P And Hp-Adaptive RKDG Methods Using The Troubled-Cell Indicators

The Runge-Kutta discontinuous Galerkin(RKDG)method is one of the main numerical methods for solving equations of hyperbolic conservation laws.It has many advantages including high-order precision,handling complex geometries and boundary conditions easily,ease of adaptation and high efficiency of parallelization.Many problems of hyperbolic conservation laws have a particularly large scale,which requires a great deal of time and computer storage space in numerical simulations.This makes adaptive methods an important research direction for the RKDG method.We firstly designed a p-adaptive RKDG algorithm using troubled-cell indicators.At each time step in this algorithm,we use a troubled-cell indicator to detect and mark the troubled cells.Then we let the troubled cells use lower-order polynomial approximations and the other cells use high-order polynomial approximations,thereby drop unnecessary degrees of freedom in the vicinity of the discontinuities so that the computer storage can be saved.We apply this algorithm to the one-dimensional Burgers,Riemann and Buckley-Leverett problems.The numerical results show that the proposed algorithm not only preserves the high-order approximation in smooth regions as the original RKDG method,but also reduces the storage effectively.Then,based on the p-adaptive RKDG algorithm,we brought in h-adaptation by refining the troubled cells and coarsening the pairs of untroubled cells,yielding an hp-adaptive RKDG method for solving the one-dimensional hyperbolic conservation laws.Numerical tests show that the new algorithm not only has the advantage of saving computer storage from p-adaptation,but also has the advantage of improving the solution quality near the discontinuities from h-adaptation.Compared with the original RKDG method with the same number of computational cells,both algorithms manage to control numerical oscillations,but the hp-adaptive algorithm gives even better numerical solutions.Finally,we worked on improving our hp-adaptive RKDG algorithm.We introduced a technique that adaptation is performed only once for every Q time steps with Q greater than 1.Numerical experiments show that when Q has an appropriate value,the improved algorithm can save a certain amount of CPU time without lowering the quality of numerical solutions significantly.