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A High-Order Method for Weakly Compressible Flows

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Equations of mathematical physics and other areas of application Basic methods in fluid mechanics Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  28 July 2017

Klaus Kaiser  and
Jochen Schütz
Klaus Kaiser*
Affiliation:
IGPM, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany
Jochen Schütz*
Affiliation:
Faculty of Sciences, Hasselt University, Agoralaan Gebouw D, BE-3590 Diepenbeek, Belgium
*
*Corresponding author. Email addresses:kaiser@igpm.rwth-aachen.de (K. Kaiser), jochen.schuetz@uhasselt.be (J. Schütz)
*Corresponding author. Email addresses:kaiser@igpm.rwth-aachen.de (K. Kaiser), jochen.schuetz@uhasselt.be (J. Schütz)

Abstract

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In this work, we introduce an IMEX discontinuous Galerkin solver for the weakly compressible isentropic Euler equations. The splitting needed for the IMEX temporal integration is based on the recently introduced reference solution splitting [32, 52], which makes use of the incompressible solution. We show that the overall method is asymptotic preserving. Numerical results show the performance of the algorithm which is stable under a convective CFL condition and does not show any order degradation.

Keywords

Asymptotic preservingisentropic compressible EulerRS-IMEXIMEX Runge-Kuttadiscontinuous Galerkinlow Mach

MSC classification

Secondary: 35Q31: Euler equations 65L06: Multistep, Runge-Kutta and extrapolation methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76M45: Asymptotic methods, singular perturbations

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 4 , October 2017 , pp. 1150 - 1174
Copyright
Copyright © Global-Science Press 2017 

