Supported by Tulane's Mathematics Department this mini series of lectures will be held in the Mathematics Department at Tulane University, New Orleans, from March 13 to 16, 2013. The principal speaker will be Professor Chi-Wang Shu from Brown University, who will deliver four lectures centered around the topic of the title. In addition we will have eight one-hour talks given by the invited speakers.

Invited Speakers

General abstract

Convection dominated partial differential equations (PDEs), including hyperbolic conservation laws, convection dominated convection-diffusion equations, convection dominated convection-dispersion equations, etc., appear often in diverse applications such as fluid dynamics, electro-magnetic waves, semi-conductor device simulations, etc. The solutions to such PDEs contain discontinuities or sharp gradient regions, posing difficulties to robust and stable schemes, especially to high order accurate schemes. In these lectures we will discuss several classes of successful schemes for solving convection dominated PDEs, including finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes, finite element discontinuous Galerkin (DG) schemes, and spectral methods. General introduction to these schemes as well as presentations on recent developments will both be covered.

Lecture titles

  • Finite volume WENO schemes

    In this lecture we will first give a general description of finite volume schemes. We will then discuss the weighted essentially non-oscillatory (WENO) reconstruction procedure which forms the foundation of high order WENO finite volume schemes. Recent developments on finite volume WENO schemes, including bound-preserving limiters and a simple WENO limiter for discontinuous Galerkin methods, will be discussed.

  • Finite difference WENO schemes

    In this lecture we will first give a general description of conservative finite difference schemes and their relationship and relative advantages / disadvantages compared with finite volume schemes. We will then discuss the weighted essentially non-oscillatory (WENO) finite difference schemes. Recent developments on finite difference WENO schemes, including an alternative flux formulation which has the advantage of ensuring a narrow stencil for Lax-Wendroff time discretization and for easy maintanance of free-streams on curvilinear meshes, and an inverse Lax-Wendroff procedure for boundary conditions, will be discussed.

  • Finite element discontinuous Galerkin schemes

    In this lecture we will first give a general description of discontinuous Galerkin methods, both for conservation laws and for convection-diffusion equations. We will then discuss several recent developments on discontinuous Galerkin methods, including the analysis of superconvergence and the design, analysis and application of DG methods for problems with solutions containing delta-singularities.

  • Spectral methods

    Spectral methods are excellent methods for problems defined on regular geometry with smooth solutions. In this lecture we are however mainly concerned with spectral methods for solving discontinuous or even unbounded solutions. We will first describe a general framework of Gegenbauer polynomial based post-processing, which allows us to recover exponential accuracy in the maximum-norm for piecewise analytic but discontinuous functions, based on their partial spectral sums. We will then discuss our recent work in generalizing this procedure to collocation for smooth functions with end-point singularities.

Registration

There is no registration fee for the lectures and everyone is welcome to attend. If you are planning to participate please send an e-mail message to kurganov@math.tulane.edu with your name and affiliation, an estimated time of arrival and departure, which will allow us to prepare a name tag for you and an information packet. For recommended hotels please see here.


Organizer