AMATH 581 F: Scientific Computing

Autumn 2021
MWF 8:30am - 9:20am / HCK 132
Section Type:
Joint Sections:
AMATH 581 A , AMATH 581 D , AMATH 581 B , AMATH 581 C , AMATH 581 E
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:
Survey of numerical techniques for differential equations. Emphasis is on implementation of numerical schemes for application problems. For ordinary differential equations, initial value problems and second order boundary value problems are covered. Methods for partial differential equations include finite differences, finite elements and spectral methods. Requires use of a scientific programming language (e.g., MATLAB or Python). Prerequisite: either a course in numerical analysis or permission of instructor. Offered: A.
Last updated:
April 11, 2024 - 10:20 am