AMATH 581 B: Scientific Computing

Autumn 2025
Meeting:
MWF 8:30am - 9:20am / SMI 205
SLN:
10262
Section Type:
Lecture
Joint Sections:
AMATH 581 E , AMATH 581 D , AMATH 581 A
Instructor:
FOR ACM CAMPUS MS STUDENTS ONLY
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
 

 

Days

Mon

Wed

Fri

Solution Methods for Differential Equations

Week 0 (Sep 24-26)

Intro

Ch 8

Python Programming

(RECORDED Materials

No In Person Lecture)

Ch 1, Ch7

Week 1 (Sep 29-Oct3)

Initial Value Problems (IVP): Forward Integration

IVP Part 1

Ch 8.1

IVP Part 1

Ch 8.1

IVP Part 1

Ch 8.2

 

Week 2 (Oct 6-10)

Initial Value Problems (IVP): Implicit Integration and Stability

IVP Part 2

Ch 8.2

IVP Part 2

Ch 8.3

IVP Part 2

Ch 8.3

Week 3 (Oct 13-17)

Boundary Values Problems (BVP): Shooting and Relaxation (Direct)

BVP 

Ch 8.4

BVP 

Ch 8.5,8.6

BVP 

Ch 8.7

Finite Difference Schemes for Partial Differential Equations

Week 4 (Oct 20-24)

Finite Difference Derivative Operators for PDE

Linear Systems

Ch 8.8

PDEs, Derivative Operators

Ch 8.9

PDE,Derivative Operators

Ch 8.9, 8.10

Week 5 (Oct 27-31)

Finite Difference: Solving Linear Problems

FD

Ch 9.1

FD

Ch 9.2

Time Stepping PDE

Ch 10.1

Week 6 (Nov 3-7)

Finite Difference: PDE Solution and Stability

Stability

Ch 10.2

Stability and FD Methods

Ch 10.2

Stability and FD Methods

Ch 10.3

Spectral Methods for Partial Differential Equations

Week 7 (Nov 10-Nov 14)

Spectral Methods for PDE

Fourier Series and Fourier Transform

Ch 11.1-11.2

Fourier Methods for PDEs

Ch 11.1-11.2

Speed and Stability of Spectral Methods

Ch 11.2

Week 8 (Nov 17- Nov 21)

Spectral Methods for PDE

Applications of Spectral Approaches to PDE

Ch. 11.8

Advanced Spectral Approaches to PDE

(Chebyshev)

Ch. 11.3

Advanced Spectral Approaches to PDE

(Chebyshev)

Ch. 11.3

Finite Element Schemes for Partial Differential Equations

Week 9 (Nov 24 - Nov 28)

Finite Elements (FE) Introduction

FE Intro

Ch 12.1

FE Intro - simplexes, complices

Ch 12.1

THANKSGIVING

NO LECTURE

Week 10 (Dec 1 - Dec 5)

Finite Elements+Review

Last Week of Instruction

Variational approach for FE application to PDE

Ch 12.2

Tools for FE approach to PDE

 

 REVIEW

Week 11 (Dec 8 - Dec 11)

Finals Week

 

 

 

 

 

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.
Credits:
5.0
Status:
Active
Last updated:
October 7, 2025 - 2:07 pm