High order schemes for convection dominated problems. Lecture 3

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 3:
DG method II: PDEs with higher order derivatives
1. Convection-diffusion equations: the local DG (LDG) scheme 2. Other types of DG discretizations for diffusion 3. Convection-dispersion equations (KdV equations) 4. Higher order PDEs 5. Recent development and applications: 1) Positivity-preserving second order DG method on general triangulations 2) Multiscale DG method for solving elliptic problems with curvilinear unidirectional rough coefficients 3) Energy conserving LDG methods for second order wave equations