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A Parallel, Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids

Published online by Cambridge University Press:  20 August 2015

Hong Luo ,
Luqing Luo ,
Amjad Ali ,
Robert Nourgaliev  and
Chunpei Cai
Hong Luo*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA
Luqing Luo*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA
Amjad Ali*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA
Robert Nourgaliev*
Affiliation:
Reactor Safety Simulation Group, Idaho National Laboratory, Idaho Falls, ID, 83415, USA
Chunpei Cai*
Affiliation:
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM, 88001, USA
*
Corresponding author.Email:hong_luo@ncsu.edu
Email:lluo2@ncsu.edu
Email:aali3@ncsu.edu
Email:robert.nourgaliev@inl.gov
Email:ccai@nmsu.edu
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Abstract

A reconstruction-based discontinuous Galerkin method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. In this method, an in-cell reconstruction is used to obtain a higher-order polynomial representation of the underlying discontinuous Galerkin polynomial solution and an inter-cell reconstruction is used to obtain a continuous polynomial solution on the union of two neighboring, interface-sharing cells. The in-cell reconstruction is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. The inter-cell reconstruction is devised to remove an interface discontinuity of the solution and its derivatives and thus to provide a simple, accurate, consistent, and robust approximation to the viscous and heat fluxes in the Navier-Stokes equations. A parallel strategy is also devised for the resulting reconstruction discontinuous Galerkin method, which is based on domain partitioning and Single Program Multiple Data (SPMD) parallel programming model. The RDG method is used to compute a variety of compressible flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results demonstrate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method, at the same time providing a better performance than the third order DG method, in terms of both computing costs and storage requirements.

