### Math 2604: Advanced Scientific Computing IV

*
Numerical solution of time-dependent advection-diffusion-reaction equations
*

**Instructor:** Catalin Trenchea

E-mail: trenchea@pitt.edu

Office: Thackeray Hall 606

**Lecture:** MWF 11:05-11:55 AM
(Web-based Class)

**Office Hours:**
Tu 10:00am-noon, Thu 10:00am-noon and by appointment.

**Textbook:**
The course will not follow closely a specific text book, and
will be self-contained as far as possible.

**Prerequisites:**
Good undergraduate background in linear algebra and advanced
calculus. Familiarity with ordinary and partial differential equations will be useful.

Lectures will adapt to diverse backgrounds. Please contact the instructor if you have questions about your preparation.

**Content:**
The course focuses on the fundamental mathematical aspects of numerical methods for the time-dependent differential equations, advection-reaction-diffusion equations, motivated by applications in: *transport chemistry* - in connection with pollution of atmospheric air, surface water and groundwater, *atmospheric* and *oceanic* sciences, and *mathematical biology* - chemotaxis problems.

**Topics to be covered:**
Advection-Diffusion-Reaction Equations, positivity and conservation laws, stability, consistency and convergence of discretizations for ordinary differential equations, spatial discretizations, boundary conditions and spatial accuracy, monotonicity properties, variable step size control.

**Homework:**
Written homework will be assigned, and there will be a in class final presentation.

Lab Instructor is
Dr. John Burkardt,
and the labs are available
online.

**Material:
**
Lecture notes
(1-2-3-4)
and research papers from published literature.

**Other references:**

Willem Hundsdorfer, Jan G. Verwer, *
Numerical Solution of Time-Dependent Advection-Diffusion Reaction Equations*, Springer 2007.

Willem Hundsdorfer,
Joke G. Blom, Jan G. Verwer, *
Numerical Time Integration for Air Pollution Models*, Centrum Wiskunde & Informatica, Wien Springer.

Dale Durran,*
Numerical Methods for Fluid Dynamics: with Applications to Geophysics*, Springer 2010.

Ernst Hairer, Syvert P. Nørsett, Gerhard Wanner
*Solving Ordinary Differential Equations I, Nonstiff Problems*. Springer.
Ernst Hairer,
Gerhard Wanner,
*Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems*. Springer.

Germund Dahlquist,
Åke Björck,* Numerical Methods on Scientific Computing*, SIAM 2008.
David F. Griffiths,
Desmond J. Higham,
*Numerical Methods for Ordinary Differential Equations: Initial Value Problems*.

Ernst Hairer,
Christian Lubich,
Gerhard Wanner,
*Geometric Numerical Integration.
Structure-Preserving Algorithms for Ordinary Differential Equations*. Springer.
Heinz-Otto Kreiss, Hedwig Ulmer Busenhart *Time-Dependent Partial Differential Equations and Their Numerical Solution*, Birkhauser 2001.

Michael J P Cullen *A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow*, Imperial College Press 2006.

Geoffrey K. Vallis *Atmospheric and Oceanic Fluid Dynamics*, Cambridge 2012.

Andrew J. Majda, Xiaoming Wang, *Nonlinear Dynamics and Statistical Theories for basic Geophysical Flows*, Cambridge 2006.

Brian Straughan *The Energy Method, Stability, and Nonlinear Convection*, Springer 2004.

Brian Straughan *Heat waves*, Springer 2011.

John Charles Butcher,
*Numerical Methods for Ordinary Differential Equations*,
Second Edition. John Wiley & Sons 2008.

John Denholm Lambert, *Numerical Methods for Ordinary Differential Equations:
The Initial Value Problem*. John Wiley & Sons.
Peter E. Kloeden,
Eckhard Platen,
*Numerical Solution of Stochastic Differential Equations*. Springer.

The course web page will be updated continuously throughout the semester.
The student is responsible for checking this web page for assignments
and policies.

If you have a disability for which you are or may be requesting an
accodomation, you are encouraged to contact both your instructor
and the Office of Disability Resources and Services, 216 William Pitt
Union, (412) 648-7890, as early as possible in the term. DRS will verify
your disability and determine reasonable accomodations for this course.