Discontinuous Galerkin methods for convection-dominated problems. (English) Zbl 0937.76049

Cockburn, Bernardo

Barth, Timothy J. (ed.) et al., High-order methods for computational physics. Berlin: Springer. Lect. Notes Comput. Sci. Eng. 9, 69-224 (1999).

Summary: We present and analyze the Runge-Kutta discontinuous Galerkin method for numerically solving nonlinear hyperbolic systems. The basic method is then extended to convection-dominated problems, yielding the local discontinuous Galerkin method. These methods are particularly attractive since they achieve formal high-order accuracy, nonlinear stability, and high parallelizability while maintaining the ability to handle complicated geometries and capture the discontinuities or strong gradients of the exact solution without producting spurious oscillations.
The discussed methods are applied to the Euler equations of gas dynamics, the shallow water equations, the equations of magnetohydrodynamics, the compressible Navier-Stokes equations at high Reynolds numbers, and to the equations of the hydrodynamic model for semiconductor devices. As a final example, we discuss an application of the discontinuous Galerkin method to the Hamilton-Jacobi equations.
For the entire collection see [Zbl 0920.00020].

MSC:

76M20	Finite difference methods applied to problems in fluid mechanics
76R99	Diffusion and convection
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs

Keywords:

Runge-Kutta discontinuous Galerkin method; nonlinear hyperbolic systems; local discontinuous Galerkin method; Euler equations; shallow water equations; magnetohydrodynamics; compressible Navier-Stokes equations; semiconductor devices; Hamilton-Jacobi equations