Faculty of Science AMSI Summer School 2013

Numerical solution of conservation laws and hyperbolic linear systems

Markus Hegland and Stephen Roberts (Australian National University)


Synopsis

This course covers advection equations such as the Euler equations, the Shallow water equations and the time-dependent Maxwell equations among others. It includes a hands-on computational lab in addition to a discussion of theoretical topics like convergence (Lax equivalence theorem), characteristics, shocks and rarefaction waves and several numerical schemes like Lax-Wendroff and Crank-Nicholson in addition to Godunov schemes. The course reviews and introduces many fundamental numerical topics including interpolation, truncation errors and stability. Both finite difference and finite volume methods are covered and there is an introduction to more modern topics of discontinuous Galerkin and least squares methods.

Course content:


Contact hours

28 hours of hours spread over the four weeks, with consultation as required.


Prerequisites

We will try to keep the pre-requisites to a minimum, All that is required is a basic level of mathematical maturity that any masters/honours student should have.

As examples will be provided in MATLAB (this is because it is a relatively easy to use language which is ideally suited to dealing with array which form the basis of the methods we will be considering), students wishing to take this course, who have no prior programming experience, should attempt to learn the basics of MATLAB prior to the summer school; Googling "Introduction to MATLAB" should provide several good sources.


Assessment

Four assignments worth 15% each, and a 3-hour exam worth 40%.


Resources

A complete set of lecture notes will be provided at the start of the summer school. These will define the assessable content of the course. Additional readings, including relevant journal articles, will be provided during the course.

There are no specific texts required for this course, though students are expected to supplement the lecture material with additional reading. Some relevant texts are:

  1. L.C. Evans, 1. Partial differential equations, American Mathematical Society, 2010, 2nd edition.
  2. E.F. Toro, Riemann solvers and numerical methods for fluid dynamics, A practical introduction, Springer 1997.
  3. C. Grossmann, H.G. Roos and M. Stynes, Numerical treatment of partial differential equations, Springer, 2007.
  4. R.J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge University Press, 2002.
  5. R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology 6, Evolution problems II, Springer 1993.
  6. E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Springer 1996.
The course does have a substantial component of implementing theory computationally. If students do have a laptop, then they should check with their home institution about installing MATLAB on it, or download and install the free Octave software.


About Markus Hegland

Markus Hegland is a member of the Computational Mathematics Group at the Institute of Advanced Studies of the Australian National University (ANU). Since 2012 he is Head of the ANU Centre for Mathematics and its Applications (CMA) and Associate Director Research of Mathematics. He received his doctoral degree at the ETH Zurich in 1988. Afterwards and until 1991, he worked as a researcher and support staff under Prof. Martin Gutknecht at the Interdisciplinary Project Centre for Supercomputing (IPS) of the ETH Zurich. In 1992, Prof. Hegland joined an HPC group at the Australian National University (ANU) where he worked on algorithms for Fujitsu’s VPP and AP series. He also continued research on the solution of ill-posed problems. In the late 1990s, he established the first data mining course at the ANU and acted as leader of a data mining group. In recent years, Prof. Hegland has been a chief investigator in the ARC (Australian Research Council) Centre of Excellence in Bioinformatics. Currently he is involved in collaborative ARC funded research with Fujitsu on fault tolerance and HPC.



About Stephen Roberts

Stephen Roberts is a member of the Computational Mathematics group at the Mathematical Sciences Institute at the Australian National University. From 2006 to 2012 he was the Head of the department of Mathematics and Associate Director of Education of Mathematics. He received his doctoral degree at the University of California, Berkeley in 1985. His research area is computational mathematics, in particular the application of efficient and robust numerical methods for the solution of partial differential equations. He is the lead developer at the ANU of the ANUGA hydrodynamic modeling software. ANUGA is a computational tool that models the impact of dam breaks, floods and tsunamis on communities. It is used extensively by councils, governments and consultant engineers.

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