)
6. Analysis of finite difference methods: consistency, stability and Von Neumann analysis for multi-steps methods.
7. Introduction of the spectral method, finite element method and finite volume method
8. Numerical methods for solving the convection-diffusion equation, Maxwell equation and Navier-Stoke equation.

• Reference：

1. Randali LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, 2007.
2. Christian Grossmann, Hans-Gorg Roos and Martin Stynes, Numerical Treatment of Partial Differential Equations, Springer, 2007.
3. Stig Larsson and Vidar Thomee, Partial Differential Equations with Numerical Method. Springer, 2005