Numerical methods are the techniques used to solve equations which for various reasons are not easy to solve analytically.
Beginning with the introductory course, FORTRAN for Scientists and Engineers (COS215), I introduce the concept of a numerical or approximate solution early on, even as the students are making their first attempts at programming.
In the second semester of this sequence, Numerical Methods with FORTRAN (COS405), I delve deeper into the techniques of finding roots, integrating and differentiating functions, finding approximate curves that fit measured or tabulated data, as well as traditional matrix techniques for solving sets of algebraic equations. This last topic provides the core for subsequent solutions of differential equations.
The final course in the sequence, Topics in Scientific Computation (COS515), is used as an arena for several advanced topics such as finite-element modeling, parallel programming and supercomputers, or computer modeling and simulation. Finite element modeling is a state-of-the-art technique for solving time-dependent or steady-state ordinary or partial differential equations on very irregular and non-uniform domains is widely used in engineering and scientific applications. Unlike courses offered in the Engineering departments, which emphasize the use of ``canned'' commercial programs, I try and stress the programming aspects so that students develop the ability to solve new and unique problems that are beyond the scope and design of the commercial programs.