Math 2604: Advanced Scientific Computing IV
Numerical Methods for Evolution Equations

Instructor: Catalin Trenchea
Phone: (412) 624-5681
Office: Thackeray 606

Lecture: MWF 9:00-9:50 THACKERAY 524

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 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: Stability, consistency and convergence for ordinary differential equations, multistep methods, implicit-explicit methods, positivity for multistep methods, stable modular splitting of advection-diffusion-reaction problems, ground-water/surface-water flow using Crank-Nicolson/Leap-Frog, time-filters in weather equations.

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

Material: Lecture notes and research papers from published literature.

Other references:
Dale R. Duran Numerical Methods for Fluid Dynamics: with Applications to Geophysics, Springer 2010.
Heinz-Otto Kreiss, Hedwig Ulmer Busenhart Time-Dependent Partial Differential Equations and Their Numerical Solution, Birkhauser 2001.
Willem Hundsdorfer, Jan G. Verwer Numerical Solution of Time-Dependent Advection-Diffusion Reaction Equations, Springer 2007.
Ernst Hairer, Gerhard Wanner Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer 2010.
Germund Dahlquist, Åke Björck Numerical Methods on Scientific Computing, SIAM 2008.
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.
J.C. Butcher Numerical Methods for Ordinary Differential Equations, Wiley 2008.

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.