COURSE DESCRIPTION
The goal of this course is to give you:
- practical experience with common computational methods in engineering, and
- enough theoretical understanding to know when those methods can go wrong
Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications.
This class is suitable for masters students, advanced undergraduates, or anyone interested in building a foundation in computational science.
Prerequisites: Calculus and some linear algebra
Text Book: Computational Science and Engineering by Gilbert Strang
Grades: 50% problem sets, 50% three in-class quizzes. Lowest problem set score will be dropped.
Problem Sets: Will be due in class.
SCHEDULE
| Event | Date | Related Documents |
|---|---|---|
| PSET 1 | Due Feb 23 | Sample code for clique problem, basic matrix problems Solutions to 1-4 Solutions to 5-6 |
| PSET 2 | Due Mar 1 | Solutions |
| Quiz 1 | Mar 8 in class | Practice problems. Solutions to practice problems. Quiz 1 with solutions |
| PSET 3 | Due Mar 22 | Solutions. Matlab code |
| PSET 4 | Due Apr 5 | Wave recording of a single note. Solutions. Code for problem 1 |
| Quiz 2 | Apr 12 in class | Practice problems with answers. Quiz 2 problems with answers |
| Makeup Quiz 2 | Due May 1 in class | |
| PSET 5 | Due Apr 26 | Solutions. Matlab code |
| PSET 6 | Due May 3 | Solutions. Finite difference code. Finite element code. Problem 5 code |
| Quiz 3 | May 10 in class | Sample problems. Solutions |
Syllabus
| Problem | Tools | Concerns |
|---|---|---|
| Resistor networks | Matrices, Backslash, LU | Don't find inverses — compute solutions to linear equations directly. Two and three dimensional problems are expensive. Use sparse matrices, if you can. |
| Stability of trusses | Null spaces, basis, QR | Don't orthogonalize vectors yourself — let QR do it. |
| Best fit polynomials | Projection, condition number | Don't solve normal equations. Beware of overfitting. |
| Problem | Tools | Concerns |
|---|---|---|
| Frequency identification | DFT, FFT | FFT is fast. Use it. |
| Blurring and Deblurring | Convolutions, deconvolutions | Be wary of division by small numbers. |
| Problem | Tools | Concerns |
|---|---|---|
| Chemical kinetics | Eigenvectors, ODEs, Forward Euler, Backward Euler | Beware of time steps with explicit schemes. |
| Mass spring systems | Midpoint rule. Reduction to first order system. | Make sure the numerics obeys the physics. Try for a second order scheme |
| Problem | Tools | Concerns |
|---|---|---|
| Hanging rope under load | Discretization, dirac delta, Green's functions | Make sure your scheme is second order. Make sure the matrix admits a solution. |
| 2d membrane under load | Laplacian stencil. Fast Poisson solvers | If matrix has Fourier modes as eigenvectors, use FFT for a fast solver. |
| Fluid flow in a triangle | Finite element method | If geometry complicated, use finite elements. |
