Superconvergence of Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws in One Space Dimension

Meng, Xiong; Shu, Chi‐Wang; Zhang, Qiang; Wu, Boying

doi:10.1137/110857635

Cited by 60 publications

(41 citation statements)

References 25 publications

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“…The result with in Theorem 2 is indeed a superconvergence result towards a particular projection of the exact solution (supercloseness) that has been established in [18], which is a starting point for proving with . For completeness, we list the superconvergence result for (zeroth order divided difference) as follows where, on each element , we have used with being a constant and .…”

Section: Norm Error Estimates For Divided Differencesmentioning

confidence: 78%

“…To do that, we start by noting that , and that , which have already been proved in [18, Appendix A.2]. Next, note also that the first order time derivative of the original error equationstill holds at for any .…”

Section: Appendixmentioning

confidence: 87%

“…Moreover, (3.37c), namely has been given in (3.4c); see [18]. To complete the proof for , we need only to establish the following relation …”

Section: Norm Error Estimates For Divided Differencesmentioning

confidence: 99%

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Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

Meng

Ryan

2016

Numer. Math.

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In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the -th order divided difference of the DG error in the norm is of order when upwind fluxes are used, under the condition that possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order in the norm for the divided differences of DG errors and thus th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.

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Section: Norm Error Estimates For Divided Differencesmentioning

confidence: 78%

Section: Appendixmentioning

confidence: 87%

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Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

Meng

Ryan

2016

Numer. Math.

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“…In [2, 5], Adjerid et al studied the ordinary differential equations and proved the (k + 2)-th order superconvergence of the DG solutions at the downwind-biased Radau points. For hyperbolic equations, the superconvergence results have been investigated by several authors [6,7,9,10,12,24,26,27]. Especially, in [26], we obtained sharp superconvergence for linear hyperbolic equations by using the dual argument, and this gives us the motivation to the prove the sharp superconvergence for linear parabolic equations.…”

mentioning

confidence: 87%

Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations

Yang

Shu

2015

JCM

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In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise k-th degree polynomials, the error between the LDG solution and the exact solution is (k + 2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is (k + 2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for P k polynomials with arbitrary k ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.Mathematics subject classification: 65M60, 65M15.The DG method was first introduced in 1973 by Reed and Hill [25], in the framework of neutron linear transport. Later, the method was applied by Johnson and Pitkäranta to a scalar linear hyperbolic equation and the L p -norm error estimate was proved [23]. Subsequently, Cockburn et al. developed Runge-Kutta discontinuous Galerkin (RKDG) methods for hyperbolic conservation laws in a series of papers [16][17][18][19]. In [20], Cockburn and Shu first introduced the LDG method to solve the convection-diffusion equation. Their idea was motivated by Bassi and Rebay [8], where the compressible Navier-Stokes equations were successfully solved.The superconvergence properties have been analyzed intensively. In [2, 5], Adjerid et al. studied the ordinary differential equations and proved the (k + 2)-th order superconvergence of the DG solutions at the downwind-biased Radau points. For hyperbolic equations, the superconvergence results have been investigated by several authors [6,7,9,10,12,24,26,27]. Especially, in [26], we obtained sharp superconvergence for linear hyperbolic equations by using the dual argument, and this gives us the motivation to the prove the sharp superconvergence for linear parabolic equations. For convection-diffusion problems, in [3,4], the authors used numerical experiments to demonstrate the superconvergence of LDG solution at the Radau points. In [11], the steady state solution was studied and the superconvergence of the numerical fluxes was proved. In [13], Cheng and Shu discussed the superconvergence property of the LDG scheme for heat equation by using piecewise linear approximations and uniform meshes. Subsequently, they proved the (k + 3 2 )-th order superconvergence when using piecewise k-th degree polynomials with arbitrary k on arbitrary regular meshes in [14]. However, the convergence rate obtained in [14] is not sharp. Numerical tests demonstrated that the error of the DG solution towards a particular projection of the exact solution is (k + 2)-th order accurate, even on highly nonuniform meshes. In [14], the framework to prove the superconvergence res...

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“…For the nonlinear problems, Meng and Shu proved that the error between the DG solution and a particular projection is (k + 3 2 )th superconvergent for the scalar nonlinear conservation laws, when the upwind fluxes are used [26].…”

Section: Introductionmentioning

confidence: 99%

The optimal error estimate and superconvergence of the local discontinuous Galerkin methods for one-dimensional linear fifth order time dependent equations

Qian

Sun

2016

Computers & Mathematics with Applications

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a b s t r a c tIn this paper, we investigate the optimal error estimate and the superconvergence of linear fifth order time dependent equations. We prove that the local discontinuous Galerkin (LDG) solution is (k + 1)th order convergent when the piecewise P k space is used. Also, the numerical solution is (k + 3 2 )th order superconvergent to a particular projection of the exact solution. The numerical experiences indicate that the order of the superconvergence is (k + 2), which implies the result obtained in this paper is suboptimal.

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Superconvergence of Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws in One Space Dimension

Cited by 60 publications

References 25 publications

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations

The optimal error estimate and superconvergence of the local discontinuous Galerkin methods for one-dimensional linear fifth order time dependent equations

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