Skip to main content Accessibility help
×
Home

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

  • Andreas Hiltebrand (a1) and Siddhartha Mishra (a2)

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.

Copyright

Corresponding author

*Corresponding author. Email addresses: andreas.hiltebrand@sam.math.ethz.ch (A. Hiltebrand), smishra@sam.math.ethz.ch (S. Mishra)

References

Hide All
[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 195285, 1999.
[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.
[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.
[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, 411435, 1989.
[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, 90113, 1989.
[6]Dafermos, C.Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, 2000.
[7]DiPerna, R. J.. Measure valued solutions to conservation laws. Arch. Rational Mech. Anal., 88(3), 223270, 1985.
[8] D. Dunavant, A.High Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle. Int. J. Numer. Method. Engr., 21, 11291148, 1985.
[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. 93139, 2009.
[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), 544573, 2012.
[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.
[12]Fjordholm, U. S.High-order accurate entropy stable numerical schemes for hyperbolic conservation laws. ETH Zürich dissertation Nr. 21025, 2013.
[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.
[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), 355368, 2008.
[15]Godlewski, E. and Raviart, P. A.Hyperbolic Systems of Conservation Laws. Mathematiques et Applications, Ellipses Publ., Paris, 1991.
[16]Hackbusch, W. and Probst, T.Downwind Gauß-Seidel smoothing for convection dominated problems. Numer. Lin. Alg. Appl., 4(2), 85102, 1997.
[17]Harten, A.Engquist, B.Osher, S. and Chakravarty, S. R.Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys., 131, 347, 1997.
[18]Hiptmair, R.Jeltsch, R. and Kressner, D.Numerische Mathematik für Studiengang Rechnergestützte Wissenschaften. Lecture Notes, ETH Zurich, 2007.
[19]Hiltebrand, A.Koley, U. and Mishra, S.An arbitrarily high order accurate convergent DG method for scalar conservation laws. In preparation, 2014.
[20]Hiltebrand, A. and Mishra, S.Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws. Numerische Mathematik, 126(1), 103151, 2014.
[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.
[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.
[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, 223234, 1986.
[24]Ismail, F. and Roe, P. L.Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks J. Comput. Phys., 228(15), 54105436, 2009.
[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), 367386, 1995.
[26]Johnson, C. and Szepessy, A.On the convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comput., 49(180), 427444, 1987.
[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), 107129, 1990.
[28]LeVeque, R. J.. Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge, 2002.
[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), 856869, 1986.
[30]Shu, C. W. and Osher, S.Efficient implementation of essentially non-oscillatory schemes – II, J. Comput. Phys., 83, 3278, 1989.
[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. 439582, 1999.
[32]Tadmor, E.The numerical viscosity of entropy stable schemes for systems of conservation laws, Math. I. Comp., 49, 91103, 1987.
[33]Tadmor, E.Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Act. Numerica, 451512, 2003.

Keywords

Related content

Powered by UNSILO

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

  • Andreas Hiltebrand (a1) and Siddhartha Mishra (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.