Footnotes

Communicated by Chi-Wang Shu

References

[1] BOOST C++ Libraries. http://www.boost.org.Google Scholar
[2] Anderson, J. D.. Fundamentals of Aerodynamics. McGraw-Hill New York, 3 rd edition, 2001.Google Scholar
[3] Ascher, U. M., Ruuth, S., and Spiteri, R.. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151–167, 1997.Google Scholar
[4] Ascher, U.M., Ruuth, S., and Wetton, B.. Implicit-Explicit methods for time-dependent partial differential equations. SIAM Journal on Numerical Analysis, 32:797–823, 1995.Google Scholar
[5] Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and Zhang, H.. PETSc users manual. Technical Report ANL-95/11 – Revision 3.1, Argonne National Laboratory, 2010.Google Scholar
[6] Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and Zhang, H.. PETSc Web page, 2011. http://www.mcs.anl.gov/petsc.Google Scholar
[7] Balay, S., Gropp, W. D., McInnes, L. C., and Smith, B. F.. Efficient management of parallelism in object oriented numerical software libraries. In Arge, E., Bruaset, A.M., and Langtangen, H. P., editors, Modern Software Tools in Scientific Computing, pages 163–202. Birkhäuser Press Boston, 1997.CrossRefGoogle Scholar
[8] Bispen, G.. IMEX Finite Volume Methods for the Shallow Water Equations. PhD thesis, Johannes Gutenberg-Universität, 2015.Google Scholar
[9] Bispen, G., Arun, K.R., Lukáčová-Medvid’ová, M., and Noelle, S.. IMEX large time step finite volume methods for low Froude number shallow water flows. Communications in Computational Physics, 16:307–347, 2014.CrossRefGoogle Scholar
[10] Bispen, G., Lukáčová-Medvid’ová, M., and Yelash, L.. Asymptotic preserving IMEX finite volume schemes for low mach number euler equations with gravitation. Journal of Computational Physics, 335:222–248, 2017.Google Scholar
[11] Boscarino, S.. Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems. SIAM Journal on Numerical Analysis, 45:1600–1621, 2007.CrossRefGoogle Scholar
[12] Boscarino, S., Pareschi, L., and Russo, G.. Implicit-explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. SIAM Journal on Scientific Computing, 35(1):A22–A51, 2013.CrossRefGoogle Scholar
[13] Cockburn, B., Hou, S., and Shu, C.-W.. The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Mathematics of Computation, 54:545–581, 1990.Google Scholar
[14] Cockburn, B., Lin, S. Y., and Shu, C.-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. Journal of Computational Physics, 84:90–113, 1989.Google Scholar
[15] Cockburn, B. and Shu, C.-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Mathematics of Computation, 52:411–435, 1988.Google Scholar
[16] Cockburn, B. and Shu, C.-W.. The Runge-Kutta local projection p 1-discontinuous Galerkin finite element method for scalar conservation laws. RAIRO Mathematical modelling and numerical analysis, 25:337–361, 1991.Google Scholar
[17] Cockburn, B. and Shu, C.-W.. The Runge-Kutta discontinuous Galerkin Method for conservation laws V: Multidimensional Systems. Mathematics of Computation, 141:199–224, 1998.Google Scholar
[18] Cockburn, B. and Shu, C. W.. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16:173–261, 2001.Google Scholar
[19] Cordier, F., Degond, P., and Kumbaro, A.. An asymptotic-preserving all-speed scheme for the Euler and Navier-Stokes equations. Journal of Computational Physics, 231:5685–5704, 2012.Google Scholar
[20] Degond, P. and Tang, M.. All speed scheme for the low Mach number limit of the isentropic Euler equation. Communications in Computational Physics, 10:1–31, 2011.CrossRefGoogle Scholar
[21] Dimarco, G. and Pareschi, L.. Asymptotic preserving implicit-explicit Runge–Kutta methods for nonlinear kinetic equations. SIAM Journal on Numerical Analysis, 51(2):1064–1087, 2013.Google Scholar
[22] Filbet, F. and Jin, S.. A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. Journal of Computational Physics, 229(20):7625–7648, 2010.CrossRefGoogle Scholar
[23] Giraldo, F. X. and Restelli, M.. High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model. International Journal for Numerical Methods in Fluids, 63(9):1077–1102, 2010.CrossRefGoogle Scholar
[24] Giraldo, F.X., Restelli, M., and Läuter, M.. Semi-implicit formulations of the Navier-Stokes equations: Application to nonhydrostatic atmospheric modeling. SIAM Journal on Scientific Computing, 32(6):3394–3425, 2010.Google Scholar
[25] Guillard, H. and Viozat, C.. On the behavior of upwind schemes in the low Mach number limit. Computers and Fluids, 28(1):63–86, 1999.Google Scholar
[26] Haack, J., Jin, S., and Liu, J.-G.. An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations. Communications in Computational Physics, 12:955–980, 2012.Google Scholar
[27] Hundsdorfer, W. and Jaffré, J.. Implicit–explicit time stepping with spatial discontinuous finite elements. Applied Numerical Mathematics, 45(2):231–254, 2003.Google Scholar
[28] Hundsdorfer, W. and Ruuth, S.-J.. IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. Journal of Computational Physics, 225(2):2016–2042, 2007.CrossRefGoogle Scholar
[29] Jaust, A., Schütz, J., and Woopen, M.. A hybridized discontinuous Galerkin method for unsteady flows with shock-capturing. AIAA Paper 2014-2781, 2014.Google Scholar
[30] Jaust, A., Schütz, J., and Woopen, M.. An HDG method for unsteady compressible flows. In Kirby, Robert M., Berzins, Martin, and Hesthaven, Jan S., editors, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, volume 106 of Lecture Notes in Computational Science and Engineering, pages 267–274. Springer International Publishing, 2015.Google Scholar
[31] Jin, S.. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review. Rivista di Matematica della Universita Parma, 3:177–216, 2012.Google Scholar
