Discontinuous Galerkin Methods with Optimal $$L^2$$ Accuracy for One Dimensional Linear PDEs with High Order Spatial Derivatives

Optimal Error Estimates of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional KdV Type Equations

“…In these studies, according to different choices of numerical fluxes, generalized Gauss-Radau (GGR) projections were proposed, and an analysis of inverse of the coefficient matrix was essential. In particular, for linear PDEs with high order derivatives, a sub-family of numerical fluxes containing average values and jumps of numerical solutions were constructed in [6], which were proved to be of optimal (k + 1)-th order by using some special projections. For the DG scheme with generalized fluxes solving wave equations, instead of analyzing the inverse of some matrices, [18] proposed an energy argument based on the coercivity of DG discretization operators and derived optimal error estimates, which can be easily extended to unstructured meshes.…”

Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

“…More recently, Frean and Ryan [18] proved that the use of SIAC filters was still able to extract the superconvergence information and obtain a globally smooth and superconvergent solution of order 2k + 1 for linear hyperbolic equations based on upwind-biased fluxes. Moreover, the αβ-fluxes, which were introduced as linear combinations of average and jumps of the solution as well as the auxiliary variables at cell interfaces, has been a hot research topic in recent years [1,12,19].…”

An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives

“…In their work, the main idea is to derive energy stability for the auxiliary variables in the LDG scheme by using the scheme and its time derivatives. In [12] Fu et al identified a sub-family of the numerical fluxes by choosing the coefficients in the linear combinations, so that the solution and some auxiliary variables of the proposed DG methods are optimally accurate in the L 2 norm. In [10] Dong and Shu proved the optimal error estimates for the higher even-order equations, including the cases both in one dimension and in multidimensional triangular meshes.…”