| Official Information | |
|---|---|
| Course Number: | Mathematics 8023.001 |
| Course Title: | Numerical Differential Equations I |
| Time: | MW 9:00-10:20 |
| Place: | 617 Wachman Hall |
| Instructor: | Benjamin Seibold |
| Instructor Office: | 518 Wachman Hall |
| Instructor Email: | seibold(at)temple.edu |
| Office Hours: | MW 10:30-11:30am |
| Official: | Course Syllabus |
| Course Textbook: | Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations - Steady State and Time Dependent Problems, SIAM, 2007 |
| 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 |
| Grading Policy | |
| The final grade consists of three parts, each counting 33.3%: | |
| Homework Problems: | Each homework assignment will be worked on for two weeks. |
| Course project: | From the third 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 announced in class. |
| Exams: | December 15, 2016. |
| 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, conservation laws. |
| Course Schedule | |
| 08/29/2016 Lec 1 | I. Fundamental Concepts: Ordinary differential equations, IVPs vs. BVPs
|
| 08/31/2016 Lec 2 | Partial differential equations, linear vs. nonlinear, well-posedness
Read:
PDE,
Well-posed problem
|
| 09/07/2016 Lec 3 | Fourier methods for linear PDE IVPs, Laplace/Poisson equation
Read:
Fourier series,
Poisson equation
|
| 09/08/2016 Lec 4 | Heat equation, transport/wave equation
|
| 09/14/2016 Lec 5 | II. Numerical methods for ODE IVPs:
fundamentals, truncation errors
|
| 09/19/2016 Lec 6 | Taylor series methods, Runge-Kutta methods
|
| 09/21/2016 Lec 7 | General Runge-Kutta methods, ERK-DIRK-IRK, order conditions
|
| 09/26/2016 Lec 8 | Embedded methods and adaptive time stepping
Read:
Adaptive stepsize
|
| 09/28/2016 Lec 9 | Linear multistep methods
Read:
Linear multistep method
|
| 10/03/2016 Lec 10 | Zero-stability and convergence
Read:
Zero-stability
|
| 10/05/2016 Lec 11 | Absolute stability
Read:
Stability
|
| 10/10/2016 Lec 12 | Stiff problems
Read:
Stiff equation
|
| 10/12/2016 Lec 13 | III. Finite difference methods for BVPs:
generalized finite difference approach
Read:
Finite difference method
|
| 10/17/2016 Lec 14 | Boundary value problems, consistency
Read:
Boundary value problem
|
| 10/19/2016 Lec 15 | Convergence and stability for BVPs, Neumann b.c.
|
| 10/24/2016 Lec 16 | 2D Poisson equation, deferred correction
|
| 10/26/2016 Lec 17 | Advection-diffusion-reaction problems, variable coefficient diffusion, anisotropic diffusion
|
| 11/02/2016 Lec 18 | Singularly perturbed problems, weak derivatives
|
| 11/04/2016 Lec 19 | Finite element method
Read:
Finite element method
|
| 11/07/2016 Lec 20 | IV. Parabolic equations:
heat equation, method of lines, stability
Read:
Method of lines
|
| 11/09/2016 Lec 21 | Accuracy, Lax-equivalence theorem
Read:
Lax equivalence theorem
|
| 11/14/2016 Lec 22 | Von Neumann stability analysis, multi-dimensional problems
|
| 11/16/2016 Lec 23 | V. Wave propagation:
advection, MOL, Lax-Wendroff, Lax-Friedrichs
Read:
Lax-Wendroff,
Lax-Friedrichs
|
| 11/28/2016 Lec 24 | Upwind methods, stability analysis
Read:
Upwind
|
| 11/30/2016 Lec 25 | Modified equation
Read:
Modified equation
|
| 12/05/2016 Lec 26 | Linear hyperbolic systems, wave equation, outlook
Read:
Hyperbolic PDE
|
| 12/07/2016 Lec 27 | Project presentations:
Belguet, Biswas, Chou, Finkelstein, Grein, Harel
|
| 12/12/2016 Lec 28 | Project presentations:
Jin, Langborg-Hansen, Li, Ramadan, Salehi
|
| 12/15/2016 | Final Examination |
| 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 | |
| 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) |
| mit18086_poisson.m |
Numerical solution of Poisson equation in 1D, 2D, and 3D Set-up of system matrices using Matlab's kron. |
| 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 Dirichlet (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. |
| Homework Problem Sets | |
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| Course Projects | |
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Project proposals due: September 16, 2016.
Project midterm reports due: October 24, 2016. Project final reports due: December 12, 2016.
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