[32] Kaiser, K., Schütz, J., Schöbel, R., and Noelle, S.. A new stable splitting for the isentropic Euler equations. Journal of Scientific Computing (in press), 2016.Google Scholar
[33] Kanevsky, A., Carpenter, M. H., Gottlieb, D., and Hesthaven, J. S.. Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. Journal of Computational Physics, 225(2):1753–1781, 2007.CrossRefGoogle Scholar
[34] Kennedy, C. A. and Carpenter, M. H.. Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Applied Numerical Mathematics, 44:139–181, 2003.CrossRefGoogle Scholar
[35] Klainerman, S. and Majda, A.. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Communications on Pure and Applied Mathematics, 34:481–524, 1981.Google Scholar
[36] Klein, R.. Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One-dimensional flow. Journal of Computational Physics, 121:213–237, 1995.Google Scholar
[37] Kröner, D.. Numerical Schemes for Conservation Laws. Wiley Teubner, 1997.Google Scholar
[38] Liu, H. and Zou, J.. Some new additive Runge–Kutta methods and their applications. Journal of Computational and Applied Mathematics, 190(1-2):74–98, 2006.CrossRefGoogle Scholar
[39] Müller, A., Behrens, J., Giraldo, F.X., and Wirth, V.. Comparison between adaptive and uniform discontinuous Galerkin simulations in dry 2d bubble experiments. Journal of Computational Physics, 235:371–393, 2013.Google Scholar
[40] Nguyen, N.C., Peraire, J., and Cockburn, B.. A hybridizable discontinuous Galerkin method for the incompressible navier-stokes equations. AIAA Paper 2010-362, 2010.CrossRefGoogle Scholar
[41] Noelle, S., Bispen, G., Arun, K.R., Lukáčová-Medvid’ová, M., and Munz, C.-D.. A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics. SIAM Journal on Scientific Computing, 36:B989–B1024, 2014.Google Scholar
[42] Pareschi, L. and Russo, G.. Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations. Recent Trends in Numerical Analysis, 3:269–289, 2000.Google Scholar
[43] Peraire, J., Nguyen, N. C., and Cockburn, Bernardo. A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations. AIAA Paper 10-363, 2010.Google Scholar
[44] Persson, P.-O.. High-order LES simulations using implicit-explicit Runge-Kutta schemes. AIAA Paper 11-684, 2011.CrossRefGoogle Scholar
[45] Di Pietro, D. and Ern, A.. Mathematical aspects of discontinuous Galerkin Methods, volume 69. Springer Science & Business Media, 2011.Google Scholar
[46] Restelli, M.. Semi-lagrangian and semi-implicit discontinuous Galerkin methods for atmospheric modeling applications. PhD thesis Politecnico di Milano, 2007.Google Scholar
[47] Russo, G. and Boscarino, S.. IMEX Runge-Kutta schemes for hyperbolic systems with diffusive relaxation. European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), 2012.Google Scholar
[48] Sang-Hyeon, L.. Cancellation problem of preconditioning method at low mach numbers. Journal of Computational Physics, 225(2):1199–1210, 2007.Google Scholar
[49] Schling, B.. The Boost C++ Libraries. XML Press, 2011.Google Scholar
[50] Schöberl, J.. Netgen - an advancing front 2d/3d-mesh generator based on abstract rules. Computing and Visualization in Science, 1:41–52, 1997.Google Scholar
[51] Schochet, S.. Fast singular limits of hyperbolic PDEs. Journal of Differential Equations, 114(2):476–512, 1994.Google Scholar
[52] Schütz, J. and Kaiser, K.. A new stable splitting for singularly perturbed ODEs. Applied Numerical Mathematics, 107:18–33, 2016.CrossRefGoogle Scholar
[53] Schütz, J. and Noelle, S.. Flux splitting for stiff equations: A notion on stability. Journal of Scientific Computing, 64(2):522–540, 2015.Google Scholar
[54] Schütz, J., Woopen, M., and May, G.. A hybridized DG/mixed scheme for nonlinear advection-diffusion systems, including the compressible Navier-Stokes equations. AIAA Paper 2012-0729, 2012.Google Scholar
[55] Sesterhenn, J., Müller, B., and Thomann, H.. On the cancellation problem in calculating compressible low mach number flows. Journal of Computational Physics, 151(2):597–615, 1999.Google Scholar
[56] Turkel, E.. Preconditioned methods for solving the incompressible and low speed compressible equations. Journal of Computational Physics, 72(2):277–298, 1987.Google Scholar
[57] Vos, P., Eskilsson, C., Bolis, A., Chun, S., Kirby, R. M., and Sherwin, S. J.. A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems. International Journal of Computational Fluid Dynamics, 25(3):107–125, 2011.Google Scholar
[58] Wang, H., Shu, C.-W., and Zhang, Q.. Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. SIAM Journal on Numerical Analysis, 53(1):206–227, 2015.Google Scholar
[59] Wesseling, P.. Principles of Computational Fluid Dynamics, volume 29 of Springer Series in Computational Mechanics. Springer Verlag, 2001.CrossRefGoogle Scholar
[60] Yelash, L., Müller, A., Lukáčová-Medvid’ová, M., Giraldo, F. X., and Wirth, V.. Adaptive discontinuous evolution Galerkin method for dry atmospheric flow. Journal of Computational Physics, 268:106–133, 2014.Google Scholar
[61] Yong, W-A.. A note on the zero Mach number limit of compressible Euler equations. Proceedings of the American Mathematical Society, 133(10):3079–3085, 2005.CrossRefGoogle Scholar
[62] Zakerzadeh, H.. Asymptotic analysis of the RS-IMEX scheme for the shallowwater equations in one space dimension. IGPM Preprint Nr. 455, 2016.Google Scholar
[63] Zakerzadeh, H. and Noelle, S.. A note on the stability of implicit-explicit flux splittings for stiff hyperbolic systems. IGPM Preprint Nr. 449, 2016.Google Scholar
[64] Zhang, H., Sandu, A., and Blaise, S.. Partitioned and implicit–explicit general linear methods for ordinary differential equations. Journal of Scientific Computing, 61(1):119–144, 2014.Google Scholar