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Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws

Published online by Cambridge University Press:  03 June 2015

Andreas Hiltebrand  and
Siddhartha Mishra
Andreas Hiltebrand*
Affiliation:
Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG J 49, Zürich -8092, Switzerland
Siddhartha Mishra
Affiliation:
Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG G 57.2, Zürich -8092, Switzerland; Center of Mathematics for Applications (CMA), University of Oslo, P. O. Box -1053, Blindern, Oslo-0316, Norway
*
*Corresponding author. Email addresses: andreas.hiltebrand@sam.math.ethz.ch (A. Hiltebrand), smishra@sam.math.ethz.ch (S. Mishra)
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Abstract

An entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper to approximate hyperbolic systems of conservation laws. This numerical scheme involves the solution of a very large nonlinear system of algebraic equations, by a Newton-Krylov method, at every time step. In this paper, we design efficient preconditioners for the large, non-symmetric linear system, that needs to be solved at every Newton step. Two sets of preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are designed. Fourier analysis of the preconditioners reveals their robustness and a large number of numerical experiments are presented to illustrate the gain in efficiency that results from preconditioning. The resulting method is employed to compute approximate solutions of the compressible Euler equations, even for very high CFL numbers.

Keywords

65M6035L65Preconditioningspace-time DGentropy stablesystems of conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 5 , May 2015 , pp. 1320 - 1359
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Barth, T. J.Numerical methods for gas-dynamics systems on unstructured meshes. In An Introduction to Recent Developments in Theory and Numerics of Conservation Laws, Lecture Notes in Computational Science and Engineering volume 5, Springer, Berlin. Kroner, D.Ohlberger, M., and Rohde, C., eds., pp 195–285, 1999.Google Scholar
[2]Barth, T. J.An introduction to upwind finite volume and finite element methods: Some unifying and contrasting themes. In VKI Lecture Series 2006–01, 34th CFD-Higher Order Discretization Methods (EUA4X), 2006.Google Scholar
[3]Chiodaroli, E.De Lellis, C., Kreml, O.. Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math., doi: 10.1002/cpa.21537, 2014.Google Scholar
[4]Cockburn, B. and Shu, C-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput., 52, 411–435, 1989.Google Scholar
[5]Cockburn, B., Lin, S-y.; Shu, C-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Phys., 84, 90–113, 1989.Google Scholar
[6]Dafermos, C.Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, 2000.Google Scholar
[7]DiPerna, R. J.. Measure valued solutions to conservation laws. Arch. Rational Mech. Anal., 88(3), 223–270, 1985.Google Scholar
[8] D. Dunavant, A.High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle. Int. J. Numer. Method. Engr., 21, 1129–1148, 1985.Google Scholar
[9]Fjordholm, U. S.Mishra, S. and Tadmor, E.Energy preserving and energy stable schemes for the shallow water equations. In Foundations of Computational Mathematics, Proc. FoCM held in Hong Kong 2008, Cucker, F., Pinkus, A. and Todd, M., eds., London Math. Soc. Lecture Notes Ser. 363, pp. 93–139, 2009.Google Scholar
[10]Fjordholm, U. S.Mishra, S. and Tadmor, E.Arbitrary order accurate essentially non-oscillatory entropy stable schemes for systems of conservation laws. SIAM J. Num. Anal., 50(2), 544–573, 2012.CrossRefGoogle Scholar
[11]Fjordholm, U. S.Käppeli, R., Mishra, S. and Tadmor, E.Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws. Preprint, arXiv:1402.0909, 2014.Google Scholar
[12]Fjordholm, U. S.High-order accurate entropy stable numerical schemes for hyperbolic conservation laws. ETH Zürich dissertation Nr. 21025, 2013.Google Scholar
[13]Fjordholm, U. S. and Mishra, S.Convergence of entropy stable finite difference schemes to measure valued solutions of hyperbolic systems of conservation laws. In preparation, 2014.Google Scholar
[14]Lim, H.Yu, Y., Glimm, J., Li, X. L. and Sharp, D. H. Chaos, transport and mesh convergence for fluid mixing. Act. Math. Appl. Sin., 24(3), 355–368, 2008.Google Scholar
[15]Godlewski, E. and Raviart, P. A.Hyperbolic Systems of Conservation Laws. Mathematiques et Applications, Ellipses Publ., Paris, 1991.Google Scholar
[16]Hackbusch, W. and Probst, T.Downwind Gauß-Seidel smoothing for convection dominated problems. Numer. Lin. Alg. Appl., 4(2), 85–102, 1997.Google Scholar
[17]Harten, A.Engquist, B.Osher, S. and Chakravarty, S. R.Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys., 131, 3–47, 1997.Google Scholar
[18]Hiptmair, R.Jeltsch, R. and Kressner, D.Numerische Mathematik für Studiengang Rechnergestützte Wissenschaften. Lecture Notes, ETH Zurich, 2007.Google Scholar
[19]Hiltebrand, A.Koley, U. and Mishra, S.An arbitrarily high order accurate convergent DG method for scalar conservation laws. In preparation, 2014.Google Scholar
[20]Hiltebrand, A. and Mishra, S.Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws. Numerische Mathematik, 126(1), 103–151, 2014.Google Scholar
[21]Hiltebrand, A. and Mishra, S.Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method, Research report 2014–17, SAM ETH Zurich.CrossRefGoogle Scholar
[22]Hiltebrand, A.Entropy-stable discontinuous Galerkin finite element methods with streamline diffusion and shock-capturing for hyperbolic systems of conservation laws. ETH Zurich dissertation No. 22279, 2014.Google Scholar
[23]Hughes, T. J. RFranca, L. P. and Mallet, M.A new finite element formulation for CFD I: Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comp. Meth. Appl. Mech. Engr., 54, 223–234, 1986.Google Scholar
[24]Ismail, F. and Roe, P. L.Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks J. Comput. Phys., 228(15), 5410–5436, 2009.Google Scholar
[25]Jaffre, J.Johnson, C. and Szepessy, A.Convergence of the discontinuous galerkin finite element method for hyperbolic conservation laws. Math. Model. Meth. Appl. Sci., 5(3), 367–386, 1995.Google Scholar
[26]Johnson, C. and Szepessy, A.On the convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comput., 49(180), 427–444, 1987.Google Scholar
[27]Johnson, C.Hansbo, P. and Szepessy, A.On the convergence of shock capturing streamline diffusion methods for hyperbolic conservation laws. Math. Comp., 54(189), 107–129, 1990.Google Scholar
[28]LeVeque, R. J.. Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge, 2002.Google Scholar
[29]Saad, Y. and Schultz, M. H.GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3), 856–869, 1986.Google Scholar
[30]Shu, C. W. and Osher, S.Efficient implementation of essentially non-oscillatory schemes – II, J. Comput. Phys., 83, 32–78, 1989.Google Scholar
[31]Shu, C. W.High-order ENO and WENO schemes for Computational fluid dynamics. In High-Order Methods for Computational Physics, Barth, T. J. and Deconinck, H. eds., Lecture notes in Computational Science and Engineering 9, Springer Verlag, pp. 439–582, 1999.Google Scholar
[32]Tadmor, E.The numerical viscosity of entropy stable schemes for systems of conservation laws, Math. I. Comp., 49, 91–103, 1987.Google Scholar
[33]Tadmor, E.Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Act. Numerica, 451–512, 2003.CrossRefGoogle Scholar