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AMATH 581 A: Scientific Computing

Meeting Time: 
MWF 8:30am - 9:20am
Location: 
LOW 216
SLN: 
10240
Joint Sections: 
AMATH 581 B, AMATH 581 D, AMATH 581 C
Instructor:
Eli Shlizerman
Eli Shlizerman

Syllabus Description:

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: 
Project-oriented computational approach to solving problems arising in the physical/engineering sciences, finance/economics, medical, social, and biological sciences. Problems requiring use of advanced MATLAB routines and toolboxes. Covers graphical techniques for data presentation and communication of scientific results. Prerequisite: either a course in numerical analysis or permission of instructor.
Credits: 
5.0
Status: 
Active
Last updated: 
October 17, 2018 - 9:00pm
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