Official Information  

Course Number:  Mathematics 8023.001 
Course Title:  Numerical Differential Equations I 
Time:  TR 11:0012:20 
Place:  617 Wachman Hall 
Instructor:  Benjamin Seibold 
Instructor Office:  428 Wachman Hall 
Instructor Email:  seibold(at)temple.edu 
Office Hours:  W 1:303:30pm 
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 191042688. (MF 9:004:45).  
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. 
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. 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, RungeKutta 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:  RungeKutta, 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. 
Schedule  
01/19/2010  Fundamental concepts: ODE 
01/21/2010  Fundamental concepts: PDE, wellposedness, 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 (5.15.5)  Numerical methods for ODE IVP, local and global truncation error 
02/09/2010 (5.65.7)  Taylor series methods, RungeKutta methods 
02/16/2010 (5.7)  Butcher tableau, embedded method 
02/18/2010 (5.85.9)  Adaptive time stepping, linear multistep methods 
02/23/2010 (Chap. 6)  Zerostability 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 (2.12.5)  Boundary value problems, consistency 
03/18/2010 (2.62.13)  Convergence and stability for BVPs, Neumann b.c. 
03/23/2010 (3.13.6, 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 (9.29.4)  Parabolic Equations, heat equation, method of lines, stability 
04/06/2010 (9.1,9.5)  Accuracy, Lax equivalence theorem 
04/08/2010 (9.69.8)  Von Neumann analysis, multidimensional problems 
04/13/2010 (10.110.3)  Wave Propagation, Advection, MOL, LaxWendroff & LaxFriedrichs methods 
04/15/2010 (10.410.8)  Upwind methods, stability analysis 
04/20/2010 (10.910.12)  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  
ode1.m ode2.m ode3.m ode4.m ode5.m 
RungeKutta 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 nth 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 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. 
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  
