| Official Information | |
|---|---|
| Course Number: | Math 5044.001 |
| CRN: | 38528 |
| Course Title: | Introduction to Numerical Analysis II |
| Times: | MW 9:00-10:20 |
| Places: | Wachman Hall 617 |
| Instructor: | Benjamin Seibold |
| Instructor Email: | seibold(at)temple.edu |
| Instructor Office: | Wachman Hall 518 |
| Instructor Office Hours: | MW 10:20-11:30 |
| Course Textbook: | Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations - Steady State and Time Dependent Problems, SIAM, 2007 |
| Further recommended reads: |
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| Official: | Course Syllabus |
| Topics Covered: | This course provides a rigorous introduction to numerical methods for ordinary differential equations, establishing both knowledge and understanding of modern and efficient methods, as well as tools of analysis to understand when and why different methods work (or fail). Particular topics: Runge-Kutta, multistep, and Taylor series methods. Deferred correction. Convergence and stability. Error analysis. Stiff problems. Boundary value problems. Finite differences. Stochastic ODEs. |
| Course Goals: | Provide a rigorous mathematical basis for numerical methods for ordinary differential equations, and lay the groundwork for more advanced courses on numerical differential equations. Provide insight and intuition to fundamental challenges inherent to many problems in computational science and engineering. |
| Attendance Policy: | Students are expected to attend every class. If a student cannot attend a class for some justifiable reason, he or she is expected to contact the instructor before class. |
| Course Grading: | Homework: 50%; exams: 50%. |
| Final Exam Date: | 05/02/2019. |
| Course Schedule | |
| 01/14/2019 Lec 1 | Introduction: Ordinary differential equations, Linear ODE
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| 01/16/2019 Lec 2 | Predator-prey model, Lipschitz continuity, IVPs vs. BVPs
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| 01/23/2019 Lec 3 | Fundamental concepts in numerical methods for ODEs
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| 01/28/2019 Lec 4 | Taylor series methods: advantages and drawbacks
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| 01/30/2019 Lec 5 | Runge-Kutta methods: Butcher tableau
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| 02/04/2019 Lec 6 | General Runge-Kutta methods, ERK-DIRK-IRK
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| 02/06/2019 Lec 7 | Order conditions
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| 02/11/2019 Lec 8 | Embedded methods, adaptive time-stepping, dense output, Richardson extrapolation
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| 02/13/2019 Lec 9 | Linear multistep methods: order conditions
Read:
Linear multistep methods
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| 02/18/2019 Lec 10 | Adams, BDF, and predictor-corrector methods
Read:
BDF,
Predictor-corrector
|
| 02/19/2019 Lec 11 | Zero-stability and convergence: failure of zero-stability, difference equations
Read:
Truncation error
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| 03/11/2019 Lec 12 | Zero-stability and convergence theorems
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| 03/12/2019 Lec 13 | Absolute stability: stability regions for different methods
Read:
Absolute Stability
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| 03/13/2019 Lec 14 | Boundary locus method, linear systems
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| 03/18/2019 Lec 15 | Stiff problems: Schemes for different stiffness types, A-stability, L-stability
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| 03/20/2019 Lec 16 | Mildly stiff problems, RK-Chebyshev methods
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| 03/25/2019 Lec 17 | Stiffness in nonlinear problems
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| 03/26/2019 Lec 18 | Order reduction
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| 03/27/2019 Lec 19 | Implementation of implicit RK methods
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| 04/01/2019 Lec 20 | Semi-implicit methods: ImEx Runge-Kutta methods
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| 04/03/2019 Lec 21 | ImEx multistep methods
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| 04/08/2019 Lec 22 | Other methods and special problems: SSP schemes, Hamiltonian ODEs
Read:
Hamiltonian mechanics
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| 04/10/2019 Lec 23 | Symplectic methods
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| 04/15/2019 Lec 24 | Deferred correction
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| 04/17/2019 Lec 25 | Stochastic differential equations: examples
Read:
SDE,
Geom. Brownian motion
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| 04/22/2019 Lec 26 | Euler-Maruyama and Milstein method, weak and strong convergence
Read:
Euler-Maruyama,
Milstein method
|
| 04/24/2019 Lec 27 | Boundary value problems: shooting methods
Read:
BVP,
Shooting method
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| 04/29/2019 Lec 28 | Finite difference discretizations, pathways to PDE
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| 05/02/2019 | Final Examination |
| Matlab Programs | |
| Introduction: | |
| temple5044_linear_ode_2x2.m |
Vector field and phase flow for linear 2x2 ODE systems Visualizes the different cases that can arise with 2x2 systems of ODE, via quiver plot, eigen-directions, and deformation of the unit circle. |
| temple_abm_population_local.m |
Population dynamics models via 2x2 nonlinear ODE systems The dynamics of two species are demonstrated in various cases: predator-prey, logistic growth, competition, and mutalism. |
| Runge Kutta methods: | |
|
ode1.m ode2.m ode3.m ode4.m ode5.m |
Runge-Kutta methods of orders 1,2,3,4, and 5 Initial value problem ODE are solved approximated equidistant time steps. |
| temple5044_rk_order_conditions.m |
Check Runge-Kutta order conditions This code verifies whether the order conditions (up to order 5) are satisfied or not for a generic Runge-Kutta scheme, provided via its Butcher tableau. Example file: temple5044_rk_order_conditions_example.m, applies the code to three example methods. |
| temple5044_voyager.m |
Simulation of jupiter swing-by of the Voyager 1 space probe A four-body problem of celestial mechanics, involving the sun, earth, jupiter, and Voyager 1, is solve numerically with a way too inaccurate RK4 discretization. |
| Absolute stability: | |
| temple5044_absolute_stability_region_rk.m |
Plot regions of absolute stability for Runge-Kutta methods Provided are 9 examples of Runge-Kutta schemes, defined via their Butcher tableaus. |
| Semi-implicit methods: | |
| temple5044_stability_region_imex_rk.m |
Plot stability regions for ImEx Runge-Kutta methods Provided are 6 examples of ImEx Runge-Kutta schemes, applied to the test problem u'=αu+iβu. |
| Stochastic differential equations: | |
| temple5044_euler_maruyama.m |
Error convergence study of Euler-Maruyama method for the stochastic differential equation (geometric Brownian motion) dX = μ X dt + σ X dW. |
| Boundary value problems: | |
| mit18336_poisson1d_error.m |
Numerical error analysis for the 1D Poisson equation A differentiable but oscillatory right hand side is considered. |
| Homework Problem Sets | |
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