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.
- (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
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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
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