Skip to main content Accessibility help

A High-Order Method for Weakly Compressible Flows


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.


Corresponding author

*Corresponding author. Email addresses: (K. Kaiser), (J. Schütz)


Hide All

Communicated by Chi-Wang Shu



Hide All
[1] BOOST C++ Libraries.
[2] Anderson, J. D.. Fundamentals of Aerodynamics. McGraw-Hill New York, 3 rd edition, 2001.
[3] Ascher, U. M., Ruuth, S., and Spiteri, R.. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151167, 1997.
[4] Ascher, U.M., Ruuth, S., and Wetton, B.. Implicit-Explicit methods for time-dependent partial differential equations. SIAM Journal on Numerical Analysis, 32:797823, 1995.
[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.
[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.
[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 163202. Birkhäuser Press Boston, 1997.
[8] Bispen, G.. IMEX Finite Volume Methods for the Shallow Water Equations. PhD thesis, Johannes Gutenberg-Universität, 2015.
[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, 16:307347, 2014.
[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, 335:222248, 2017.
[11] Boscarino, S.. Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems. SIAM Journal on Numerical Analysis, 45:16001621, 2007.
[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):A22A51, 2013.
[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, 54:545581, 1990.
[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, 84:90113, 1989.
[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, 52:411435, 1988.
[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, 25:337361, 1991.
[17] Cockburn, B. and Shu, C.-W.. The Runge-Kutta discontinuous Galerkin Method for conservation laws V: Multidimensional Systems. Mathematics of Computation, 141:199224, 1998.
[18] Cockburn, B. and Shu, C. W.. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16:173261, 2001.
[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, 231:56855704, 2012.
[20] Degond, P. and Tang, M.. All speed scheme for the low Mach number limit of the isentropic Euler equation. Communications in Computational Physics, 10:131, 2011.
[21] Dimarco, G. and Pareschi, L.. Asymptotic preserving implicit-explicit Runge–Kutta methods for nonlinear kinetic equations. SIAM Journal on Numerical Analysis, 51(2):10641087, 2013.
[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):76257648, 2010.
[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):10771102, 2010.
[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):33943425, 2010.
[25] Guillard, H. and Viozat, C.. On the behavior of upwind schemes in the low Mach number limit. Computers and Fluids, 28(1):6386, 1999.
[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, 12:955980, 2012.
[27] Hundsdorfer, W. and Jaffré, J.. Implicit–explicit time stepping with spatial discontinuous finite elements. Applied Numerical Mathematics, 45(2):231254, 2003.
[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):20162042, 2007.
[29] Jaust, A., Schütz, J., and Woopen, M.. A hybridized discontinuous Galerkin method for unsteady flows with shock-capturing. AIAA Paper 2014-2781, 2014.
[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 267274. Springer International Publishing, 2015.
[31] Jin, S.. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review. Rivista di Matematica della Universita Parma, 3:177216, 2012.
[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.
[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):17531781, 2007.
[34] Kennedy, C. A. and Carpenter, M. H.. Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Applied Numerical Mathematics, 44:139181, 2003.
[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, 34:481524, 1981.
[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, 121:213237, 1995.
[37] Kröner, D.. Numerical Schemes for Conservation Laws. Wiley Teubner, 1997.
[38] Liu, H. and Zou, J.. Some new additive Runge–Kutta methods and their applications. Journal of Computational and Applied Mathematics, 190(1-2):7498, 2006.
[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, 235:371393, 2013.
[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.
[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:B989B1024, 2014.
[42] Pareschi, L. and Russo, G.. Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations. Recent Trends in Numerical Analysis, 3:269289, 2000.
[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.
[44] Persson, P.-O.. High-order LES simulations using implicit-explicit Runge-Kutta schemes. AIAA Paper 11-684, 2011.
[45] Di Pietro, D. and Ern, A.. Mathematical aspects of discontinuous Galerkin Methods, volume 69. Springer Science & Business Media, 2011.
[46] Restelli, M.. Semi-lagrangian and semi-implicit discontinuous Galerkin methods for atmospheric modeling applications. PhD thesis Politecnico di Milano, 2007.
[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.
[48] Sang-Hyeon, L.. Cancellation problem of preconditioning method at low mach numbers. Journal of Computational Physics, 225(2):11991210, 2007.
[49] Schling, B.. The Boost C++ Libraries. XML Press, 2011.
[50] Schöberl, J.. Netgen - an advancing front 2d/3d-mesh generator based on abstract rules. Computing and Visualization in Science, 1:4152, 1997.
[51] Schochet, S.. Fast singular limits of hyperbolic PDEs. Journal of Differential Equations, 114(2):476512, 1994.
[52] Schütz, J. and Kaiser, K.. A new stable splitting for singularly perturbed ODEs. Applied Numerical Mathematics, 107:1833, 2016.
[53] Schütz, J. and Noelle, S.. Flux splitting for stiff equations: A notion on stability. Journal of Scientific Computing, 64(2):522540, 2015.
[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.
[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):597615, 1999.
[56] Turkel, E.. Preconditioned methods for solving the incompressible and low speed compressible equations. Journal of Computational Physics, 72(2):277298, 1987.
[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):107125, 2011.
[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):206227, 2015.
[59] Wesseling, P.. Principles of Computational Fluid Dynamics, volume 29 of Springer Series in Computational Mechanics. Springer Verlag, 2001.
[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, 268:106133, 2014.
[61] Yong, W-A.. A note on the zero Mach number limit of compressible Euler equations. Proceedings of the American Mathematical Society, 133(10):30793085, 2005.
[62] Zakerzadeh, H.. Asymptotic analysis of the RS-IMEX scheme for the shallowwater equations in one space dimension. IGPM Preprint Nr. 455, 2016.
[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.
[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):119144, 2014.


MSC classification


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