Keywords

65M6065M9976M2576M10Discontinuous Galerkin methodsleast-squares reconstruction methodscompressible Navier-Stokes equations
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 2 , February 2011 , pp. 363 - 389
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Reed, W. H. and Hill, T. R., Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report, LA-UR-73-479, 1973.Google Scholar
[2]Cockburn, B., Hou, S. and Shu, C. W., TVD Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Mathematics of Computation, Vol. 55, pp. 545–581, 1990.Google Scholar
[3]Cockburn, B. and Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional system, Journal of Computational Physics, Vol. 141, pp. 199–224, 1998.Google Scholar
[4]Cockburn, B., Karniadakis, G. and Shu, C. W., The development of discontinuous Galerkin method, in: Discontinuous Galerkin Methods, Theory, Computation, and Applications, Cockburn, B., Karniadakis, G. E. and Shu, C. W. (Eds.), Lecture Notes in Computational Science and Engineering, Springer-Verlag, New York, 2000, Vol. 11, pp. 5–50, 2000.Google Scholar
[5]Bassi, F. and Rebay, S., High-order accurate discontinuous finite element solution of the 2D Euler equations, Journal of Computational Physics, Vol. 138, pp. 251–285, 1997.Google Scholar
[6]Atkins, H. L. and Shu, C. W., Quadrature free implementation of discontinuous Galerkin method for hyperbolic equations, AIAA Journal, Vol. 36, No. 5, 1998.Google Scholar
[7]Bassi, F. and Rebay, S., GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations, Discontinuous Galerkin Methods, Theory, Computation, and Applications, Cockburn, B., Karniadakis, G. E. and Shu, C. W. (Eds.), Lecture Notes in Computational Science and Engineering, Springer-Verlag, New York, 2000, Vol. 11, pp. 197–208, 2000.Google Scholar
[8]Warburton, T. C. and Karniadakis, G. E., A discontinuous Galerkin method for the viscous MHD equations, Journal of Computational Physics, Vol. 152, pp. 608–641, 1999.CrossRefGoogle Scholar
[9]Hesthaven, J. S. and Warburton, T., Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Texts in Applied Mathematics, Vol. 56, 2008.Google Scholar
[10]Rasetarinera, P. and Hussaini, M. Y., An efficient implicit discontinuous spectral Galerkin method, Journal of Computational Physics, Vol. 172, pp. 718–738, 2001.Google Scholar
[11]Helenbrook, B. T., Mavriplis, D. and Atkins, H. L., Analysis of p-multigrid for continuous and discontinuous finite element discretizations, AIAA Paper, 2003-3989, 2003.Google Scholar
[12]Fidkowski, K. J., Oliver, T. A., Lu, J. and Darmofal, D. L., p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, Journal of Computational Physics, Vol. 207, No. 1, pp. 92–113, 2005.Google Scholar
[13]Luo, H., Baum, J. D. and Löhner, R., A discontinuous Galerkin method using Taylor basis for compressible flows on arbitrary grids, Journal of Computational Physics, Vol. 227, No. 20, pp. 8875–8893, 2008.Google Scholar
[14]Luo, H., Baum, J. D. and Löhner, R., On the computation of steady-state compressible flows using a discontinuous Galerkin method, International Journal for Numerical Methods in Engineering, Vol. 73, No. 5, pp. 597–623, 2008.CrossRefGoogle Scholar
[15]Luo, H., Baum, J. D. and Löhner, R., A Hermite WENO-based limiter for discontinuous galerkin method on unstructured grids, Journal of Computational Physics, Vol. 225, No. 1, pp. 686–713, 2007.Google Scholar
[16]Luo, H., Baum, J. D. and Löhner, R., A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids, Journal of Computational Physics, Vol. 211, No. 2, pp. 767–783, 2006.CrossRefGoogle Scholar
[17]Luo, H., Baum, J. D. and Löhner, R., A fast, p-multigrid discontinuous Galerkin method for compressible flows at all speeds, AIAA Journal, Vol. 46, No. 3, pp. 635–652, 2008.Google Scholar
[18]Dumbser, M., Balsara, D. S., Toro, E. F. and Munz, C. D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, Journal of Computational Physics, Vol. 227, pp. 8209–8253, 2008.CrossRefGoogle Scholar
[19]Dumbser, M. and Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, Journal of Computational Physics, Vol. 228, pp. 6991–7006, 2009.Google Scholar
[20]Dumbser, M., Arbitrary high order PNPM Schemes on unstructured meshes for the compressible Navier-Stokes equations, Computers & Fluids, Vol. 39, pp. 60–76, 2010.Google Scholar
[21]Bassi, F. and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, Journal of Computational Physics, Vol. 131, pp. 267–279, 1997.CrossRefGoogle Scholar
[22]Bassi, F. and Rebay, S., Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations, Journal of Computational Physics, Vol. 34, pp. 507–540, 2005.Google Scholar
[23]Cockburn, B. and Shu, C. W., The local discontinuous Galerkin method for time-dependent convection-diffusion system, SIAM Journal of Numerical Analysis, Vol. 35, pp. 2440–2463, 1998.CrossRefGoogle Scholar
[24]Baumann, C. E. and Oden, J. T., A discontinuous hp finite element method for the Euler and Navier-Stokes equations, International Journal for Numerical Methods in Fluids, Vol. 31, pp. 79–95, 1999.Google Scholar
[25]Peraire, J. and Persson, P. O., The compact discontinuous Galerkin method for elliptic problems, SIAM Journal on Scientific Computing, Vol. 30, pp. 1806–1824, 2008.Google Scholar
[26]Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, Vol. 39, No. 5, pp. 1749–1779, 2002.Google Scholar
[27]Gassner, G., Lorcher, F. and Munz, C. D., A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, Journal of Computational Physics, Vol. 224, No. 2, pp. 1049–1063, 2007.Google Scholar
[28]Liu, H. and Xu, K., A Runge-Kutta discontinuous Galerkin method for viscous flow equations, Journal of Computational Physics, Vol. 224, No. 2, pp. 1223–1242, 2007.CrossRefGoogle Scholar
[29]Luo, H., Luo, L. and Xu, K., A discontinuous Galerkin method based on a BGK scheme for the Navier-Stokes equations on arbitrary grids, Advances in Applied Mathematics and Mechanics, Vol. 1, No. 3, pp. 301–318, 2009.Google Scholar
[30]van Leer, B. and Nomura, S., Discontinuous Galerkin method for diffusion, AIAA Paper, 2005-5108, 2005.Google Scholar
[31]van Leer, B. and Lo, M., A Discontinuous Galerkin method for diffusion based on recovery, AIAA Paper, 2007-4083, 2007.Google Scholar
[32]Raalte, M. and van Leer, B., Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion, Communication of Computational Physics, Vol. 5, No. 2-4, pp. 683–693, 2009.Google Scholar
[33]Nourgaliev, R., Park, H. and Mousseau, V., Recovery discontinuous Galerkin Jacobian-free Newton-Krylov method for multiphysics problems, Computational Fluid Dynamics Review 2009, 2009, to appear.CrossRefGoogle Scholar
[34]Huynh, H.T , A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion, AIAA Paper, 2009-0403, 2009.Google Scholar
[35]Luo, H., Luo, L., Nourgaliev, R. and Mousseau, V., A reconstructed discontinuous Galerkin method for the compressible Euler equations on arbitrary grids, AIAA Paper, 2009-3788, 2009.CrossRefGoogle Scholar
[36]Luo, H., Luo, L., Norgaliev, R., Mousseau, V. A. and Dinh, N., A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, Journal of Computational Physics, Vol. 229, No. 19, pp. 6961–6978, 2010.Google Scholar
[37]Colella, P. and Woodward, P. R., The piecewise parabolic method (PPM) for gas-dynamical simulations, Journal of Computational Physics, Vol. 54, No. 1, pp. 115–173, 1984.CrossRefGoogle Scholar
[38]Karypis, G. and Kumar, V., A fast and high quality scheme for partitioning irregular graphs, SIAM Journal on Scientific Computing, Vol. 20, pp. 359–392, 1999.Google Scholar