Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection–diffusion problems

Baccouch, Mahboub

doi:10.1016/j.amc.2013.10.026

Cited by 18 publications

(17 citation statements)

References 47 publications

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“…We provided a mathematical justification in our earlier studies for one-dimensional convection [21,63,64], convection-diffusion [65], and wave equation [66]. Indeed, for one-dimensional problems, we proved that the error estimates without the time change of e u converge to the true errors under mesh refinement.…”

Section: Remarkmentioning

confidence: 80%

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

Baccouch

2014

Computers & Mathematics with Applications

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Section: Remarkmentioning

confidence: 80%

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

Baccouch

2014

Computers & Mathematics with Applications

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“…In this paper, a new IBM to solve the second-order partial equation applied to the local discontinuous Galerkin method (LDG) is presented and we analyzed the causes of error variation for the adaptive Cartesian grid. The LDG method [17][18][19][20] means it easy to achieve high accuracy in space and time and provides useful mathematical properties with respect to conservation, stability, and super convergence. In particular, the LDG method can use the mesh with the hanging node [21,22] for calculation, and it is convenient to apply the method for simulating flows in complex geometries.…”

Section: Introductionmentioning

confidence: 99%

New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method

Zhang

Zhu

Yan

et al. 2018

Chin. J. Mech. Eng.

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Currently, many studies on the local discontinuous Galerkin method focus on the Cartesian grid with low computational efficiency and poor adaptability to complex shapes. A new immersed boundary method is presented, and this method employs the adaptive Cartesian grid to improve the adaptability to complex shapes and the immersed boundary to increase computational efficiency. The new immersed boundary method employs different boundary cells (the physical cell and ghost cell) to impose the boundary condition and the reconstruction algorithm of the ghost cell is the key for this method. The classical model elliptic equation is used to test the method. This method is tested and analyzed from the viewpoints of boundary cell type, error distribution and accuracy. The numerical result shows that the presented method has low error and a good rate of the convergence and works well in complex geometries. The method has good prospect for practical application research of the numerical calculation research. which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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“…LDG methods for diffusion problems were investigated by Adjerid and Klauser who constructed efficient and accurate a posteriori error estimates. More recently, the author constructed and analyzed new superconvergence‐based a posteriori error estimates for the LDG methods in for convection‐diffusion equations, for the KdV equation, and for the fourth‐order Euler–Bernoulli equations. The first superconvergence‐based a posteriori LDG error estimates for two‐dimensional convection‐diffusion and wave problems on Cartesian grids were developed in for convection‐diffusions problems and for the wave equation.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation

Baccouch

2014

Numer. Methods Partial Differential Eq.

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In this article, we analyze a residual-based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one-dimensional second-order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L 2 error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862-901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L 2 -norm under mesh refinement. The order of convergence is proved to be p + 3/2, when p-degree piecewise polynomials with p ≥ 1 are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(h p+3/2 ) superconvergent solutions. Our computational results show higher O(h p+2 ) convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L 2 -norm converge to unity at O(h 1/2 ) rate while numerically they exhibit O(h 2 ) and O(h) rates, respectively. Numerical experiments are shown to validate the theoretical results.

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Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection–diffusion problems

Cited by 18 publications

References 47 publications

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method

Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation

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