Course 8023 - Numerical Differential Equations I - Spring 2010

Official Information
Course Number:Mathematics 8023.001
Course Title:Numerical Differential Equations I
Time:TR 11:00-12:20
Place:617 Wachman Hall
Instructor: Benjamin Seibold
Instructor Office:428 Wachman Hall
Instructor Email: seibold(at) 
Office Hours:W 1:30-3:30pm
Official Links: Course registration website
  Course Syllabus
Course Textbook: Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations - Steady State and Time Dependent Problems, SIAM, 2007
  Book on the SIAM website. List price $63.00. Temple students can purchase the book for $44.10 directly from the publisher. SIAM is located at 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688. (M-F 9:00-4:45).
Further Reading: L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, V. 19, American Mathematical Society, 1998
L.N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000
Randall J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002
C.A.J. Fletcher, Computational Techniques for Fluid Dynamics I, Springer
C.G. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer, 2007
Grading Policy
The final grade consists of three parts, each counting 33.3%:
Homework Problems:There are six homework assignments, each of which are to be worked on for two weeks.
Course project:From the fifth until the second to last week, each participant works on a course project. Students can/should suggest projects themselves. 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. Many modern and efficient approaches are presented, after fundamentals of numerical approximation are established. Of particular focus are a qualitative understanding of the considered ordinary and partial differential equation, Runge-Kutta and multistep methods, fundamentals of finite difference, finite volume, and finite element methods, and important concepts such as stability, convergence, and error analysis.
Fundamentals:Stability, convergence, error analysis, Fourier approaches.
Methods:Runge-Kutta, multistep, finite difference, finite volume, finite element methods.
Problems:Dynamical systems, boundary value problems, Poisson equation, advection, diffusion, wave propagation problems, conservation laws, flow problems.
01/19/2010 Fundamental concepts: ODE
01/21/2010 Fundamental concepts: PDE, well-posedness, notions of solutions
01/26/2010 Fourier methods for linear PDE IVP
01/28/2010    (Evans 2.2) Laplace and Poisson equation
02/02/2010    (Evans 2.1,2.3,2.4) Heat equation, transport equation, wave equation
02/04/2010    (5.1-5.5) Numerical methods for ODE IVP, local and global truncation error
02/09/2010    (5.6-5.7) Taylor series methods, Runge-Kutta methods
02/16/2010    (5.7) Butcher tableau, embedded method
02/18/2010    (5.8-5.9) Adaptive time stepping, linear multistep methods
02/23/2010    (Chap. 6) Zero-stability and convergence
02/25/2010    (Chap. 7) Absolute stability
03/02/2010    (Chap. 8) Stiff systems
03/04/2010    (Chap. 1) Finite Difference Methods for BVP and elliptic equations, generalized finite difference approach
03/16/2010    (2.1-2.5) Boundary value problems, consistency
03/18/2010    (2.6-2.13) Convergence and stability for BVPs, Neumann b.c.
03/23/2010    (3.1-3.6, 2.15, 2.20.3) 2D Poisson equation, deferred correction
03/25/2010    (2.15, 2.17) General linear second order elliptic equations, singular perturbations and boundary layers
03/30/2010     Finite element method
04/01/2010    (9.2-9.4) Parabolic Equations, heat equation, method of lines, stability
04/06/2010    (9.1,9.5) Accuracy, Lax equivalence theorem
04/08/2010    (9.6-9.8) Von Neumann analysis, multidimensional problems
04/13/2010    (10.1-10.3) Wave Propagation, Advection, MOL, Lax-Wendroff & Lax-Friedrichs methods
04/15/2010    (10.4-10.8) Upwind methods, stability analysis
04/20/2010    (10.9-10.12) Modified equation, hyperbolic systems, wave equation
04/22/2010 Higher space dimensions, conclusions
04/27/2010 Project presentations
04/29/2010 Project presentations
Matlab Programs
Chapter I: Fundamental Concepts
mit18086_linpde_fourier.m Four linear PDE solved by Fourier series
Shows the solution to the IVPs u_t=u_x, u_t=u_xx, u_t=u_xxx, and u_t=u_xxxx, with periodic b.c., computed using Fourier series. The initial condition is given by its Fourier coefficients. In the example a box function is approximated.
Chapter II: Numerical Methods for ODE Initial Value Problems
Runge-Kutta methods of orders 1,2,3,4, and 5
Initial value problem ODE are solved approximated equidistant time steps.
Chapter III: Finite Difference Methods for Boundary Value Problems and Elliptic Equations
mit18086_stencil_stability.m Compute stencil approximating a derivative given a set of points and plot von Neumann growth factor
Computes the stencil weights which approximate the n-th derivative for a given set of points. Also plots the von Neumann growth factor of an explicit time step method (with Courant number r), solving the initial value problem u_t = u_nx.
Example for third derivative of four points to the left:
>> mit18086_stencil_stability(-3:0,3,.1)
mit18336_poisson1d_error.m Perform numerical error analysis for the Poisson equation
A differentiable but oscillatory right hand side is considered.
Chapter IV: Parabolic Equations
mit18086_fd_heateqn.m Finite differences for the heat equation
Solves the heat equation u_t=u_xx with Dichlet (left) and Neumann (right) boundary conditions.
Chapter V: Wave Propagation
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.
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.
Many more great Matlab programs can be found on
Additional Course Materials
Homework Problem Sets
Course Projects
  • Soroush Assari: Fluid flow simulation around an immersed body
  • Peter de Blanc: Modeling atmospheric escape using statistical mechanics
  • Shimao Fan: Meshfree finite differences for the Poisson equation with moving least square approach
  • Zhiyong Feng: Numerical simulation of incompressible Navier-Stokes equations for a rotating fluid in a cylindrical pipe
  • Scott Ladenheim: Non-linear conservation laws: Modeling traffic lights
  • Kaveh Laksari: 2D model of impact to the head and what happens to brain
  • Michael McPhellin: Green Roofs and the management of untreated sewage releases
  • John Ratanaprasatporn: Simulation of brushfires using the level set method
  • Stephen Shank: The Black-Scholes equation and basic numerical methods in stochastic differential equations
  • Kirk Soodhalter: Semi-automated measurement of aortic diameter for diagnosis of aortic aneurysm
  • Dong Zhou: Reconstruction of patterns based on reaction-diffusion systems
Project presentations in 617 Wachman on
  • 04/27/2010: Soroush Assari, Peter de Blanc, Shimao Fan, Zhiyong Feng, Kaveh Laksari, Michael McPhellin
  • 04/29/2010: John Ratanaprasatporn, Stephen Shank, Kirk Soodhalter, Scott Ladenheim, Dong Zhou