Math 700 - Special Topics in Applied and Numerical Analysis:
Numerical Methods for Hyperbolic and Parabolic Conservation Laws
Fall 2013 - Andreas Meister
- Andreas Meister,
Math/Psyc 240, (410) 455-3167, firstname.lastname@example.org,
office hours: TuTh 02:30-03:30 or by appointment
- Classes: room MP 401,
Tuesdays 10:00-11:15 and Wednesdays 09:30-10:45
- Prerequisites: Math 620 and 630; familiarity with a high-level
procedural programming language such as Matlab, C, or Fortran;
or consent of instructor
There is no required textbook.
Several books are recommended for various parts of the course, namely
- A. Meister, J. Struckmeier:
Hyperbolic Partial Differential Equations, Vieweg, 2002.
- C. Hirsch:
Numerical Computation of Internal and External Flows,
Part 1 (Reprint 2001) and 2 (Reprint 2002), Wiley.
- E. F. Toro:
Riemann Solvers and Numerical Methods for Fluid Dynamics ,
Springer, 3rd Edition, 2009.
- R. J. LeVeque:
Finite Volume Methods for Hyperbolic Problems ,
Cambridge University Press, 2004.
- D. Kröner:
Numerical Schemes for Conservation Laws ,
- A. J. Chorin, J. E. Marsden:
A Mathematical Introduction to Fluid Mechanics ,
3rd Edition, Springer, 1993.
- Grading policy:
|| Class Project or Oral Exam
The homework includes
the computer assignments that are vital to understanding
the course material.
A late assignment accrues a deduction of
up to 10% of the possible score
for each day late until my receiving it;
I reserve the right to exclude any problem from scoring
on late homework, for instance, if we discuss it in class.
The graded participation component rewards
your professional behavior and active involvement
in all aspects of the course.
Examples of expected professional behavior include
attending class regularly,
reading assigned material when requested,
cooperating with formal issues such as
submitting requested material on time, and
participating constructively in class.
It is increasingly important
at this point in your education to learn
how to work on a larger project on your own
(with guidance by the instructor)
and to present your results in the form of a
professional-grade type-set report
and a prepared oral class presentation.
The class project will include all these components:
substantial work on an individual project;
a written report (in the form of a technical report);
and an oral class presentation.
Real life situations are in almost all cases modelled by systems of partial differential equations. Well known representatives of such mathematical models are the nonlinear Euler- and the Navier-Stokes equations governing inviscid and viscous flow fields, respectively. Consequently, these models are always employed for the simulation of flow fields around air foils, reentry vehicles or even usual cars, trains and much more. Analogously, the so-called shallow water equations represent a sub-model of the above mentioned equations and are applicable to describe water flows in a wide variety of scenarios including channels as well as oceans.
Beside linear models, which can be used to determine the transport of mass or even simple leaves at the surface of a sea at constant velocity usually nonlinear partial differential equations are of particular interest in the context of real life applications. Unfortunately, analytic solution of a system of nonlinear partial differential equations are generally not available. Hence, numerical methods are necessary to obtain a deeper insight in the behaviour of the constituents under consideration. However, the development of appropriate, accurate and reliable schemes for the numerical approximation of solutions of nonlinear partial differential equations requires a huge knowledge of general properties of the particular mathematical model under consideration.
Due to the reasons discussed above we will focus on two main topics within the course:
- Analytic investigation of partial differential equations with respect to their classification and the corresponding properties of the underlying solutions.
- Development of numerical methods for different types of partial differential equations as well as their analysis concerning stability, reliability, consistency and convergence
The main classes of numerical methods design for the above mentioned mathematical models are finite
difference, finite volume and finite element methods, and we will discuss examples of all
methods. We will use this as the basis for
discussing the associated issues of discretizing the time-direction
and solving large sparse systems of linear equations efficiently
with respect to memory and computing time.
One specific goal of this course is to understand the
method of lines approach to transient advection-diffusion equations
including all numerical techniques necessary to deal with the
spatial and time discretizations as well as non-linear and linear solvers.
For the finite volume methods, we will write our own code;
we will use MATLAB
for this purpose because of its ease of programming.
By the end of this course, you should:
understand and remember the key ideas, concepts, definitions,
and theorems of the subject.
Examples include classification of partial differential equations
and their key properties, the fundamental ideas of
finite difference, finite volume and finite element methods,
the main results concerning stability, consistency and convergence
for types of schemes,
and basic issues of computer implementations of these methods.
More broadly, you should also understand the purpose of
numerical methods and some of their major applications and you should be able to understand the main ingredients of commercial code in the actual field of computational fluid dynamics. Thus, you should have the knowledge to discuss with scientist as well as members of companies in the field of numerical methods for problems in fluid dynamics such that you are well prepared to write a PhD-thesis or to work in company in this very important area.
have experience using a professional software package,
writing code in it, and understanding how some of its functions work.
We will use Matlab in this course,
which is professional-grade packages in this field.
Writing code in this context includes the requirements to deliver code
in a form required, such as writing code to stated specifications and implementing a requested method.
This course will introduce the basic aspects of partial different equations as well as their numerical treatment.
- Analysis of partial differential equations
- Heat Equation
- Linear Advection Equation
- Wave Equation
- Shallow Water Equation
- Method of characteristics
- Strong and weak solutions of partial differential equations
- Numerical methods for partial differential equations
- Finite difference-, finite-volume- and finite-element-methods
- Consistency, stability and convergence of numerical schemes
- Central schemes
- Upwind schemes
- Approximate Riemann-solver
- Methods of higher order
- Introduction to Matlab at UMBC
(maintained by CIRC)
introduction to LaTeX including a sample file and
a template file for project reports.
UMBC Academic Integrity Policy
By enrolling in this course, each student assumes the responsibilities of
an active participant in UMBC's scholarly community in which everyone's
academic work and behavior are held to the highest standards of honesty.
Cheating, fabrication, plagiarism, and helping others to commit these acts
are all forms of academic dishonesty, and they are wrong.
Academic misconduct could result in disciplinary action that may include,
but is not limited to, suspension or dismissal.
To read the full Student Academic Conduct Policy, consult the
UMBC Student Handbook, the Faculty Handbook, the UMBC Integrity
UMBC Undergraduate Student Academic Conduct Policy
for undergraduate students,
University of Maryland Graduate School, Baltimore (UMGSB)
Policy and Procedures for Student Academic Misconduct
for graduate students.
Copyright © 2013 by Andreas Meister. All Rights Reserved.
This page version 1.7, November 2013.