In this paper, we review the development of the Runge-Kutta discontinuous Galerkin (RBLDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge-Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RBLDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.
|Original language||English (US)|
|Number of pages||89|
|Journal||Journal of Scientific Computing|
|State||Published - 2001|
Bibliographical noteFunding Information:
B.C. was supported in part by NSF Grant D MS-9807491 and by the University of Minnesota Supercomputing Institute. C.-W.S. was supported in part by ARO Grant DAAD19-00-1-0405, NSF Grants DMS-9804985 and ECS-9906606, NASA Langley Grant NCC1-01035 and Contract NAS1-97046 while this author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23681-2199, and AFOSR Grant F49620-99-1-0077.
- Convection-diffusion equations
- Discontinuous Galerkin methods
- Non-linear conservation laws