Research On Generalization Of Fu-Shu Troubled-Cell Indicator

The Runge-Kutta discontinuous Galerkin(RKDG)method for solving hyperbolic conservation law equations is one of the mainstream methods in computational fluid dynamics nowadays.It has become a research frontier and hotspot of high-resolution methods in scientific computing and has been successfully applied to many other fields.Troubled-cell indicators and adaptive methods are always important research directions of RKDG method.However,very few troubled-cell indicators can be applied to h-adaptive meshes,and they need further improvements.In 2017,Fu and Shu proposed a new troubled-cell indicator.This indicator is simple,compact,and does not contain problem-dependent parameters with satisfying numerical performance.This paper mainly works on its generalization to h-adaptive meshes.We first directly generalize the Fu-Shu troubled-cell indicator on triangular meshes to uniform rectangular meshes.Numerical tests using classical examples of the two-dimensional hyperbolic systems of Euler equations demonstrate its effectiveness.Compared with the KXRCF troubled-cell indicator,the Fu-Shu indicator is overall better,especially for the high-order cases where it can detect discontinuities more accurately and hence has a clear advantage.Then we generalize the Fu-Shu troubled-cell indicator to rectangular h-adaptive meshes.The generalized indicator still keeps the good properties of simplicity and compactness without problem-dependent parameters.Numerical tests show that the generalized Fu-Shu troubled-cell indicator performs well on rectangular h-adaptive meshes.It can capture the shocks accurately and produce nonoscillatory numerical solutions.Finally,we use the generalized Fu-Shu troubled-cell indicator to generate h-adaptive meshes for the RKDG method.In such adaptive meshes,fine meshes are used at the discontinuities and coarse meshes elsewhere so that the computational cost is saved and the solution quality near the discontinuities is improved.The numerical experiments on the two-dimensional Euler equations verify the effectiveness of the proposed method.We also show the advantage of the generalized Fu-Shu troubled-cell indicator by comparing with the commonly used KXRCF troubled-cell indicator.