Course 9200 - Topics in Numerical Analysis I: Computational Methods for Flow Problems - Fall 2022

Official Information
Course Number:	Mathematics 9200.001
Course Title:	Topics in Numerical Analysis I: Computational Methods for Flow Problems
Time:	TR 2:00-3:20
Place:	617 Wachman Hall

Instructor:	Benjamin Seibold
Instructor Office:	518 Wachman Hall
Instructor Email:	seibold(at)temple.edu
Office Hours:	TR 3:30-4:30pm
Official:	Course Syllabus

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
Grading Policy
The final grade consists of two parts, each counting 50%: homework problem sets and final examination.
Outline
This course provides an overview of numerical methods for many important types of flow problems, ranging from passive convection, over incompressible fluids (Navier-Stokes equations), shock problems (such as the compressible Euler equations), kinetic equations (Boltzmann equation, radiative transfer), to network flows (such as traffic on roads). 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 methods for their solution. The computational approaches include: finite volume methods, finite difference methods, meshfree and particle methods, moment methods. The goal of this course is provide a broad perspective on these important types of flow problems, their connections, and how to tackle them computationally. This course provides the needed familiarity with each topic to enable the participants to engage into further studies via literature.
Course Schedule
08/23/2022 Lec 1	I. Fundamentals of Flows: Different types of flow problems Read: CFD, Rayleigh-Taylor instability, re-entry
08/25/2022 Lec 2	reference frames, transport equations, incompressibility, vorticity Read: material derivative, reference frames, continuity equation, advection equations, incompressible flow, vorticity
08/30/2022 Lec 3	stream function, field lines Read: stream function, streamlines etc.
09/01/2022 Lec 4	potential flow Read: potential flow, conformal map
09/06/2022 Lec 5	II. Passive Convection: Particle methods Read: Voronoi tessellation, scattered data interpolation, kernel density estimation, rejection sampling
09/08/2022 Lec 6	Semi-Lagrangian methods: fundamentals Read: semi-Lagrangian scheme
09/13/2022 Lec 7	Semi-Lagrangian methods: high order Read: jet schemes
09/15/2022 Lec 8	Eulerian finite difference methods: convergence, truncation errors, stability Read: finite difference method, truncation error, von Neumann stability
09/20/2022 Lec 9	Method of lines, upwind, Lax-Wendroff, Lax-Friedrichs Read: method of lines, stiff equation, upwind scheme
09/22/2022 Lec 10	Advection-diffusion-reaction problems Read: convection-diffusion equation, reaction-diffusion system, operator splitting
09/27/2022 Lec 11	III. Hyperbolic conservation laws: Examples, characteristics, weak solution, Riemann problem, entropy Read: hyperbolic PDE, characteristics, weak solution, Riemann problem
09/29/2022 Lec 12	Finite volume methods: Godunov's method Read: finite volume method, Godunov's scheme
10/04/2022 Lec 13	Cell-transmission model Read: cell-transmission model
10/06/2022 Lec 14	Network flows, theory and numerics Read: Braess's paradox
10/11/2022 Lec 15	Particle methods for conservation laws Read: particleclaw
10/13/2022 Lec 16	Linear hyperbolic systems Read: examples, tutorial notes by A. Bressan
10/18/2022 Lec 17	Nonlinear hyperbolic systems Read: Rankine-Hugoniot, shallow water eqns., Euler eqns.
10/25/2022 Lec 18	Approximate Riemann solvers, higher dimensions Read: Riemann solver, Roe solver
10/26/2022 Lec 19	IV. Incompressible viscous flows: Calculus of variations, Stokes problem Read: calculus of variations, Stokes flow
10/27/2022 Lec 20	Saddle-point problems, inf-sup condition, staggered grids Read: Saddle point Stokes problem, inf-sup condition
11/01/2022 Lec 21	Finite element methods for the Stokes problem (guest lecture by Kiera Kean) Read: finite element method
11/03/2022 Lec 22	Navier-Stokes equations, derivation, Reynolds number Read: Navier-Stokes equations, derivation, Reynolds number
11/07/2022 Lec 23	Finite differences for the Navier-Stokes equations Read: staggered grids
11/08/2022 Lec 24	Pseudospectral methods for Navier-Stokes Read: spectral method, pseudo-spectral method, DFT, FFT
11/10/2022 Lec 25	Turbulence and turbulence models (guest lecture by Kiera Kean) Read: turbulence, turbulence model
11/28/2022 Lec 26	V. Kinetic equations: Vlasov and Boltzmann equation Read: Vlasov equation, Boltzmann equation
11/29/2022 Lec 27	Moment methods for radiative transfer, StaRMAP (guest lecture by Rujeko Chinomona) Read: radiative transfer, StaRMAP
12/01/2022 Lec 28	Moment methods, asymptotic preserving methods
12/12/2022	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 Particleclaw method for flows on networks. Solver file: particleclaw_network_solver.m; and diamond network example file: particleclaw_network_ex_diamond.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) FEniCS: finite element package (wikipedia article) StaRMAP: moment method solver for radiation transport
Homework Problem Sets
Problem set 1 (due 09/13/2022) Problem set 2 (due 09/22/2022) Problem set 3 (due 10/04/2022) Problem set 4 (due 10/18/2022) Problem set 5 (due 11/10/2022)