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 disper...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.展开更多
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 ...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.展开更多
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.On...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.展开更多
基金This work was sponsored by the Air Force Office of Scientific Research(AFOSR),Air Force Material Command,USAF,under grant number FA8655-09-1-3017
文摘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.
基金This work was motivated by discussions with Dr.Venke Sankaran(Edwards Air Force Research Lab,USA)and was performed while visiting the Applied Mathematics group at HeinrichHeine University,Düsseldorf,Germany.Research supported by the Air Force Ofce of Scientifc Research(AFOSR)Computational Mathematics Program(Program Manager:Dr.Fariba Fahroo)under Grant numbers FA9550-18-1-0486 and FA9550-19-S-0003.
文摘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.
基金Funding for this work was partially supported by the National Natural Science Foundation of China(NSFC)under Grant no.11801062.
文摘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.
