A New Troubled-Cell Indicator For Discontinuous Galerkin Methods Using K-Means Clustering

The Runge-Kutta discontinuous Galerkin(RKDG)method is one of the mainstream methods for solving nonlinear hyperbolic conservation laws.It is a research frontier and hotspot of high-resolution methods in the scientific computing and has been successfully applied to many other fields.The troubled-cell indicator is an important research direction of the RKDG method.At present,there is no troubled-cell indicator that has excellent performance for all problems;Many troubled-cell indicators contain parameters that depend on the problem to be solved,which is very inconvenient to use in the numerical simulations.The majority of troubled-cell indicators cannot be directly used for some special meshes,such as unstructured meshes and adaptive meshes.In this paper we design a new troubled-cell indicator using K-means clustering,which is convenient to use because it is free of problem-dependent parameters,and has good performance for most classic numerical examples.Also,it can be generalized to special meshes.We first use K-means clustering to design a new troubled-cell indicator that does not contain problem-dependent parameters,and can work with various indication variables,and can be used on special meshes.The values of a troubled-cell indication variable are in general of different magnitudes between smooth and discontinuous regions,so the values from a local region are grouped into clusters by a K-means clustering algorithm so that each cluster consists of either the troubled cells or the untroubled cells.A classification criterion is also proposed to discriminate what kind of cells that each cluster contains.We apply this new troubled-cell indicator to uniform meshes to test its effectiveness.Extensive numerical tests in both one and two dimensions show that the new troubled-cell indicator can detect the troubled cells accurately using the KXRCF indication variable,and it is in general better than the original KXRCF troubled-cell indicator.Then we study the sensitivity of the new troubled-cell indicator to different stencil sizes.The numerical results verify that the new troubled-cell indicator is not sensitive to different stencil sizes when it satisfies the minimum stencil size requirements.This property makes the indicator very promising to be extended to unstructured meshes and irregular meshes whose stencil size is difficult to remain constant.Furthermore,we find that the smaller the stencil is,the less the CPU time is costed.Finally,we generalize the new troubled-cell indicator using K-means clustering to rectangular h-adaptive meshes.It still keeps the good properties of containing no problem-dependent parameters.Numerical tests show that the new troubled-cell indicator performs well on rectangular h-adaptive meshes.It can capture the shocks accurately and produce nonoscillatory numerical solutions.