Finite Differences for the Heat Equation

Lecture Description

Course Description

This graduate-level course is a continuation of Computational Science and Engineering I. Topics include numerical methods; initial-value problems; network flows; and optimization.

Course Index

1. Difference Methods for Ordinary Differential Equations
2. Finite Differences, Accuracy, Stability, Convergence
3. The One-way Wave Equation and CFL / von Neumann Stability
4. Comparison of Methods for the Wave Equation
5. Second-order Wave Equation (including leapfrog)
6. Wave Profiles, Heat Equation / point source
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 (1D, 2D, 3D)
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 / Multigrid Continued
19. Conjugate Gradient Method
20. Fast Poisson Solver
21. Optimization with constraints
22. Weighted Least Squares
23. Calculus of Variations / Weak Form
24. Error Estimates / Projections
25. Saddle Points / Inf-sup condition
26. Two Squares / Equality Constraint Bu = d
27. Regularization by Penalty Term
28. Linear Programming and Duality
29. Duality Puzzle / Inverse Problem / Integral Equations