Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations

ArticleinJournal of Computational Physics 302:251-266 · September 2015with 83 Reads 
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  • Preprint
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    In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L^2$ stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method.
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Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the high-resolution and shock-capturing capabilities of the news schemes are significantly enhanced within the cell-by-cell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. 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  • Quadratic velocity/linear pressure Stokes elements, in: Advances in Computer Methods for Partial Differential Equations—VII
    • D Arnold
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    D. Arnold, J. Qin, Quadratic velocity/linear pressure Stokes elements, in: Advances in Computer Methods for Partial Differential Equations—VII, IMACS, 1992, pp. 28–34.
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