| Official Information | |
|---|---|
| Course Number: | Mathematics 8024.001 |
| Course Title: | Numerical Differential Equations II |
| Time: | TR 2:00-3:20 |
| Place: | 617 Wachman Hall |
| Instructor: | Benjamin Seibold |
| Instructor Office: | 428 Wachman Hall |
| Instructor Email: | seibold(at)temple.edu |
| Office Hours: | W 2:00-4:00 |
| Official Links: | Course Syllabus |
| Core Textbooks: |
This course will cover many modern ideas, and various textbooks will be used.
The following two are recommended to own or have access to when the
respective material is covered.
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| Further Reading: |
The following textbooks are recommended for further reading and for obtaining
background knowledge.
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| Grading Policy | |
| The final grade consists of three parts, each counting 33.3%: | |
| Homework Problems: | There are five homework assignments, each of which are to be worked on for two weeks. |
| Course project: | Each participant works on a course project. Students can/should suggest projects themselves. Projects started in 8023 may be continued, as long as clear research goals can be given. The instructor is quite flexible with project topics as long as they are new, interesting, and relate to the topics of this course. The project grade involves a midterm report (20%) and a final report (50%), and a final presentation (30%). Due dates are anounced in class. |
| Exams: | Oral examinations in the final exams week. Schedule provided in class. |
| Outline | |
| This course is designed for graduate students of all areas who are interested in numerical methods for differential equations, with focus on a rigorous mathematical basis. It continues last semester's course 8023. Topics covered this fall include nonlinear hyperbolic conservation laws, finite volume methods, ENO/WENO, SSP Runge-Kutta schemes, wave equations, spectral methods, interface problems, level set method, Hamilton-Jacobi equations, Stokes problem, Navier-Stokes equation, and pseudospectral approaches for fluid flow. Further topics can be requested by the students. | |
| Schedule | |
| 08/31/2010 | Review of 8023 |
| 09/02/2010 | Review of 8023 |
| 09/07/2010 | Hyperbolic conservation laws: Theory, examples, weak solutions |
| 09/09/2010 | Shocks, entropy conditions |
| 09/14/2010 project proposal | Finite difference methods, linear advection with discontinuous solutions |
| 09/16/2010 | Truly nonlinear problems |
| 09/21/2010 | Finite volume methods |
| 09/23/2010 pset 1 | Non-convex flux functions |
| 09/28/2010 | Nonlinear stability theory |
| 09/30/2010 | High resolution schemes, limiters |
| 10/05/2010 | Linear hyperbolic systems |
| 10/07/2010 pset 2 | Nonlinear systems |
| 10/12/2010 | Higher space dimensions, semidisrete methods |
| 10/14/2010 | SSP Runge-Kutta schemes, ENO/WENO |
| 10/19/2010 | Wave equations, staggered grids, operator splitting |
| 10/21/2010 midterm report | Interface problems: Evolution of curves and surfaces |
| 10/26/2010 | Level set method |
| 10/28/2010 pset 3 | Hamilton-Jacobi equations |
| 11/02/2010 | Spectral methods: Idea, theory |
| 11/03/2010 | Periodic problems, fast Fourier transform |
| 11/09/2010 | Non-periodic problems |
| 11/11/2010 pset 4 | Applications |
| 11/18/2010 | Fluid flows: Calculus of variations, Stokes problem |
| 11/23/2010 | Saddle point problems, staggered grid approaches |
| 11/30/2010 pset 5 | Navier-Stokes equations |
| 12/01/2010 | Semispectral methods for the Navier-Stokes equations |
| 12/02/2010 | Project presentations |
| 12/07/2010 final report | Project presentations |
| Matlab Programs | |
| Hyperbolic conservation laws | |
| mit18086_fd_transport_growth.m |
Finite differences for the one-way wave equation, additionally plots
von Neumann growth factor Approximates solution to u_t=u_x, which is a pulse travelling to the left. The methods of choice are upwind, downwind, centered, Lax-Friedrichs, Lax-Wendroff, and Crank-Nicolson. For each method, the corresponding growth factor for von Neumann stability analysis is shown. |
| mit18086_fd_transport_limiter.m |
Nonlinear finite differences for the one-way wave equation with
discontinuous initial conditions Solves u_t+cu_x=0 by finite difference methods. Of interest are discontinuous initial conditions. The methods of choice are upwind, Lax-Friedrichs and Lax-Wendroff as linear methods, and as a nonlinear method Lax-Wendroff-upwind with van Leer and Superbee flux limiter. |
| temple8024_weno_claw.m |
WENO finite volume code for one-dimensional scalar conservation laws Solves u_t+f(u)_x = 0 by a semidiscrete approach, in which 5th order WENO is used for the reconstruction of the Riemann states at cell boundaries, and the 3rd order SSP Shu-Osher scheme is used for the time stepping. |
| mit18086_fd_waveeqn.m |
Finite differences for the wave equation Solves the wave equation u_tt=u_xx by the Leapfrog method. The example has a fixed end on the left, and a loose end on the right. |
| Interface problems | |
| mit18086_levelset_front.m |
Level set method for front propagation under a given front velocity field First order accurate level set method with reinitialization to compute the movement of fronts in normal direction under a given velocity. |
| Spectral methods (all direct links to Nick Trefethen's codes) | |
| cheb.m | |
| p4.m | Periodic spectral differentiation using matrices |
| p5.m | Periodic spectral differentiation using FFT |
| p13.m | Solving a linear BVP |
| p14.m | Solving a nonlinear BVP |
| p15.m | Solving an eigenvalue problem |
| p16.m | Solving the 2D Poisson equation |
| p17.m | Solving the 2D Helmholtz equation |
| Fluid flows | |
| mit18086_navierstokes.m |
Finite differences for the incompressible Navier-Stokes equations in a box Solves the incompressible Navier-Stokes equations in a rectangular domain with prescribed velocities along the boundary. The standard setup solves a lid driven cavity problem. This Matlab code is compact and fast, and can be modified for more general fluid computations. You can download a Documentation for the program. |
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Many more great Matlab programs can be found on
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| Additional Course Materials | |
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| Homework Problem Sets | |
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| Course Projects | |
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