Numerical PDEs

Course Information

• Numerical Solutions of Partial Differential Equations (1107016008)
• This is an English taught course for students ready for both master and doctor degree.
• The application used to demonstarte the live codes, interactive computing during lecture is call Jupyter Notebook. The notebooks for this course are available to be viewed on link.
• Teaching QQ group (slides, lecture notes, recording): 700601089

Purpose of the Course:

This course presents the fundamentals of modern numerical techniques for a wide range of equations. The emphasis is on building an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods. The goals are:

• To understand the fundamental mathematics theory and algorithms of finite difference methods;
• To be able to implement finite difference methods for simple 1d and 2d problems as well as to evaluate and to interpret the numerical results;
• To be able to solve some engineering problems by using known algorithms.

Textbook:

• Randall J. LeVeque. Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.

  Reference:

• K. W. Morton and D. F. Mayers. Numerical Solution of Partial Differential Equations, Cambridge University Press 2005.
• Alfio Quarteroni and Alberto Valli. Numerical Approximation of Partial Differential Equations, Springer 1994.
• Susanne C. Brenner, L. Ridgway Scott. The Mathematical Theory of Finite Element Methods, 3rd edition, Springer.

Assessment Method:

Grade for this course is determined by the performance in computer projects and final exams, which are designed based on the course objectives.

Exercise

• Requirement and samples. Exercise Sample (PDF) Latex Sample (PDF) tex
• Exercise 1. PDF tex
• Exercise 2. PDF tex
• Sample Problems PDF

Teaching Contents:

• Chapter 0. Introduction Slide
• Scientific Computing with Python Notebook
• Chapter 1. Finite Difference Approximations [Notebook]
• Chapter 2. Steady States and Boundary Value Problems
• The centered finite difference method for 1d linear BVP. Code Notebook
• The eigenvalues of the matrix from the FDM for 1d linear BVP. Note
• Singularly Perturbed Equations Notebook
• Chapter 3. Elliptic Equations
• Laplace’s Equations Notebook
• Chapter 4. Iterative Methods for Sparse Linear Systems
• Chapter 5. The Initial Value Problem for Ordinary Differential Equations
• Chapter 6. Zero-Stability and Convergence for Initial Value Problems
• Chapter 7. Absolute Stability for Ordinary Differential Equations
• Chapter 8. Stiff Ordinary Differential Equations
• Chapter 9. Diffusion Equations and Parabolic Problems
• Chapter 10. Advection Equations and Hyperbolic Systems
• Advection Equation Notebook
• Chapter 11. Mixed Equations Slide
• Discontinuous Galerkin Methods