Two New Troubled-Cell Indicators For RKDG Method

The Runge-Kutta discontinuous Galerkin(RKDG)method is one of the mainstream methods for solving nonlinear hyperbolic conservation law equations in many engineering problems.The solutions to hyperbolic conservation law equations are often discontinuous,and oscillations may occur near the discontinuities in numerical simulations,making the numerical scheme unstable.The RKDG method needs to detect the locations of the discontinuities via a troubled-cell indicator,and reconstruct the numerical solution to control the oscillations.Therefore,how to quickly and accurately identify the troubled cells that cause these oscillations has become an important research direction of the RKDG method.Zhu et al.proposed a K-means clustering troubled-cell indicator in2021.This indicator detects troubled-cells quickly and accurately,which can not only use different indication variables,but also work well on various complex grids.This article presents work in three aspects.Firstly,based on the K-means clustering troubled-cell indicator,we introduce data range to reduce the number of tunable parameters.Secondly,we abandon the K-means clustering method,and use a simple classification method to quickly divide the cells in each indication stencil into two categories.Thirdly,we investigate the performance of the new troubled-cell indicators applied to an h-adaptive RKDG method.First of all,the K-means clustering troubled-cell indicator has many advantages.It can accurately capture the discontinuities,and its parameters are relatively insensitive to test problems.In addition,it can work well with various complex meshes and different indication variables.However,it has two adjustable real parameters,which brings inconvenience to numerical applications.To overcome this difficulty,we improve the K-means clustering troubled-cell indicator.We first introduce efficient and simple techniques of normalization and data range to quickly rule out the stencils that consist of all untroubled cells,and then single out the troubled cells in the rest of the stencils via K-means clustering.The improved indicator only contains one adjustable parameter,which reduces the complexity of the determination of the parameters and facilitates practical applications.Secondly,considering that K-means clustering method essentially divides the cells in each stencil into two categories,we propose a simple classification method to further reduce the computational time of the algorithm.We first filter out the stencils that consist only of untroubled cells,and then use a simple classification method to divide cells in each of the remaining stencils into two categories.Finally,we compare the mean values of the two categories to distinguish the troubled and untroubled cells.The new simple classification troubled-cell indicator performs well,which can effectively identify the troubled cells near the discontinuities and produce numerical results without oscillations.Finally,based on the two new troubled-cell indicators,we modify an h-adaptive RKDG algorithm by using the new indicators to generate adaptive meshes.Through refining the troubled cells and coarsening the untroubled ones,we can obtain adaptive meshes which are fine in discontinuous regions and coarse in continuous regions.As a result,the numerical solution in the discontinuous regions is improved.In the meanwhile,we also use the same troubled-cell indicator in the limiter to mark the troubled cells.We numerically test the h-adaptive algorithm,which is based on the new troubled-cell indicators,via classical test problems.The results indicate that the new indicators can detect troubled cells accurately.It not only generates high-quality adaptive meshes,but also controls the oscillations in the numerical solutions.