| Official Information | |
|---|---|
| Course Number: | Mathematics 8024.001 |
| Course Title: | Numerical Differential Equations II |
| Time: | TR 11:00-12:20 |
| Place: | 527 Wachman Hall |
| Instructor: | Benjamin Seibold |
| Instructor Office: | 518 Wachman Hall |
| Instructor Email: | seibold(at)temple.edu |
| Office Hours: | T 12:20-13:30, R 10:00-11:00 |
| Official: | Course Syllabus |
| Course Textbook: |
There is no single textbook for the course. The material is inspired by the following resources (in decreasing order of relevance):
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| 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: | 05/08/2017 |
| 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. This course continues last semester's 8023. Topics covered 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 possible upon request. | |
| Course Schedule | |
| 01/17/2017 Lec 1 | Review of 8023
|
| 01/19/2017 Lec 2 | I. Hyperbolic conservation laws:
derivation, examples, weak solutions
Read:
Conservation law,
Hyperbolic PDE
|
| 01/24/2017 Lec 3 | Riemann problem, shocks, entropy solution
Read:
Riemann problem,
Shock
|
| 01/26/2017 Lec 4 | Finite difference methods for discontinuous solutions
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| 01/31/2017 Lec 5 | Failure of finite different methods for nonlinear problems
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| 02/02/2017 Lec 6 | Consistent finite difference schemes, finite volume methods
Read:
Finite volume method
|
| 02/07/2017 Lec 7 | Godunov's methods, non-convex flux functions
Read:
Godunov's method
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| 02/14/2017 Lec 8 | Nonlinear stability theory
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| 02/16/2017 Lec 9 | High-order methods, limiters
Read:
Flux limiter,
Godunov's theorem
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| 02/17/2017 Lec 10 | Linear hyperbolic systems
Read:
Examples of equations
|
| 02/21/2017 Lec 11 | Nonlinear hyperbolic systems, approximate Riemann solvers
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| 02/23/2017 Lec 12 | Higher dimensions, semidiscrete methods
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| 02/24/2017 Lec 13 | SSP time stepping, ENO/WENO
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| 03/07/2017 Lec 14 | Operator splitting, stiff source terms, staggered grids
Read:
Strang splitting,
Arakawa grids
|
| 03/09/2017 Lec 15 | II. Interface problems: Front propagation
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| 03/21/2017 Lec 16 | Numerical interface representation
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| 03/23/2017 Lec 17 | Hamilton-Jacobi equations
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| 03/24/2017 Lec 18 | Numerical schemes for HJ equations
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| 03/30/2017 Lec 19 | III. Spectral methods: periodic problems
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| 04/04/2017 Lec 20 | Fast Fourier transform
Read:
FFT
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| 04/06/2017 Lec 21 | Non-periodic problems
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| 04/07/2017 Lec 22 | Applications
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| 04/11/2017 Lec 23 | IV. Fluid flows: Calculus of variations, Stokes problem
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| 04/13/2017 Lec 24 | Saddle point problems, staggered grid approaches
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| 04/14/2017 Lec 25 | Navier-Stokes equations
Read:
Navier-Stokes equations
|
| 04/18/2017 Lec 26 | Semispectral methods for the Navier-Stokes equations
|
| 04/25/2017 Lec 27 | Project presentations
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| 04/27/2017 Lec 28 | Project presentations
|
| 05/08/2017 | Final Examination |
| Matlab Programs | |
| Chapter I: 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_godunov_shallow_water.m |
Godunov's method for the shallow water equations Godunov's method for the shallow water equations, using an approximate Riemann solver with Roe averaging. No entropy fix is implemented. |
| 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. |
| Chapter II: 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. |
| Chapter III: Spectral methods (all direct links to Nick Trefethen's codes) | |
| cheb.m | |
| 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 |
| Chapter IV: 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. |
| Homework Problem Sets | |
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| Course Projects | |
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Project proposals due: January 31, 2017.
Project midterm reports due: March 23, 2017. Project final reports due: May 1, 2017.
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