Superconvergence and the Numerical Flux: a Study Using the Upwind-Biased Flux in Discontinuous Galerkin Methods

One of the beneficial properties of the discontinuous Galerkin method is the accurate wave propagation properties.That is,the semi-discrete error has dissipation errors of order 2k+1(≤Ch2k+1)and order 2k+2 for dispersion(≤Ch2k+2).Previous studies have concentrated on the order of accuracy,and neglected the important role that the error constant,C,plays in these estimates.In this article,we show the important role of the error constant in the dispersion and dissipation error for discontinuous Galerkin approximation of polynomial degree k,where k=0,1,2,3.This gives insight into why one may want a more centred flux for a piecewise constant or quadratic approximation than for a piecewise linear or cubic approximation.We provide an explicit formula for these error constants.This is illustrated through one particular flux,the upwind-biased flux introduced by Meng et al.,as it is a convex combination of the upwind and downwind fluxes.The studies of wave propagation are typically done through a Fourier ansatz.This higher order Fourier information can be extracted using the smoothness-increasing accuracy-conserving(SIAC)filter.The SIAC filter ties the higher order Fourier information to the negative-order norm in physical space.We show that both the proofs of the ability of the SIAC filter to extract extra accuracy and numerical results are unaffected by the choice of flux.

Capitalizing on Superconvergence for More Accurate Multi‑Resolution Discontinuous Galerkin Methods

This article focuses on exploiting superconvergence to obtain more accurate multi-resolution analysis. Specifcally, we concentrate on enhancing the quality of passing of information between scales by implementing the Smoothness-Increasing Accuracy-Conserving (SIAC) fltering combined with multi-wavelets. This allows for a more accurate approximation when passing information between meshes of diferent resolutions. Although this article presents the details of the SIAC flter using the standard discontinuous Galerkin method, these techniques are easily extendable to other types of data.

How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Higher order accuracy is one of the well-known beneficial properties of the discontinu-ous Galerkin(DG)method.Furthermore,many studies have demonstrated the supercon-vergence property of the semi-discrete DG method.One can take advantage of this super-convergence property by post-processing techniques to enhance the accuracy of the DG solution.The smoothness-increasing accuracy-conserving(SIAC)filter is a popular post-processing technique introduced by Cockburn et al.(Math.Comput.72(242):577-606,2003).It can raise the convergence rate of the DG solution(with a polynomial of degree k)from order k+1 to order 2k+1 in the L2 norm.This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction.The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features,such as extra smoothness.Second,we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving(or even improving)its ability to enhance the accuracy of the DG solution.We prove the superconvergence error estimate of the new SIAC filters.Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.

Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.