Numerical Methods II is the second half of a two-course series meant to introduce graduate students to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. We will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE.
Topics covered include:
- numerical quadrature
- two-point boundary value problems
- methods for ordinary differential equations
- methods for elliptic partial differential equations
- fast solvers, fast transforms, and multigrid methods
- parabolic partial differential equations
- hyperbolic partial differential equations
- convergence analysis and mesh refinement strategies
- Numerical linear algebra, some ODE, PDE
TextbookThe course textbook is Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. Leveque.
- Numerical Mathematics (2nd ed.) by A. Quarteroni, R. Sacco and F. Saleri
- Scientific Computing - An Introduction Using Maple and MATLAB by W. Gander, M.J. Gander, and F. Kwok
- Approximation Theory and Approximation Practice by L. N. Trefethen
- Spectral Methods in Matlab by L. N. Trefethen