Numerical Solutions of Partial Differential Equations

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.

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