Physics motivated algorithms for partial differential equations
San Martin, Luis Emilio
This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.
Permalink
https://hdl.handle.net/2142/30854
Description
Title
Physics motivated algorithms for partial differential equations
Author(s)
San Martin, Luis Emilio
Issue Date
1998
Doctoral Committee Chair(s)
Oono, Yoshitsugu
Department of Study
Physics
Discipline
Physics
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
partial differential equations
PDE
hyperbolic equations
Language
en
Abstract
Many nonequilibrium phenomena are spatially extended, and the most popular means to model them is the partial differential equation (PDE). Resultant PDEs are, however,
often nonlinear, defying analytical approaches. Thus, devising efficient numerical algorithms to solve PDEs is important for the study of nonequilibrium systems, and has traditionally been considered a major branch of applied mathematics.
A computationally efficient model that captures the crucial physics of a system can
be an efficient numerical solver of the PDE describing the system. This general idea
will be illustrated in terms of solvers for hyperbolic equations, such as those describing advection in fluids and linear wave propagation. We demonstrate in this thesis that a conscious pursuit of physics essence can lead to useful numerical algorithms. From this point of view, the development of solvers for physically meaningful PDEs can be considered a branch of applied physics.
Our strategy for deriving new algorithms is to implement the crucial physics, as
faithfully as possible, in order to reproduce the phenomenon inside the computer.
The solution of the PDE is obtained, in this approach, as a by-product of the correct
implementation of the physics of the problem. After explaining the derivation
of algorithms for the solution of advection in fluids, we present a new methodology
to derive algorithms for wave propagation problems, based on the modeling of Huygens'
principle. The new methodology can be used to derive higher-order algorithms systematically. We explain why these algorithms are advantageous in comparison to
standard higher-order finite-difference algorithms, and present tests and evaluations
of the new schemes. We give new algorithms for the wave equation and Maxwell's
equations, including the implementation of some types of boundary conditions. We conclude by suggesting extensions of the method to related problems.
Use this login method if you
don't
have an
@illinois.edu
email address.
(Oops, I do have one)
IDEALS migrated to a new platform on June 23, 2022. If you created
your account prior to this date, you will have to reset your password
using the forgot-password link below.
