摘要
Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution,a direct discontinuous Galerkin(DDG)method for diffusion problems was introduced in[H.Liu and J.Yan,SIAM J.Numer.Anal.47(1)(2009),475-698].In this work,we show that higher order(k≥4)derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method;still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all p^(k) elements.The refined DDG method with such numerical fluxes enjoys the optimal(k+1)th order of accuracy.The developed method is also extended to solve convection diffusion problems in both one-and two-dimensional settings.A series of numerical tests are presented to demonstrate the high order accuracy of the method.
基金
partially supported by the National Science Foundation under the Kinetic FRG Grant DMS07-57227 and the Grant DMS09-07963
partially supported by the National Science Foundation Grant DMS-0915247.
