AMATH 301 C: Beginning Scientific Computing

Winter 2023
Meeting:
MWF 3:30pm - 4:20pm / KNE 210
SLN:
10215
Section Type:
Lecture
Instructor:
PYTHON
Syllabus Description (from Canvas):

We will begin with ODE solvers applied to both initial and boundary value problems. Our application will be to find the eigenstates of a quantum mechanical problem or of an optical waveguide.

We will introduce the idea of finite-differencing of differential operators. Our application will be to two problems: vibrating modes of a drum and the evolution of potential vorticity in an advection-diffusion problem of fluid mechanics.

Transform methods for PDEs will be introduced with special emphasis given to the Fast-Fourier Transform. We will revisit the potential vorticity in an advection-diffusion problem of fluid mechanics by using these spectral techniques.

For subtle computational domains, the use of a finite element scheme is compulsory. The steady-state flow of a fluid over various airfoils will be considered.

 

(1) Solution Methods for Differential Equations: (2 weeks)
(a) Initial value problems
(b) Euler method, 2nd- and 4th-order Runge-Kutta, Adams-Bashford
(c) Stability and time stepping issues
(d) Boundary values problems: shooting/collocation/relaxation
 
(2) Finite Difference Schemes for Partial Differential Equations: (3 weeks)
(a) Collocation
(b) Stability and CFL conditions
(c) Time and space stepping routines
(d) Tri-diagonal matrix operations
 
(3) Spectral Methods for Partial Differential Equations: (3 weeks)
(a) The Fast-Fourier transform (FFT)
(b) Chebychev transforms
(c) Time and space stepping routines
(d) Numerical filtering algorithms
 
(4) Finite Element Schemes for Partial Differential Equations: (2 weeks)
(a) Mesh generation
(b) Advanced matrix operations
(c) Boundary conditions
 

 

Catalog Description:
Introduction to the use of computers to solve problems arising in the physical, biological, and engineering sciences. Application of mathematical judgment, programming architecture, and flow control in solving scientific problems. Introduction to MATLAB or Python routines for numerical programming, computation, and visualization. Prerequisite: either MATH 125, Q SCI 292, or MATH 135. Offered: AWSpS.
GE Requirements Met:
Natural Sciences (NSc)
Credits:
4.0
Status:
Active
Last updated:
May 25, 2024 - 7:04 pm