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Entropy Stable Scheme on Two-Dimensional Unstructured Grids for Euler Equations

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  17 May 2016

Deep Ray ,
Praveen Chandrashekar ,
Ulrik S. Fjordholm  and
Siddhartha Mishra
Deep Ray*
Affiliation:
Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bengaluru, India
Praveen Chandrashekar*
Affiliation:
Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bengaluru, India
Ulrik S. Fjordholm*
Affiliation:
Department of Mathematical Sciences, NTNU, Trondheim 7491, Norway
Siddhartha Mishra*
Affiliation:
Seminar for Applied Mathematics, ETH Zurich and Center of Mathematics for Applications, University of Oslo, Norway
*
*Corresponding author. Email addresses:[email protected] (D. Ray), [email protected] (P. Chandrashekar), [email protected] (U. S. Fjordholm), [email protected] (S. Mishra)
*Corresponding author. Email addresses:[email protected] (D. Ray), [email protected] (P. Chandrashekar), [email protected] (U. S. Fjordholm), [email protected] (S. Mishra)
*Corresponding author. Email addresses:[email protected] (D. Ray), [email protected] (P. Chandrashekar), [email protected] (U. S. Fjordholm), [email protected] (S. Mishra)
*Corresponding author. Email addresses:[email protected] (D. Ray), [email protected] (P. Chandrashekar), [email protected] (U. S. Fjordholm), [email protected] (S. Mishra)
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Abstract

We propose an entropy stable high-resolution finite volume scheme to approximate systems of two-dimensional symmetrizable conservation laws on unstructured grids. In particular we consider Euler equations governing compressible flows. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. High resolution is achieved based on a linear reconstruction procedure satisfying a suitable sign property that helps to maintain entropy stability. The proposed scheme is demonstrated to robustly approximate complex flow features by a series of benchmark numerical experiments.

Keywords

Entropy stabilityEuler equationsunstructured gridssign propertyhigher-order accuracy

