Math 700 - Special Topics in Applied and Numerical Analysis:
Numerical Methods for Hyperbolic and Parabolic Conservation Laws

Fall 2013 - Andreas Meister

Basic Information

Course Description

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:

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.

Learning Goals

By the end of this course, you should:

Detailed schedule

This course will introduce the basic aspects of partial different equations as well as their numerical treatment.


  1. Analysis of partial differential equations
    • Laplace-Equation
    • Poisson-Equation
    • Heat Equation
    • Linear Advection Equation
    • Burgers-Equation
    • Wave Equation
    • Shallow Water Equation
    • Euler-Equation
    • Navier-Stokes-Equation
    • Method of characteristics
    • Strong and weak solutions of partial differential equations
  2. 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

Additional materials

Other Information

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 webpage, the UMBC Undergraduate Student Academic Conduct Policy (PDF) for undergraduate students, or the University of Maryland Graduate School, Baltimore (UMGSB) Policy and Procedures for Student Academic Misconduct (PDF) for graduate students.

Copyright © 2013 by Andreas Meister. All Rights Reserved.
This page version 1.7, November 2013.