| 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)temple.edu |
| 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. |
| 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, 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. |
| Schedule | |
| 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 | |
| ode1.m ode2.m ode3.m ode4.m ode5.m |
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. |
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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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