# IMEX finite volume evolution Galerkin scheme for three-dimensional weakly compressible flows

## Main Article Content

## Abstract

In this paper we will derive an implicit-explicit (IMEX) finite volume evolution Galerkin scheme for three-dimensional Euler equations. We will in particular concentrate a singular limit of weakly compressible flows when the Mach number is about ${\cal O}(10^{-2}) - {\cal O}(10^{-6}).$ In order to efficiently resolve slow dynamics we split the whole nonlinear system in a stiff linear part governing the acoustic and gravitational waves and a non-stiff nonlinear part that models nonlinear advection effects. We use stiffly accurate second order IMEX scheme for time discretization to approximate stiff linear operator implicitly and the non-stiff nonlinear operator explicitly. Furthermore in order to take multdimensional effects of flow propagation into account we apply three-dimensional evolution Galerkin operator.

## Article Details

How to Cite

BISPEN, Georgij; LUKÁČOVÁ-MEDVIĎOVÁ, Mária; YELASH, Leonid.
IMEX finite volume evolution Galerkin scheme for three-dimensional weakly compressible flows.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 62-73, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/395>. Date accessed: 22 sep. 2017.
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## References

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[2] G. Bispen, IMEX finite volume schemes for the shallow water equations, PhD-thesis, 2015.

[3] G. Bispen, K. R. Arun, M. Lukáčová-Medviďováand S. Noelle, IMEX large time step finite volume methods for low Froude number shallow water flows, Comm. Comput. Phys., 16 (2014), pp. 307–347.

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[7] M. Feistauer, V. Dolejší, and V. Kučera, On the Discontinuous Galerkin Method for the Simulation of Compressible Flow with Wide Range of Mach Numbers, Computing and Visualization in Science, 10 (2007), pp. 17–27.

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[9] F.X. Giraldo, and M. Restelli, A Study of Spectral Element and Discontinuous Galerkin Methods for the Navier-Stokes Equations in Nonhydrostatic Mesoscale Atmospheric Modeling: Equation Sets and Test Cases, J. Comput. Phys., 227 (2008), pp. 3849–3877.

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[11] D.R. van der Heul, C. Viuk, and P. Wesseling, A conservative pressure-correction method for flow at all speeds Computers & Fluids, 32(8) (2003), pp. 1113–1132.

[12] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34(4) (1981), pp. 481–524.

[13] R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I. One-dimensional flow, J. Comput. Phys., 121(2) (1995), pp. 213–237.

[14] M. Lukáčová-Medviďová, and K.W. Morton, Finite volume evolution Galerkin methodsa survey, Indian J. Pure Appl. Math., 41 (2010), pp. 329–361.

[15] M. Lukáčová-Medviďová, K.W. Morton, and G. Warnecke, Finite volume evolution Galerkin methods for hyperbolic systems, J. Sci. Comput., 26(1) (2004), pp. 1–30.

[16] M. Lukáčová-Medviďová, A. Mu ̈ller, V. Wirth, and L. Yelash, Adaptive discontinuous evolution Galerkin method for dry atmospheric flow. J. Comput. Phys., 268 (2014), pp. 106–133.

[17] C.-D. Munz, S. Roller, R. Klein, K.J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime Computers & Fluids, 32(2) (2003), pp. 173–196.

[18] A. Muller, J. Behrens, F.X., Giraldo, and V. Wirth, Comparison between adaptive and uniform discontinuous Galerkin simulations in dry 2D bubble experiments, J. Comput. Phys., 235 (2013), pp. 371–393.

[19] S. Noelle, G. Bispen, K.R. Arun, M. Lukáčová-Medviďová, C.-D. Munz, A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics, SIAM J. Sci. Comput., 36(6) (2014), pp. 989–1024.

[20] J. H. Park and C.-D. Munz, Multiple pressure variables methods for fluid flow at all Mach numbers, Internat. J. Numer. Methods Fluids, 49(8) (2005), pp. 905–931.

[21] Y. Sun, and Y.-X. Ren, The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws, J. Comput. Phys., 228(13) (2009), pp. 4945–4960.

[2] G. Bispen, IMEX finite volume schemes for the shallow water equations, PhD-thesis, 2015.

[3] G. Bispen, K. R. Arun, M. Lukáčová-Medviďováand S. Noelle, IMEX large time step finite volume methods for low Froude number shallow water flows, Comm. Comput. Phys., 16 (2014), pp. 307–347.

[4] G. Bispen and M. Lukáčová-Medviďová, Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations, in preparation.

[5] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys., 10(1) (2011), pp. 1–31.

[6] E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system, Commun. Math. Phys., 321 (2013), pp. 605–628.

[7] M. Feistauer, V. Dolejší, and V. Kučera, On the Discontinuous Galerkin Method for the Simulation of Compressible Flow with Wide Range of Mach Numbers, Computing and Visualization in Science, 10 (2007), pp. 17–27.

[8] M. Feistauer, V. Kučera, On a robust discontinuous Galerkin technique for the solution of compressible flow, J. Comput. Phys., 224 (2007), pp. 208–221.

[9] F.X. Giraldo, and M. Restelli, A Study of Spectral Element and Discontinuous Galerkin Methods for the Navier-Stokes Equations in Nonhydrostatic Mesoscale Atmospheric Modeling: Equation Sets and Test Cases, J. Comput. Phys., 227 (2008), pp. 3849–3877.

[10] J. Haack, S. Jin, and J.-G. Liu, An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations, Commun. Comput. Phys., 12 (2012), pp. 955–980.

[11] D.R. van der Heul, C. Viuk, and P. Wesseling, A conservative pressure-correction method for flow at all speeds Computers & Fluids, 32(8) (2003), pp. 1113–1132.

[12] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34(4) (1981), pp. 481–524.

[13] R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I. One-dimensional flow, J. Comput. Phys., 121(2) (1995), pp. 213–237.

[14] M. Lukáčová-Medviďová, and K.W. Morton, Finite volume evolution Galerkin methodsa survey, Indian J. Pure Appl. Math., 41 (2010), pp. 329–361.

[15] M. Lukáčová-Medviďová, K.W. Morton, and G. Warnecke, Finite volume evolution Galerkin methods for hyperbolic systems, J. Sci. Comput., 26(1) (2004), pp. 1–30.

[16] M. Lukáčová-Medviďová, A. Mu ̈ller, V. Wirth, and L. Yelash, Adaptive discontinuous evolution Galerkin method for dry atmospheric flow. J. Comput. Phys., 268 (2014), pp. 106–133.

[17] C.-D. Munz, S. Roller, R. Klein, K.J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime Computers & Fluids, 32(2) (2003), pp. 173–196.

[18] A. Muller, J. Behrens, F.X., Giraldo, and V. Wirth, Comparison between adaptive and uniform discontinuous Galerkin simulations in dry 2D bubble experiments, J. Comput. Phys., 235 (2013), pp. 371–393.

[19] S. Noelle, G. Bispen, K.R. Arun, M. Lukáčová-Medviďová, C.-D. Munz, A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics, SIAM J. Sci. Comput., 36(6) (2014), pp. 989–1024.

[20] J. H. Park and C.-D. Munz, Multiple pressure variables methods for fluid flow at all Mach numbers, Internat. J. Numer. Methods Fluids, 49(8) (2005), pp. 905–931.

[21] Y. Sun, and Y.-X. Ren, The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws, J. Comput. Phys., 228(13) (2009), pp. 4945–4960.