Numerical methods for conservation laws

The modeling of many problems in the applied sciences and engineering is based on concepts of conservation of mass, momentum and energy, leading to systems of conservation laws. Prominent examples are the Maxwell equations of electromagnetics, the Euler and Navier-Stokes equations of fluid dynamics and equations of elasticity and the systems of magnetohydrodynamics of plasma physics.

In this course we shall develop, analyze and apply computational methods suitable for solving systems of conservation laws. We shall begin to discussing fundamental properties of conservation laws, including their ability to generate non-smooth solutions - shocks - from smooth initial conditions, leading to the introduction of weak solutions and entropy conditions.

After an initial discussion of finite difference methods for conservation laws, we introduce finite volume methods as the first major class of methods to study, including accuracy and stability of these methods. We discuss the importance of the numerical flux and approximate Riemann solvers and the extension of finite volume methods to general grids.

For larger and more complex problems, the ability to increase the order of the method is important and we discuss such extensions and the new challenges these introduce. This sets the stage for the development of nonlinear schemes, beginning with MUSCL schemes and continuing with the development of essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) methods. Higher order in time is achieved through the development of strongly stable Runge-Kutta methods (SSP-RK).

As an alternative but closely related techniques we use the last part of the class on the development and analysis of discontinuous Galerkin methods as a very general and robust high-order accurate extension of finite volume methods. 

Throughout the course there will be an emphasis on mastering mathematical as well as computational aspects of the methods.

We will be using the text

J.S. Hesthaven, 2017, Numerical Methods for Conservation Laws: From Analysis to Algorithms. SIAM Publishing.

The text is available for free download through the EPFL Library.

Lecturer:

Martin Licht

Assistant:

Massimo Rizzuto

Class times:

Lectures: Wednesdays 13:15-15:00 (Room MA A3 31)

Exercises: Fridays 15:15-17:00 (Room MA A1 10)

Grading scheme

The final grade will be the best of the following three options

- 100% final exam (presumably an oral exam as in last years)
- 90% final exam and 10% best project
- 80% final exam and 10% each project

Readings: pages 1-10, 29-32.