| Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu
This paper provides an overview of the development of discontinuous Galerkin (DG) methods since their introduction in 1973 by Reed and Hill for neutron transport problems. It discusses how these methods have evolved to become widely used in computational fluid dynamics and various other applications. The paper reviews the theoretical and algorithmic aspects of DG methods, including their application to nonlinear conservation laws, compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations. Key features of DG methods include their ability to handle discontinuities accurately without spurious oscillations, high parallelizability, and suitability for complex geometries. The paper also covers the evolution of DG methods for nonlinear hyperbolic systems, the development of Runge-Kutta DG (RKDG) methods, and the extension of DG methods to convection-diffusion systems. It highlights the importance of numerical fluxes and slope limiters in ensuring stability and accuracy. The paper concludes with a discussion on parallelization, adaptivity, and future developments in DG methods.This paper provides an overview of the development of discontinuous Galerkin (DG) methods since their introduction in 1973 by Reed and Hill for neutron transport problems. It discusses how these methods have evolved to become widely used in computational fluid dynamics and various other applications. The paper reviews the theoretical and algorithmic aspects of DG methods, including their application to nonlinear conservation laws, compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations. Key features of DG methods include their ability to handle discontinuities accurately without spurious oscillations, high parallelizability, and suitability for complex geometries. The paper also covers the evolution of DG methods for nonlinear hyperbolic systems, the development of Runge-Kutta DG (RKDG) methods, and the extension of DG methods to convection-diffusion systems. It highlights the importance of numerical fluxes and slope limiters in ensuring stability and accuracy. The paper concludes with a discussion on parallelization, adaptivity, and future developments in DG methods.