Computational Mechanics and Optimization Laboratory

Notre Dame courses

AME60714: Advanced Numerical Methods

Lecture

Location: 138 DeBartolo Hall
Meeting time: MW 2:20pm - 3:35pm

Office Hours

Matthew J. Zahr, 300B Cushing Hall, MW 3:35p-5p (virtual)

Description

Theory and implementation of advanced numerical methods to solve and optimize linear and nonlinear partial differential equations (PDEs) with particular emphasis on hyperbolic PDEs and (compressible) computational fluid dynamics. In the first part of the course, the foundational theory of hyperbolic partial differential equations including conservation laws, characteristics, Riemann problems, and Rankine-Hugoniot jump conditions, will be discussed. These concepts will be used to construct numerical methods for approximating solutions of hyperbolic PDEs including finite volume and high-order discontinuous Galerkin methods. The second part of the course will build upon the theory and methods developed in the first part to delve into more advanced topics including Arbitrary Lagrangian-Eulerian (ALE) formulations to solve PDEs on moving domains, optimization problems constrained by partial differential equations including boundary control and shape optimization, and model reduction methods to reduce the computational cost of solving PDEs. Aspects of software design (version control, continuous integration) and verification will also be discussed, as well as advanced programming tools such as automatic and symbolic differentiation and code generation.

Course Details

Course syllabus (pdf)

Lecture slides

Introduction to AME60714 (pdf)

Introduction to PDE-constrained optimization (pdf)

Homework

Homework 1 (pdf) - due 9/14

Homework 2 (pdf, code) - due 10/5

Homework 3 (pdf, code) - due 10/26

Homework 4 (pdf, code) - due 11/11

Final project

Project (pdf) - due 10/23, 11/25