Lectures
- 1. Difference Methods for Ordinary Differential Equations
- 2. Finite Differences
- 3. One-way Wave Equation and CFL, von Neumann Stability
- 4. Comparison of Methods for the Wave Equation
- 5. Second-order Wave Equation
- 6. Wave Profiles, Heat Equation
- 7. Finite Differences for the Heat Equation
- 8. Convection-Diffusion, Conservation Laws
- 9. Conservation Laws, Analysis, Shocks
- 10. Shocks and Fans from Point Source
- 11. Level Set Method
- 12. Matrices in Difference Equations
- 13. Elimination with Reordering: Sparse Matrices
- 14. Financial Mathematics, Black-Scholes Equation
- 15. Iterative Methods and Preconditioners
- 16. General Methods for Sparse Systems
- 17. Multigrid Methods
- 18. Krylov Methods
- 19. Conjugate Gradient Method
- 20. Fast Poisson Solver
- 21. Optimization with constraints
- 22. Weighted Least Squares
- 23. Calculus of Variations
- 24. Error Estimates, Projections
- 25. Saddle Points
- 26. Two Squares, Equality Constraint
- 27. Regularization by Penalty Term
- 28. Linear Programming and Duality
- 29. Duality Puzzle, Inverse Problem, Integral Equations
Mathematical Methods for Engineers II
Course Summary
This course is based on 18.086 Mathematical Methods for Engineers II, Spring 2006 made available by Massachusetts Institute of Technology: MIT OpenCourseWare under the Creative Commons BY-NC-SA license.
This graduate-level course is a continuation of Computational Science and Engineering I (Mathematical Methods for Engineers I). The two major topics of this course are Initial Value Problems and Solution of Large Linear Systems. Initial Value Problems cover topics such as Wave Equation, Heat Equation, Convection Equation, Conservation Laws, Navier-Stokes Equation, Finite Difference Methods, Lax Equivalence Theorem, Fourier Analysis, Separation of Variables and Spectral Methods. The topics under Solution of Large Linear Systems are Finite Differences, Finite Elements, Optimization, Direct Methods, Iterative Methods and Preconditioning, Inverse Problems and Regularization.
Reading Material
1. Textbook: Computational Science and EngineeringStrang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.
2. Finite Differences and Fast Poisson Solvers
Section 3.5 - Finite Differences and Fast Poisson Solvers
3. Section 5.1- Finite Difference Methods
Section 5.1- Finite Difference Methods
4. Accuracy and Stability
Section 5.2 - Accuracy and Stability for $u_t= cu_x$
5. The Wave Equation and Staggered Leapfrog
Section 5.3 - The Wave Equation and Staggered Leapfrog
6. The Heat Equation and Convection-Diffusion
Section 5.4 - The Heat Equation and Convection-Diffusion
7. Difference Matrices and Eigenvalues
Section 5.5 - Difference Matrices and Eigenvalues
8. Nonlinear Flow and Conservation Laws
Section 5.6 - Nonlinear Flow and Conservation Laws
9. Level Sets and the Fast Marching Method
Section 5.7 - Level Sets and the Fast Marching Method
10. Elimination with Reordering
Section 6.1 - Elimination with Reordering
11. Iterative Methods
Section 6.2 - Iterative Methods
12. Multigrid Methods
Section 6.3 - Multigrid Methods
13. Krylov Subspaces and Conjugate Gradients
Section 6.4 - Krylov Subspaces and Conjugate Gradients
14. The Saddle Point Stokes Problem
Section 6.5 - The Saddle Point Stokes Problem
15. One Fundamental Example
Section 7.1 - One Fundamental Example
16. Calculus of Variations
Section 7.2 - Calculus of Variations
Course Material
1. Approximate and Incomplete FactorizationsChan, Tony, and Henk A. Van der Vorst. "Approximate and Incomplete Factorizations." (272 KB pdf file)
2. Numerical Solution of Saddle Point Problems
Benzi, M., G. H. Golub, and J. Liesen. "Numerical Solution of Saddle Point Problems." Acta Numerica 14 (2005): 1-137. (1.0 MB pdf file)
