Math 5415 - Reduced Order Modeling of Fluids

## Traian Iliescu

Professor
Department of Mathematics
428 McBryde Hall
Virginia Tech
Blacksburg, VA 24061

Class Meeting Time:
Monday, Wednesday 2:30pm - 3:45pm, McBryde 212

Office Hours:
Monday, Wednesday 4:00pm - 5:00pm

Course Goals and Objectives:
This course is a general introduction to reduced order modeling of fluid flows.
It focuses on both mathematical and computational topics.

The main goal is to provide a research playground for students interested in using reduced order modeling in the numerical simulation of fluid flows.

The objectives are:
(1) to introduce the basic mathematical and computational tools used in nonlinear reduced order modeling;
(2) to help each student in his/her research.

At the end of the course, every student should be able to:
(i) develop a reduced order model for building block partial differential equations; and
(ii) analyze the resulting reduced order model.

Besides mathematics students, students in engineering, geosciences and computer science are strongly encouraged to take this course.

This course introduces students to the reduced order modeling of fluid flows. Assuming minimal background, the course emphasizes the strong connection among the modeling, analysis, and computations of various reduced order modeling approaches for engineering and geophysical flows. The course is divided into two main areas: fluids and reduced order models (ROMs). Topics in the fluids area include the Navier-Stokes equations, the Boussinesq equations, the primitive equations (PE), the quasi-geostrophic equations (QGE), and the shallow water equations (SWE). These equations are developed and their connections to the physical and computational worlds are emphasized. Topics in the ROM area include the proper orthogonal decomposition (POD) and the reduced basis method (RBM) for parametrized systems. Both the development and the numerical analysis of the POD and RBM are covered. Finally, the different ROMs are used in the efficient and relatively accurate numerical simulation of engineering and geophysical flows.

Prerequisites: Very basic background in partial differential equations and numerical methods. Students should be able to use a programming language (such as MATLAB, FORTRAN, or C) to complete their computational assignments.

Suggested References:
Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Holmes, Lumley, Berkooz, 1996.
Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Hesthaven, Rozza, Stamm 2015.
Reduced Basis Methods for Partial Differential Equations: An Introduction, Quarteroni, Manzoni, Negri 2015.
Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Vallis, 2006.
Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Cushman-Roisin, Beckers, 2011.

Evaluation Criteria and Grading:
Homework assignments (35% of the final grade) and final project (65% of the final grade).
A score of 90% will guarantee an A.

Missed Work:
If a student fails to to hand in an assignment on time, his/her score is zero unless the reasons for the failure are serious and beyond the student's control (``subc''). The instructor reserves the right to verify that the reasons are serious and beyond the student's control. It is to the student's advantage to inform the instructor of such reasons before missing the work. When work is missed for ``subc'' reasons, the instructor, after consultation with the student, will decide how to handle the missed work.

Honor:
Students work on in-class exams and the final exam alone. There will be NO books, notes, calculators, etc. allowed on the exam. The honor code applies to all graded work in this course. The Undergraduate Honor Code pledge that each member of the university community agrees to abide by states: “As a Hokie, I will conduct myself with honor and integrity at all times. I will not lie, cheat, or steal, nor will I accept the actions of those who do.” Students enrolled in this course are responsible for abiding by the Honor Code. A student who has doubts about how the Honor Code applies to any assignment is responsible for obtaining specific guidance from the course instructor before submitting the assignment for evaluation. Ignorance of the rules does not exclude any member of the University community from the requirements and expectations of the Honor Code. For additional information about the Honor Code, please visit: https://www.honorsystem.vt.edu/

Special Needs:
If you need adaptations or special accommodations because of a disability (learning disability, attention deficit disorder, psychological, physical, etc.), if you have emergency medical information to share with me, or if you need special arrangements in case the building must be evacuated, please make an appointment with me as soon as possible.

IMPORTANT:
If you are not on the class roll that comes out after the last add date, immediately check your schedule at a terminal and start attending the proper section. For no foreseeable reason (computer and registrar personnel mistakes included) will you be allowed to stay in the wrong section or to drop a section for which you are actually enrolled after the last drop date. By simply attending a section you will not be placed on its roll.