MIT
Open Course Ware contains links to lecture notes on a great variety of topics.

Southern Methodist University
Prof. Lawrence F. Shampine
is an authority on numerical solution of ODEs using software packages such as Matlab. Click on his link to "some current work" and browse the topics.

University of Washington
Prof. Nathan Kutz,
an expert in nolinear wave equations, teaches a course on  scientific computing (see its notes here).

Northwestern University
Prof. Hermann Riecke,
an expert in the reaction-diffusion equations, computational neuroscience and such, teaches several courses on numerical methods. Among them is a course on spectral methods for PDEs, and a few others. Check out this useful and impressive website!

Prof. David Chopp
is another expert on numerical methods at Northwestern. He maintains a collection of his course notes under the rubric "Available Course Notes". Topics of his notes are similar to the topics of Prof. Riecke's notes, but, of course, the notes themselves are different. Prof. Chopp asks you to enter your name and email address before you are given access to his notes. Don't be scared -- this is safe, and the access is provided immediately once you have entered the requested information.

Royal Institute of Technology (Sweden)
A concise on-line course on numerical PDEs with some examples of codes
by Prof. Andre Jaun.

Rensselaer Polytechnic Institute
Prof. Mark Holmes
is an expert in asymptotic/perturbation methods and computational applied mathematics. He teaches a course on numerical methods for ODEs and PDEs and wrote a book based on the notes for that course. While the book's or notes' content is not available on-line, there are many example codes posted on his book's website (linked to Prof. Holmes' website).

University of Maryland at College Park
Prof. Dianne O'Leary
teaches courses on large-scale computation (mostly Linear Algebra techniques for large matrices).