A Conservative Runge–Kutta Discontinuous Galerkin ConRKDG Method for Inviscid Compressible Flows in One-Dimensional Computational Fluid Dynamics Simulations

This article proposes a novel conservative ConRKDG method for one-dimensional hyperbolic conservation laws with applications in computational fluid dynamics simulations. A DG local solution is reconstructed over each element based on the sub-cell solution averages with a newly proposed set of shape functions. In this virtue, the conservation property of the problem is naturally imposed for the numerical DG solution. In addition, the availability of finite-volume sub-cell solution averages without any DG-to-FV transformation or vice versa facilitates a direct and robust technique for detecting troubled elements, in which the unlimited DG local solution is deemed unstable. A new WENO-type smoothness measurement based on sub-cell solution averages is introduced to assess whether a DG local solution is admissible or unstable, thereby determining whether an element is good or troubled. For the latter case, a secondary finite-volume WENO method is invoked in an \emph{a posteriori} phase to recalculate the sub-cell averages to sustain numerical stability by essentially suppressing non-physical spurious oscillations in the vicinity of shocks or discontinuities at troubled elements. The performance of the ConRKDG method with different secondary finite-volume WENO methods is compared for both problems with smooth solutions and those with shocks and discontinuities.