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An implicit discontinuous Galerkin method for the unsteady compressible Navier–Stokes equations

Hong Luo, Hidehiro Segawa, Miguel R. Visbal
2012 Computers & Fluids  
Most of these numerical methods are based on the semi-discrete approach: discontinuous Galerkin finite element methods are used for the spatial discretization, rendering the original partial differential  ...  The compactness also allows for structured and simplified coding for the method; 5) It can easily handle adaptive strategies, since refining or coarsening a grid can be achieved without considering the  ...  Acknowledgments The first author would like to acknowledge the partial support for this work provided by the Air Force Summer faculty Fellowship Program, while he was in residence at Computational Sciences  ... 
doi:10.1016/j.compfluid.2011.10.009 fatcat:3qwgawmmpjh4fcc4u2mtywtnhy

Efficient Variable-Coefficient Finite-Volume Stokes Solvers

Mingchao Cai, Andy Nonaka, John B. Bell, Boyce E. Griffith, Aleksandar Donev
2014 Communications in Computational Physics  
We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems.  ...  Building on the success of using the classical projection method as a preconditioner for the coupled velocity pressure system[B. E. Griffith, J. Comp.  ...  In our implementation, the multigrid coarsening factor is 2, and coarsening continues until the problem domain represented on the coarsest grid contains two grid points (with respect to cell-centers) in  ... 
doi:10.4208/cicp.070114.170614a fatcat:jbb7zkn3fjhnzduxpnxfvpvqsu

Perspective on unstructured grid flow solvers

V. Venkatakrishnan
1996 AIAA Journal  
The integration of these components into an overall framework to solve practical problems is addressed.  ...  This survey paper assesses the status of compressible Euler and Navier-Stokes solvers on unstructured grids.  ...  In a cell-centered setting, they extended much of what was established for finite volume schemes on structured grids [74, 67] to triangular grids.  ... 
doi:10.2514/3.13101 fatcat:ihw66pcsxfa65k4okgt3ssmxkm

Unstructured-Mesh Discretizations and Solvers for Computational Aerodynamics

D. J. Mavriplis
2008 AIAA Journal  
With regards to solution techniques, implicit and multigrid methods are discussed, as well as efficient implementation on massively parallel computer hardware.  ...  Cell-centered versus vertex-centered discretizations are discussed, as well as issues of grid alignment with flow features, higher-order reconstruction, and viscous term formulation.  ...  and efficient unstructured mesh solution mechanism compared with structured multigrid methods.  ... 
doi:10.2514/1.34681 fatcat:2gvob7pydzcc7bvgumdqhcdxvm

Multigrid solution of the Navier–Stokes equations at low speeds with large temperature variations

Peter M. Sockol
2003 Journal of Computational Physics  
The resulting methods have been tested on four 2D laminar problems over a range of Reynolds numbers on both uniform and highly stretched grids.  ...  Multigrid methods for the Navier-Stokes equations at low speeds and large temperature variations are investigated.  ...  From the present studies it is evident that an efficient robust multigrid solver can be based on either LBGS or BILU with either full or semi-coarsening, depending on the problem.  ... 
doi:10.1016/j.jcp.2003.07.033 fatcat:qckh7wtfnvfsdn4xsv3bed3hhy

Multigrid relaxation methods for systems of saddle point type

C.W. Oosterlee, F.J. Gaspar
2008 Applied Numerical Mathematics  
In this paper, we give an overview of multigrid methods for two systems of equations, namely the Stokes equations and the incompressible poroelasticity equations.  ...  For general systems of equations it is a challenge to design an efficient distributive relaxation scheme. This paper may help in finding one.  ...  problems with discontinuous coefficients.  ... 
doi:10.1016/j.apnum.2007.11.014 fatcat:kvhdqqqmtbf5rdeyiemh2ezwry

A two-level variational multiscale method for convection-dominated convection–diffusion equations

Volker John, Songul Kaya, William Layton
2006 Computer Methods in Applied Mechanics and Engineering  
This paper studies the error in, the efficient implementation of and time stepping methods for a variational multiscale method (VMS) for solving convectiondominated problems.  ...  Our tests show that these choices lead to an efficient VMS whose complexity is further reduced if a (locally) L 2 -orthogonal basis for L H is used.  ...  Based on the results obtained in this paper, we will use discontinuous finite elements for the space L H and the semi-implicit approach.  ... 
doi:10.1016/j.cma.2005.10.006 fatcat:z4rl4a74b5gtresgo47t6iebv4

Mode Multigrid - A novel convergence acceleration method [article]

