Tareq/tareq/Cockburn’s review paper on RKDG

Reading notes

Abstract:

  • used in convection-dominated problems. convection-diffusion equations
  • RKDG is DG that has special class of RK time discretization,
  • combines numerical fluxes and slope limiters
  • Stable, high order, and highly parallelizable
  • can handle complicated geometries and boundaries

  • Some applications:

  • Non-linear conservation laws
  • compressible and incompressible Navier-Stokes equations

  • Hamilton-Jacobi-like equation

  1. Introduction:
  • Mainly used for computational fluid dynamics
  • Currently applied in purely elliptic systems

Applications: ‘‘Practical problems in which convection plays an important role arise in applications as diverse as meteorology, weather-forecasting, oceanography, gas dynamics, turbomachinery, turbulent flows, granular flows, oil recovery simulation, modeling of shallow waters, transport of contaminant in porous media, viscoelastic flows, semiconductor device simulation, magneto-hydrodynamics, and electro-magnetism, among many others. This is why devising robust, accurate and efficient methods for numerically solving these problems is of considerable importance.’’

Reasons for success of RKDG methods:

  1. Non-linear conservation laws are enforced locally
  2. when the solution u is not piecewise-constant, the stability of the method does not follow from the form of the numerical fluxes anymore and has to be enforced by means of flux or slope limiters.

Notes:

  • Highly parallel because it uses local data at each element (information from elements sharing edges)
  • DG used for spacial discretization and RK used for time discretization
  • The last section “ongoing work and open problems” presents several research topics in RKDG, but I have very little understanding as I am not familiar with most of the terms and concepts.

Concepts to know:

  • Runge-Kutta methods
  • Discontinuous Galerkin
  • slope limiters
  • (approximate) Riemann Solver
  • numerical fluxes
  • total variation boundedness (which is essential in many practical simulation)

Issues:

  • There are many unfamiliar terms and concepts that I do not know. Should I spend time knowing all? Which of them are essential in the topic?
  • Algorithm steps are very short in the introduction. I am not sure if the details inside the paper are enough
  • When the paper shows the main features of RKDG methods, they refer to problems’ cases that I have no clue about them