In this course, we study finite difference, finite element, and finite volume methods for the numerical solution of elliptic, parabolic, and hyperbolic partial differential equations. The course will concentrate on the key ideas underlying the derivation of numerical schemes and a study of their stability and accuracy. Students will have the opportunity to gain computational experience with numerical methods with a minimal of programming by the use of Matlab's PDE Toolbox software.
Since there are many sophisticated computer packages available for solving partial differential equations, one might think that a thorough understanding of the numerical methods employed is no longer necessary. A striking example of why naive use of such codes can lead to disastrous results is the sinking of the Sleipner A offshore oil platform in Norway in 1991, resulting in an economic loss of about $700 million. The post accident investigation traced the problem to inaccurate finite element approximation of the linear elastic model of the structure (using the popular finite element program NASTRAN). The shear stresses were underestimated by 47%, leading to insufficient design. More careful finite element analysis, made after the accident, predicted that failure would occur with this design at a depth of 62m, which matches well with the actual occurrence at 65m.
Sections Taught This Semester:
- Numerical Solution of Partial Differential Equations 642:575 -- Spring 2006