Stable Runge–Kutta methods of 2nd and 3rd order for gas dynamics simulation using discontinuous Galerkin method

The Runge–Kutta methods with stability region including the circle of maximum radius are considered. The methods are used to approximate gas dynamics system of equations using the discontinuous Galerkin method. Three second-order and one third-order methods fulfiling maximum disk property are proposed and investigated. Test problems for the 1D and 2D gas dynamics are considered. The ability of the obtained methods to maintain the stability and monotonicity of the numerical solution at the maximum avaliable time steps is studied.