Yilang Liu, Weiwei Zhang, Jiaqing Kou
2018 arXiv   pre-print
This paper proposes a mode multigrid (MMG) method, and applies it to accelerate the convergence of the steady state flow on unstructured grids.  ...  Therefore, it is very convenient to be applied to any numerical schemes with just little change for the flow solver, which is also suitable for unstructured grids and easy for parallel computing.  ...  Mulder [18] and Pierce [19] proposed the semi-coarsening (or J-coarsening) method in order to apply it on the anisotropic structured grid, which has been widely used for the viscous flows on the highly  ... 
arXiv:1802.08962v1 fatcat:dc3odmonhrg23kevtn2rfskh5u

A Parallel Multigrid Solver on a Structured Triangulation of a Hexagonal Domain [chapter]

Kab Seok Kang
2014 Lecture Notes in Computational Science and Engineering  
Tskhakaya for helpful discussions.  ...  at Computational Simulation Centre of International Fusion Energy Research Centre (IFERC-CSC), Aomori, Japan, under the Broader Approach collaboration between Euratom and Japan, implemented by Fusion for  ...  The non-overlapping DDM is a natural method for problems which have discontinuous coefficients or many parts and are akin to being implemented on distributed memory computers.  ... 
doi:10.1007/978-3-319-05789-7_76 fatcat:h4hklmyjabgtzci2hh65eqoo4q

Multigrid algorithms for compressible flow calculations [chapter]

Antony Jameson
1986 Lecture notes in mathematics  
and contact discontinuities, treatment of viscous effects, and computational efficiency.  ...  to non-elliptic problems securely anchored on a firm theoretical foundation.  ...  Computational methods may then finally fulfill their proper role as a reliable guide for the design of aeroplanes, cars, and any other devices whose performance significantly depends on aerodynamic efficiency  ... 
doi:10.1007/bfb0072647 fatcat:7p36lyzojjhbroa3rzvkwp5a2a

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

Jeffrey Lee Hellrung, Luming Wang, Eftychios Sifakis, Joseph M. Teran
2012 Journal of Computational Physics  
Keywords: Elliptic interface problems Embedded interface methods Virtual node methods Variational methods Multigrid methods a b s t r a c t We present a numerical method for the variable coefficient Poisson  ...  For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump  ...  We also wish to thank Russell Howes and Alexey Stomakhin for submitting typographical errors and providing comments on the later drafts of this paper.  ... 
doi:10.1016/j.jcp.2011.11.023 fatcat:xvkrotmfwvfnlhojsapto7poam

A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

Hong Luo, Luqing Luo, Robert Nourgaliev, Vincent A. Mousseau, Nam Dinh
2010 Journal of Computational Physics  
A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids.  ...  The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility.  ...  DE-AC07-05ID14517 (INL/CON-09-16255) with the U.S. Department of Energy.  ... 
doi:10.1016/j.jcp.2010.05.033 fatcat:iw2fbidxkzd7bazqtgqv6fxotm

A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids

Hong Luo, Joseph D. Baum, Rainald Löhner
2007 Journal of Computational Physics  
A weighted essential non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for a discontinuous Galerkin finite element method on unstructured grids.  ...  The developed HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and timeaccurate compressible flow problems on unstructured grids.  ...  Cockburn at University of Minnesota for many helpful, instructive, and fruitful discussions about DG and WENO.  ... 
doi:10.1016/j.jcp.2006.12.017 fatcat:su5p2w575zggtdyaplkzddq2ia

Higher Order Discontinuous Galerkin Methods for Flow and Transport in Porous Media [chapter]

Peter Bastian
2003 Lecture Notes in Computational Science and Engineering  
Based on a scheme developed by Oden, Babuška and Baumann we present a method for the solution of the elliptic flow equation and describe a multigrid method for the fast solution of the arising algebraic  ...  In this paper we describe higher order discontinuous Galerkin (DG) methods for application in subsurface transport.  ...  For an introduction to multigrid methods we refer to [21, 22] Here, we will derive the coarse grid correction of a two-grid method in variational form.  ... 
doi:10.1007/978-3-642-19014-8_1 fatcat:o7ij6hwqdzcl7cuba4vpjgxgy4

Computational transonics

Antony Jameson
1988 Communications on Pure and Applied Mathematics  
With properly optimized coefficients the multi-stage time stepping scheme is a very efficient driver of the multigrid process.  ...  These variations are illustrated in Figure 2 for a two dimensional case. Figures 2(a) and 2(b) show cell centered schemes on rectilinear and triangular meshes [13−16] .  ... 
doi:10.1002/cpa.3160410503 fatcat:uixf3lcxfvdlvlptpjbizdrqsa
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