Course 9200 - Topics in Numerical Analysis I: Computational Methods for Flow Problems - Spring 2015

Official Information
Course Number:	Mathematics 9200.001
Course Title:	Topics in Numerical Analysis I: Computational Methods for Flow Problems
Times:	MW 1:00-2:40
Places:	Wachman 527

Instructor:	Benjamin Seibold
Instructor Office:	518 Wachman Hall
Instructor Email:	seibold(at)temple.edu
Instructor Office Hours:	M 2:20-3:20, W 12:00-1:00

Course Textbooks:	There is no single textbook for this course. The materials come from a variety of books and other sources. Recommended resources: Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations - Steady State and Time Dependent Problems, SIAM, 2007 Randall J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002 Alexandre J. Chorin, Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd ed., Springer 1993. C.A.J. Fletcher, Computational Techniques for Fluid Dynamics I, Springer

Official:	Course Syllabus
Prerequisites:	see Math Course Listings
Topics Covered:	This course provides an overview of many important flow problems, ranging from incompressible fluids (Navier-Stokes equations), over shock problems (such as the compressible Euler equations) and front propagation problems, to kinetic equations (Boltzmann equation, radiative transfer) and network flows (traffic flow). One third of the course will be devoted to the modeling, derivation, and mathematical/physical properties of the equations and their solutions; and two thirds to the design of efficient and robust numerical approaches for their solution on compute infrastructures. The computational approaches include: finite volume methods, finite difference methods, particle methods, spectral methods, level set methods, moment methods. The purpose of this course is provide a broad perspective on these important types of flow problems, their connections, and how to tackle them computationally. Participants will be provided with sufficient familiarity with each topic to enable them to engage into further studies via literature. Course Grading: Homework problems sets and final examination.
Course Goals:	Provide students knowledge and a solid big picture perspective about important flow problems that arise in many fields of science and engineering applications, and in particular effective computational methods to approximate their solutions.
Attendance Policy:	Students are expected to attend every class. If a student cannot attend a class for some justifiable reason, he or she is expected to contact the instructor before class (if possible).
Course Grading:	Homework problems sets and final examination.
Exam Date:	04/30/2015 (oral exam schedule via email)
Course Schedule
01/12/2015 Lec 1	Fundamentals of flows: Examples of flows Read: CFD, re-entry
01/14/2015 Lec 2	reference frames, transport equations, incompressibility, vorticity Read: material derivative, reference frames, continuity equation, advection equations, incompressible flow, vorticity
01/21/2015 Lec 3	stream function, flow lines Read: stream function, streamlines etc.
01/26/2015 Lec 4	potential flow Read: potential flow, conformal map
01/28/2015 Lec 5	Particle methods Read: Voronoi tessellation, scattered data interpolation, kernel density estimation, rejection sampling
02/02/2015 Lec 6	Software development and best practices Read: software licenses
02/06/2015 Lec 7	Semi-Lagrangian methods: fundamentals Read: semi-Lagrangian scheme
02/09/2015 Lec 8	High-order semi-Lagrangian methods Read: jet schemes
02/11/2015 Lec 9	Finite difference methods: Consistency, stability, convergence, truncation errors Read: finite difference method, truncation error
02/16/2015 Lec 10	Von Neumann stability, semi-discretization Read: von Neumann stability analysis, method of lines, stiff equation
02/18/2015 Lec 11	Upwind, Lax-Wendroff, Lax-Friedrichs methods Read: upwind scheme, Lax-Wendroff, Lax-Friedrichs
02/23/2015 Lec 12	Advection-reaction-diffusion problems Read: convection-diffusion equation, reaction-diffusion system
02/25/2015 Lec 13	Operator splitting Read: MIT OCW, 2005 talk
02/27/2015 Lec 14	Hyperbolic conservation laws: Examples, characteristics (scalar 1d) Read: conservation law, hyperbolic PDE, method of characteristics
03/09/2015 Lec 15	weak solutions, Riemann problem, entropy Read: weak solution, Riemann problem
03/11/2015 Lec 16	Finite volume methods: Godunov's method Read: finite volume method, Godunov's scheme
03/23/2015 Lec 17	High-order methods, limiters Read: TVD, Godunov's theorem, limiter
03/25/2015 Lec 18	Semi-discrete methods, MUSCL, SSP time stepping Read: high-resolution scheme, MUSCL scheme
03/27/2015 Lec 19	ENO/WENO, hyperbolic systems Read: shallow water equations
03/30/2015 Lec 20	Incompressible viscous flows: Calculus of variations Read: calculus of variations
04/01/2015 Lec 21	Stokes problem, saddle-point structure Read: Stokes flow
04/06/2015 Lec 22	Finite elements for Stokes problem [guest lecture by Scott Ladenheim] Read: finite element method, mixed finite elements
04/08/2015 Lec 23	Implementation in deal.II [guest lecture by Scott Ladenheim] Read: deal.II on Wikipedia, deal.II website
04/13/2015 Lec 24	Orders of convergence for advection with discontinous solutions Read: modified equation
04/15/2015 Lec 25	Staggered grid finite differences and Navier-Stokes equations Read: staggered grids, Navier-Stokes equations
04/20/2015 Lec 26	Pseudospectral methods for Navier-Stokes Read: spectral method, pseudo-spectral method, DFT, FFT
04/22/2015 Lec 27	Kinetic equations: Vlasov and Boltzmann equation Read: Vlasov equation, Boltzmann equation
04/27/2015 Lec 28	Moment methods for radiative transfer Read: radiative transfer
04/30/2015	Final Examination
Matlab Programs
Example of an unsteady velocity field obtained from a stream function. Both are plotted, and Lagrangian markers are moved with the flow: temple9200_velocity_field.m 1d advection-reaction problem with a variable advection velocity field, and a bi-stable reaction term. This example shows how the moving particles eventually fail to resolve the solution, if no particle management is applied: temple9200_particle_1d_advection_reaction.m Interpolation routine that applies a simple cubic interpolant in each cell, based on the four data points around that cell: interp1cubic.m Nonlinear finite differences for the one-way wave equation with discontinuous initial conditions. Solves u_t+cu_x=0 by finite difference methods. 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_transport_limiter.m WENO finite volume code for one-dimensional scalar conservation laws. Solves u_t+f(u)_x = 0 by a semidiscrete approach, in which WENO5 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: temple8024_weno_claw.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: mit18086_navierstokes.m (Documentation of the approach and code) Examples to solve the Stokes equations using the finite elements software deal.II: temple9200_lid_driven_cavity.tar, temple9200_manufactured_solution.tar Pseudospectral code to solve the Navier-Stokes equations on a 2D torus: mit18336_spectral_ns2d.m
Software Used in the Course
MATLAB: high-level programming language (wikipedia article) particleclaw: characteristic particle method for scalar hyperbolic conservation laws Clawpack: finite volume package for hyperbolic problems deal.II: finite element package (wikipedia article)
Homework Problem Sets
Problem set 1 (due 01/28/2015) Problem set 2 (due 02/11/2015) Problem set 3 (due 02/25/2015) Problem set 4 (due 03/11/2015) Problem set 5 (due 03/25/2015) Problem set 6 (due 04/08/2015) Problem set 7 (due 04/22/2015)