Official Information  

Course Number:  Mathematics 8024.001 
Course Title:  Numerical Differential Equations II 
Time:  TR 2:003:20 
Place:  617 Wachman Hall 
Instructor:  Benjamin Seibold 
Instructor Office:  428 Wachman Hall 
Instructor Email:  seibold(at)temple.edu 
Office Hours:  W 2:004:00 
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.

Further Reading: 
The following textbooks are recommended for further reading and for obtaining
background knowledge.

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 RungeKutta schemes, wave equations, spectral methods, interface problems, level set method, HamiltonJacobi equations, Stokes problem, NavierStokes 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  Nonconvex 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 RungeKutta 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  HamiltonJacobi equations 
11/02/2010  Spectral methods: Idea, theory 
11/03/2010  Periodic problems, fast Fourier transform 
11/09/2010  Nonperiodic 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  NavierStokes equations 
12/01/2010  Semispectral methods for the NavierStokes 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 oneway 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, LaxFriedrichs, LaxWendroff, and CrankNicolson. For each method, the corresponding growth factor for von Neumann stability analysis is shown. 
mit18086_fd_transport_limiter.m 
Nonlinear finite differences for the oneway 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, LaxFriedrichs and LaxWendroff as linear methods, and as a nonlinear method LaxWendroffupwind with van Leer and Superbee flux limiter. 
temple8024_weno_claw.m 
WENO finite volume code for onedimensional 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 ShuOsher 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 NavierStokes equations in a box Solves the incompressible NavierStokes 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. 
Many more great Matlab programs can be found on
 
Additional Course Materials  
 
Homework Problem Sets  
 
Course Projects  
