High order schemes for convection dominated problems. Lecture 2

Convection dominated partial differential equations are used extensively in applications including fluid dynamics, astrophysics, electro-magnetism, semi-conductor devices, and biological sciences. High order accurate numerical methods are efficient for solving such partial differential equations, however they are difficult to design because solutions may contain discontinuities and other singularities or sharp gradient regions. In this series of lectures we will give a general survey of several types of high order numerical methods for such problems, including weighted essentially non-oscillatory (WENO) finite difference methods, WENO finite volume methods, and discontinuous Galerkin (DG) finite-element methods. We will discuss essential ingredients, properties and relative advantages of each method, and comparisons among these methods. Recent development and applications of these methods will also be discussed.
Lecture 2:
DG method I: hyperbolic conservation laws
1. The first DG scheme in 1973 for neutron transport 2. Setup of Runge-Kutta DG schemes 3. Properties of DG schemes 4. Systems and multi-dimensions, unstructured meshes 5. Remarks on implementations: matrix form for linear equations, quadratures and quadrature-free, CPR schemes 6. Recent development and applications: 1) A simple WENO limiter for DG schemes 2) Positivity-preserving DG and finite volume schemes 3) DG method for problems involving delta-singularities