MSC classification

Secondary: 65M08: Finite volume methods 35L65: Conservation laws 35Q31: Euler equations
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1111 - 1140
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Kyle Anderson, W.. A grid generation and flow solution method for the euler equations on unstructured grids. J. Comput. Phys., 110(1):23–38, 1994.Google Scholar
[2]Angrand, F. and Lafon, F. C.. Flux formulation using a fully 2D approximate Roe Riemann solver. In Donato, A. and Oliveri, F., editors, Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects, volume 43 of Notes on Numerical Fluid Mechanics (NNFM), pages 15–22. Vieweg+Teubner Verlag, 1993.Google Scholar
[3]Arminjon, P., Dervieux, A., Fezoui, L., Steve, H., and Stoufflet, B.. Non-oscillatory schemes for multidimensional Euler calculations with unstructured grids. In Ballmann, J. and Jeltsch, R., editors, Nonlinear Hyperbolic Equations Theory, Computation Methods, and Applications, volume 24 of Notes on Numerical Fluid Mechanics, pages 1–10. Vieweg+Teubner Verlag, 1989.Google Scholar
[4]Barth, T. J.. Numerical methods for gasdynamic systems on unstructured meshes. In An Introduction to Recent Developments in Theory and Numerics for Conservation Laws (Freiburg/Littenweiler, 1997), volume 5 of Lect. Notes Comput. Sci. Eng., pages 195–285. Springer, Berlin, 1999.Google Scholar
[5]Barth, T. J. and Jespersen, D. C.. The Design and Application of Upwind Schemes on Unstructured Meshes. AIAA-89-0366, 1989.Google Scholar
[6]Billey, V., Périaux, J., Perrier, P., and Stoufflet, B.. 2-D and 3-D Euler computations with finite element methods in aerodynamics. In Nonlinear Hyperbolic Problems (St. Etienne, 1986), volume 1270 of Lecture Notes in Math., pages 64–81. Springer, Berlin, 1987.Google Scholar
[7]Blazek, J.. Computational Fluid Dynamics: Principles and Applications. Elsevier Science, Oxford, second edition, 2005.Google Scholar
[8]Chandrashekar, P.. Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations. Commun. Comput. Phys., 14(5):1252–1286, 2013.Google Scholar
[9]Cockburn, B. and Shu, C.-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp., 52(186):411–435, 1989.Google Scholar
[10]Colella, P.. A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Statist. Comput., 6(1):104–117, 1985.Google Scholar
[11]Dafermos, C. M.. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 2010.Google Scholar
[12]Deconinck, H., Roe, P. L., and Struijs, R.. A multidimensional generalization of Roe's flux difference splitter for the Euler equations. Comput. & Fluids, 22(2-3):215–222, 1993.Google Scholar
[13]Désidéri, J.-A. and Dervieux, A.. Compressible flow solvers using unstructured grids. In Computational Fluid Dynamics, Vol. 1, 2, volume 88 of von Karman Inst. Fluid Dynam. Lecture Ser., page 115. von Karman Inst. Fluid Dynamics, Rhode-St-Genèse, 1988.Google Scholar
[14]Durlofsky, L. J., Engquist, B., and Osher, S.. Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comput. Phys., 98(1):64–73, 1992.Google Scholar
[15]Fjordholm, U. S., Mishra, S., and Tadmor, E.. Energy preserving and energy stable schemes for the shallow water equations. In Cucker, F., Pinkus, A. and Todd, M., editors, Proc. FoCM (Hong Kong, 2008), London Math. Soc. Lecture Notes Ser. 363, pages 93–139, 2009.Google Scholar
[16]Fjordholm, U. S., Mishra, S., and Tadmor, E.. Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal., 50(2):544–573, 2012.Google Scholar
[17]Fjordholm, U. S.. Mishra, S., and Tadmor, E.. ENO reconstruction and ENO interpolation are stable. Foundations of Computational Mathematics 13(2):139–159, 2013.Google Scholar
[18]Fjordholm, U. S., Käppeli, R., Mishra, S., and Tadmor, E.. Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws. Foundations of Computational Mathematics, pages 1–65, 2015.Google Scholar
[19]Godlewski, E. and Raviart, P.-A.. Numerical Approximation of Hyperbolic Systems of Conservation Laws, volume 118 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.Google Scholar
[20]Godunov, S. K.. An interesting class of quasilinear systems. Dokl. Akad. Nauk. SSSR, 139:521–523, 1961.Google Scholar
[21]Gottlieb, S., Shu, C.-W., and Tadmor, E.. Strong stability-preserving high-order time discretization methods. SIAM Rev., 43(1):89–112 (electronic), 2001.Google Scholar
[22]Harten, A.. On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys., 49(1):151–164, 1983.Google Scholar
[23]Hughes, T. J. R., Franca, L. P., and Mallet, M.. 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(2):223–234, 1986.Google Scholar
[24]Ismail, F. and Roe, P. L.. Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. J. Comput. Phys., 228(15):5410–5436, 2009.Google Scholar
