Course 8024 - Numerical Differential Equations II - Fall 2010

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) 
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.
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.
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.
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)
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.
Many more great Matlab programs can be found on
Additional Course Materials
Homework Problem Sets
Course Projects
  • Soroush Assari: Simulation of incompressible fluid flow around a rigid immersed body located inside a cavity
  • Shimao Fan: Simulation of a traffic model for rotaries
  • Zhiyong Feng: Searching for the best grids for one-dimensional convection-diffusion problem
  • Scott Ladenheim: Modeling and simulation of muscle contraction
  • Kaveh Laksari: Simulation of wave propagation in a nonlinear viscoelastic bar
  • Stephen Shank: Wave propagation in anisotropic media with applications to geophysics
  • Kirk Soodhalter: Semi-automated measurement of aortic diameter for diagnosis of aortic aneurysm
  • Dong Zhou: A study on the optimal way of reinitialization