Runge–Kutta discontinuous Galerkin methods for convection-dominated problems.

*(English)*Zbl 1065.76135Summary: We review the development of Runge-Kutta discontinuous Galerkin (RKDG) methods for nonlinear 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 nonlinearly 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 RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, compressible and incompressible Navier-Stokes equations, and Hamilton-Jacobi equation.