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a 3-day (July 13-15, 2009) Short Course on

 

SERRI (http://serri.org/) is a program managed by Oak Ridge National Laboratory (ORNL) for the US Department of Homeland Security to assist local, state and tribal leaders in developing the tools and methods required to anticipate and forestall terrorist events and to enhance disaster response.

Systems of hyperbolic partial differential equations (such as hyperbolic conservation laws) have important applications in all branches of science and engineering and their numerical solution has long been a subject of intensive research. One of the important aspects of nonlinear hyperbolic partial differential equations is the fact that even smooth initial conditions can give rise to discontinuities propagating at finite speeds. The challenge in developing numerical schemes for solving systems of hyperbolic partial differential equations resides in approximating smooth solution regions with high spatial accuracy while capturing discontinuities as sharply as possible without any oscillations. In the last couple of decades important advances have been made in this area with the introduction of finite-volume high-resolution upwind methods.

This three-day short course intends to provide participants with an overview of modern numerical methods developed for solving systems of hyperbolic partial differential equations, with special emphasis on shallow water equations, which have a wide range of applicability; such as free-surface hydraulics and environmental flows.

This short-course is designed for scientists, engineers from academic institutions, industry, research laboratories, state and federal agencies, who are interested in solving practical problems involving hyperbolic partial differential equations in areas ranging from fluid dynamics, aerodynamics, hydraulics, geophysics, to elasticity, acoustics, explosions, astrophysics, electro-magnetic waves, magneto-gas dynamics, crystal growth, and traffic problems, etc.

In the morning, basic concepts and the theory will be introduced in two 90-minute sessions separated by a coffee break. Practical tutorial sessions in the afternoon will be held in a room equipped with computers, where participants will gain hands-on experience by carrying out numerical experiments using simple numerical codes provided as part of the course material. The participants may also bring their own laptops for these practical tutorial sessions.

08:30-10:00     Lecture 1: Notions on hyperbolic equations. Eigenvalues, eigenvectors, hyperbolicity, characteristic variables. The Riemann problem for linear systems. Example: the linearized shallow water equations. Non-linear equations.

10:30-12:00     Lecture 2: Basics on numerical methods for hyperbolic equations. Finite differences and properties of schemes. The finite volume approach. Numerical fluxes and numerical sources. Examples. Upwind methods and the Riemann problem for model problems.

13:30-15:00     Practical tutorial I: Numerical experiments using well-known numerical methods for the linear advection equation, for Burgers equation and for the linearized shallow water equations.

10:30-12:00     Lecture 4:   The shallow water equations . Approximate Riemann solvers and numerical fluxes, HLL and HLLC. FORCE-type centred (non-upwind) fluxes.

13:30-15:00     Practical tutorial II: Numerical experiments with (i) the exact Riemann solver for the shallow water equations (ii) Godunov’s method using the HLL, HLLC and the FORCE fluxes.

08:30-10:00     Lecture 5: Godunov’s theorem. Non-linear TVD and ENO methods for model problems. The ADER approach for non-linear systems. The MUSCL-Hancock method. Application to the shallow water equations with source terms.

Participants will receive 2 CEUs or 20 PDHs. A certificate of attendance signed by the lecturer and the organizer will be provided to all participants.

Professor Eleuterio F. Toro has made important contributions in the field of high resolution numerical methods for solving hyperbolic conservation laws, and is regarded as one of the leading applied mathematicians and authorities in this field. His research interests span over numerical methods for partial differential equations, with particular emphasis on methods for hyperbolic equations and design and application of new algorithms. Professor Toro is particularly interested in mathematical modeling and simulation of physico/chemical models of various types of processes and applications of models and methods to a wide range of real problems such as propulsion in the aerospace industry, nuclear reactor safety, accidental collapse of dams, seismic waves, generation and propagation of tsunamis, application of shock waves in medicine and industrial problems.

Professor Toro is currently a full professor of numerical analysis the University of Trento, Italy, and carries out his research and teaching activities in the Laboratory of Applied Mathematics. Previously, he held the position of Full Professor of Applied Mathematics at the Manchester Metropolitan University, United Kingdom.

Professor Toro is author of two books (see below); editor of two books of contributed research papers, and author/co-author of more than 220 scientific publications. He has taught short courses and presented invited lectures in various countries around the world. He has received numerous distinctions and honorary degrees: Officer of the Most Excellent Order of the British Empire (2000); Honorary Citizen of Carahue, Chile (2001); Fellow of the Society for Shock Wave Research (2005); Doctor Honoris Causa, Universidad de Santiago de Chile (2008).

The short course will be held in Room 213 (Computer Lab) of the Carrier Hall, The University of Mississippi. The lunch will be served in the buffet-style cafeteria located in Johnson Commons, West.

The course fee includes attendance to lectures and practical tutorials, course notes and the numerical codes for exercises, coffee breaks and lunches.