High order schemes for convection dominated problems. Lecture 1

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 1: WENO finite volume and finite difference schemes
The following topics will be discussed:
1. Setup of finite volume framework
2. WENO reconstruction
3. Time discretization
4. Multi-dimensions and unstructured meshes
5. Setup of conservative finite difference framework
6. Relationship between finite difference and finite volume schemes
7. Recent development and applications:
1) Inverse Lax-Wendroff type boundary treatments
2) Free-stream preserving finite difference schemes on curvilinear meshes
3) A homotopy method based on WENO schemes for solving steady state problems
4) Application: Shock-vortex and vortex-vortex interactions