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 a 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
- spectral methods
- 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
Prerequisites
- Numerical linear algebra, some ODE, PDE
Textbook
The course textbook is Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. Leveque. It is available for download in PDF format . Matlab and LaTex files are available on the author's website.Recommended Texts
- 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