Optimal Convergence of the Original DG Method on Special Meshes for Variable Transport Velocity

Cockburn, Bernardo; Dong, Bo; Guzmán, Johnny; Qian, Jianliang

doi:10.1137/080740805

Cited by 25 publications

(28 citation statements)

References 19 publications

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“…We show that for the HDG methods using polynomial degree k ≥ 1 with a suitably chosen stabilization function, we have, for general meshes, that 5) and, for meshes (almost) aligned with the direction of β, that 6) where C is a constant independent of and h. Note that if ≤ O(h 2 ), we obtain optimal convergence for u h − u L 2 (Ω) in (1.6), which can be considered as an extension of a similar result for the pure hyperbolic case [15,17]. We also show that, with a suitably chosen stabilization function, the condition number of the global matrix for the scaled numerial traces is O(h −2 ), independent of .…”

Section: Introductionmentioning

confidence: 55%

An analysis of HDG methods for convection-dominated diffusion problems

Qiu

Zhang

2015

ESAIM: M2AN

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Abstract. We present the first a priori error analysis of the h-version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection-dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L 2 -error of the scalar variable converges with order k + 1/2 on general conforming quasi-uniform simplicial meshes, just as for conventional DG methods. We also show that the method achieves the optimal L 2 -convergence order of k + 1 on special meshes. Moreover, we discuss a new way of implementing the HDG methods for which the spectral condition number of the global matrix is independent of the diffusion coefficient. Numerical experiments are presented which verify our theoretical results.

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Section: Introductionmentioning

confidence: 55%

An analysis of HDG methods for convection-dominated diffusion problems

Qiu

Zhang

2015

ESAIM: M2AN

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“…where the last inequality holds similar to [17]. Now, recall a standard estimate for the L 2 -projection,…”

Section: It Remains To Estimate the Terms Tmentioning

confidence: 99%

“…We remark that the accuracy order (k + 1 2 ) obtained below is optimal under general meshes [41]. The error estimates could be improved to (k + 1)th order, if special restrictions on the mesh and special projections are used; see [45,19,16,17] for related discussions, and we will not pursue it in this paper.…”

Section: Proof Let the Test Function Wmentioning

confidence: 99%

“…However, this is not true when α depends on x and v. In this case, similar to [17], we define an average of α on each simplex K as a constant vector α 0 , such that…”

Section: It Remains To Estimate the Terms Tmentioning

confidence: 99%

“…The main difficulty is that we are treating a variable coefficient equation with a collisional integral. The averaging technique in the proof of [17] will be adopted to take care of the variable coefficient α, while the collision term will be bounded using Lemma 7. We remark that the accuracy order (k + 1 2 ) obtained below is optimal under general meshes [41].…”

Section: Proof Let the Test Function Wmentioning

confidence: 99%

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Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

2012

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Abstract. We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of the numerical solutions in high dimensions. Our work extends the maximum-principle-satisfying scheme for scalar conservation laws in a recent work by X. Zhang and C.-W. Shu to include the linear Boltzmann collision term. The DG schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy. A discussion of the standard semi-discrete DG schemes for the BTE are included as a foundation for the stability and error estimates for this new scheme. Numerical results of the relaxation models are provided to validate the method.

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Discontinuous Galerkin Methods for Computational Fluid Dynamics

Cockburn

2017

Encyclopedia of Computational Mechanics Second Edition

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The discontinuous Galerkin (DG) methods are locally conservative, high‐order accurate, robust methods that can easily handle elements of arbitrary shapes, irregular meshes with hanging nodes, and polynomial approximations of different degrees in different elements. These properties, which render them ideal for hp ‐adaptivity in domains of complex geometry, have brought them to the main stream of computational fluid dynamics. We study the properties of the DG methods as applied to a wide variety of problems arising in fluid dynamics with a special emphasis on linear, symmetric positive hyperbolic systems, the Euler equations of gas dynamics, purely elliptic problems, and the incompressible and compressible Navier–Stokes equations. In each instance, we discuss the main properties of the methods, display the mechanisms that make them work so well, and present numerical experiments showing their performance.

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Optimal Convergence of the Original DG Method on Special Meshes for Variable Transport Velocity

Cited by 25 publications

References 19 publications

An analysis of HDG methods for convection-dominated diffusion problems

An analysis of HDG methods for convection-dominated diffusion problems

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

Discontinuous Galerkin Methods for Computational Fluid Dynamics

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