Spacetime discontinuous Galerkin methods for solving convection-diffusion systems

May, Sandra

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1755-1781.

In this paper, we present two new approaches for solving systems of hyperbolic conservation laws with correct physical viscosity and heat conduction terms such as the compressible Navier–Stokes equations. Our methods are extensions of the spacetime discontinuous Galerkin method for hyperbolic conservation laws developed by Hiltebrand and Mishra [26]. Following this work, we use entropy variables as degrees of freedom and entropy stable fluxes. For the discretization of the diffusion term, we consider two different approaches: the interior penalty approach, resulting in the ST-SIPG and the ST-NIPG method, and a variant of the local discontinuous Galerkin method, resulting in the ST-LDG method. We show entropy stability of the ST-NIPG and the ST-LDG method when applied to the compressible Navier–Stokes equations. For the ST-SIPG method, this result holds under an assumption on the computed solution. All schemes incorporate shock capturing terms. Therefore, the schemes can handle both regimes of underresolved and fully resolved physical diffusion. We present a numerical comparison of the three methods in one space dimension.

Classification : 65M12, 65M60
Mots clés : Discontinuous Galerkin method, entropy stability, convection-diffusion systems, compressible Navier–Stokes equations

D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. | DOI | MR | Zbl

D.N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. | DOI | MR | Zbl

T.J. Barth, Numerical methods for gasdynamic systems on unstructured meshes. In An introduction to recent developments in theory and numerics of conservation laws. Edited by D. Kröner, M. Ohlberger and C. Rohde. Vol. 5 of Lect. Notes Comput. Sci. Eng. Springer (1999) 195–285. | MR | Zbl

F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131 (1997) 267–279. | DOI | MR | Zbl

F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti and M. Savini, A high-order accurate discontinuous finite element method for inviscid turbomachinery flows. In Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics. Antwerp, Belgium (1997) 99–108.

C.E. Baumann and J.T. Oden, A discontinuous $hp$ finite element method for the Euler and Navier–Stokes equations. Int. J. Numer. Methods Fluids 31 (1999) 79–95. | DOI | MR | Zbl

S. Brdar, A. Dedner and R. Klöfkorn, Compact and stable discontinuous Galerkin methods for convection-diffusion problems. SIAM J. Sci. Comput. 34 (2012) A263–A282. | DOI | MR | Zbl

J. Česenek, M. Feistauer and A. Kosík, DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM Z. Angew. Math. Mech. 93 (2013) 387–402. | DOI | MR | Zbl

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. | DOI | MR | Zbl

V. Dolejíš, On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations. Int. J. Numer. Methods Fluids 45 (2004) 1083–1106. | DOI | MR | Zbl

V. Dolejíš, Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows. Commun. Comput. Phys. 4 (2008) 231–274. | MR

V. Dolejíš and M. Feistauer, Discontinuous Galerkin Methods. Springer (2015). | MR

M. Feistauer, V. Dolejíš and V. Kučera, On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers. Comput. Visual. Sci. 10 (2007) 17–27. | DOI | MR

U.S. Fjordholm, S. Mishra and E. Tadmor, Energy preserving and energy stable schemes for the shallow water equations. In Foundations of Computational Mathematics. Math. Soc. Lect. Notes Ser. 363 (2009) 93–139 | MR | Zbl

U.S. Fjordholm, S. Mishra and E. Tadmor, Arbitrarily high order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50 (2012) 544–573. | DOI | MR | Zbl

K.O. Friedrichs and P.D. Lax, Systems of conservation laws with a convex extension. Proc. Nat. Acad. Sci. U.S.A. 68 (1971) 1686–1688. | DOI | MR | Zbl

G. Gassner, F. Lörcher and C.-D. Munz, A discontinuous Galerkin scheme based on a space-time expansion II. Viscous flow equations in multi dimensions. J. Sci. Comput. 34 (2008) 260–286. | DOI | MR | Zbl

S.K. Godunov, An interesting class of quasilinear systems. Dokl. Acad. Nauk. SSSR 139 (1961) 521–523. | MR | Zbl

A. Harten, On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49 (1983) 151–164. | DOI | MR | Zbl

A. Harten and P.D. Lax. A random choice finite difference scheme for hyperbolic conservation laws. SIAM J. Numer. Anal. 18 (1981) 289–315. | DOI | MR | Zbl

R. Hartmann, Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids 51 (2006) 1131–1156. | DOI | MR | Zbl

R. Hartmann and P. Houston, Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: Method formulation. Int. J. Numer. Anal. Model 3 (2006) 1–20. | MR | Zbl

R. Hartmann and P. Houston, An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier–Stokes equations. J. Comput. Phys. 227 (2008) 9670–9685. | DOI | MR | Zbl

K. Hillewaert, Development of the discontinuous Galerkin method for high-resolution, large scale CFD and acoustics in industrial geometries. Ph.D. thesis, Université catholique de Louvain, École Polytechnique de Louvain (2013).

A. Hiltebrand, Entropy-stable discontinuous Galerkin finite element methods with streamline diffusion and shock-capturing for hyperbolic systems of conservation laws. Ph.D. thesis, Seminar for Applied Mathematics, ETH Zurich (2014).

A. Hiltebrand and S. Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws. Numer. Math. 126 (2014) 103–151. | DOI | MR | Zbl

A. Hiltebrand and S. Mishra, Efficient preconditioners for a shock capturing space-time discontinuous Galerkin method for systems of conservation laws. Commun. Comput. Phys. 17 (2015) 1360–1387. | DOI | MR | Zbl

T.J.R. Hughes, L.P. Franca and M. Mallet, A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier–Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Engrg. 54 (1986) 223–234. | DOI | MR | Zbl

F. Ismail and P.L. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. J. Comput. Phys. 228 (2009) 5410–5436. | DOI | MR | Zbl

C.M. Klaij, J.J.W. Van Der Vegt and H. Van Der Ven, Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations. J. Comput. Phys. 217 (2006) 589–611. | DOI | MR | Zbl

P.G. LeFloch, Hyperbolic Systems of Conservation Laws. Lectures in Mathematics. ETH Zürich. Birkhäuser (2002). | MR | Zbl

M.S. Mock. Systems of conservations laws of mixed type. J. Differ. Equ. 37 (1980) 70–88. | DOI | MR | Zbl

J.A. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. | DOI | MR | Zbl

J.T. Oden, I. Babuška and C.E. Baumann, A discontinuous $hp$ finite element method for diffusion problems. J. Comput. Phys. 146 (1998) 491–519. | DOI | MR | Zbl

P.-O. Persson and J. Peraire, Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier–Stokes equations. SIAM J. Sci. Comput. 30 (2008) 2709–2733. | DOI | MR | Zbl

B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM (2008). | MR | Zbl

F. Shakib, T.J.R. Hughes and Z. Johan, A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg. 89 (1991) 141–219. | DOI | MR

E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. Math. Comput. 49 (1987) 91–103. | DOI | MR | Zbl

E. Tadmor and W. Zhong, Entropy stable approximations of Navier–Stokes equations with no artificial numerical viscosity. J. Hyperbolic Differ. Equ. 3 (2006) 529–559. | DOI | MR | Zbl

T. Warburton and J.S. Hesthaven, On the constants in $hp$-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Engrg. 192 (2003) 2765–2773. | DOI | MR | Zbl