[25]Jameson, A., Schmidt, W., and Turkel, E.. Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes. In AIAA Paper 1981-1259, page 1, 1981.Google Scholar
[26]Jameson, A. and Mavriplis, D.. Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh. AIAA Journal, 24:611–618, 1986.Google Scholar
[27]Jameson, A.. Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes. J. Sci. Comput., 34(2):188–208, 2008.Google Scholar
[28]Katz, A. and Sankaran, V.. An efficient correction method to obtain a formally third-order accurate flow solver for node-centered unstructured grids. J. Sci. Comput., 51(2):375–393, 2012.Google Scholar
[29]Kröner, D., Noelle, S., and Rokyta, M.. Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numerische Mathematik, 71(4):527–560, 1995.Google Scholar
[30]Kröner, D. and Rokyta, M.. Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal., 31(2):324–343, 1994.Google Scholar
[31]Lefloch, P. G., Mercier, J. M., and Rohde, C.. Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal., 40(5):1968–1992 (electronic), 2002.Google Scholar
[32]Madrane, A., Fjordholm, U. S., Mishra, S., and Tadmor, E.. Entropy conservative and entropy stable finite volume schemes for multi-dimensional conservation laws on unstructured meshes. Technical Report 2012-31, Seminar for Applied Mathematics, ETH Zürich, 2012.Google Scholar
[33]Mavriplis, D. J.. Adaptive mesh generation for viscous flows using delaunay triangulation. J. Comput. Phys., 90(2):271–291, 1990.Google Scholar
[34]Mock, M. S.. Systems of conservation laws of mixed type. J. Differential Equations, 37(1):70–88, 1980.Google Scholar
[35]Perthame, B. and Qiu, Y.. A variant of van Leer's method for multidimensional systems of conservation laws. J. Comput. Phys., 112(2):370–381, June 1994.Google Scholar
[36]Robinet, J.-Ch., Gressier, J., Casalis, G., and Moschetta, J.-M.. Shock wave instability and the carbuncle phenomenon: Same intrinsic origin? J. Fluid Mech., 417:237–263, 2000.Google Scholar
[37]Roe, P.L., Nishikawa, H., Ismail, F., and Scalabrin, L.. On carbuncles and other excrescences. In 17th AIAA Computational Fluid Dynamics Conference (Toronto), AIAA Paper 2005-4872, 2005.Google Scholar
[38]Roe, P. L.. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43(2):357–372, 1981.Google Scholar
[39]Rostand, P. and Stoufflet, B.. TVD schemes to compute compressible viscous flows on unstructured meshes. In Nonlinear Hyperbolic Equations — Theory, Computation Methods, and Applications (Aachen, 1988), volume 24 of Notes Numer. Fluid Mech., pages 510–520. Vieweg, Braunschweig, 1989.Google Scholar
[40]Shu, C.-W. and Osher, S.. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77(2):439–471, 1988.Google Scholar
[41]Sod, G. A.. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys., 27(1):1–31, 1978.Google Scholar
[42]Stoufflet, B.. Implicit finite element methods for the Euler equations. In Numerical Methods for the Euler Equations of Fluid Dynamics (Rocquencourt, 1983), pages 409–434. SIAM, Philadelphia, PA, 1985.Google Scholar
[43]Tadmor, E.. Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp., 43(168):369–381, 1984.Google Scholar
[44]Tadmor, E.. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer., 12:451–512, 2003.Google Scholar
[45]van Albada, G. D., van Leer, B., and Roberts, W. W. Jr.. A comparative study of computational methods in cosmic gas dynamics. In Hussaini, M. Y., van Leer, B., and Van Rosendale, J., editors, Upwind and High-Resolution Schemes, pages 95–103. Springer Berlin Heidelberg, 1997.Google Scholar
[46]van Leer, B.. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method [J. Comput. Phys., 32(1):101-136,1979]. J. Comput. Phys., 135(2):227–248, 1997. With an introduction by Ch. Hirsch, Commemoration of the 30th anniversary of J. Comput. Phys..Google Scholar
[47]Venkatakrishnan, V.. On the Accuracy of Limiters and Convergence to Steady State Solutions. AIAA-93-0880, 1993.Google Scholar
[48]Venkatakrishnan, V.. Convergence to steady state solutions of the euler equations on unstructured grids with limiters. J. Comput. Phys., 118(1):120–130, 1995.Google Scholar
[49]Vijayasundaram, G.. Transonic flow simulations using an upstream centered scheme of Godunov in finite elements. J. Comput. Phys., 63(2):416–433, 1986.Google Scholar
[50]Whitaker, D. L., Grossman, B., and Löhner, R.. Two-Dimensional Euler Computations on a Triangular Mesh Using an Upwind, Finite-Volume Scheme. AIAA-89-0365, 1989.Google Scholar
[51]Woodward, P. and Colella, P.. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54(1):115–173, 1984.Google Scholar