Startseite Entropy stable non-oscillatory fluxes: An optimized wedding of entropy conservative flux with non-oscillatory flux
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Entropy stable non-oscillatory fluxes: An optimized wedding of entropy conservative flux with non-oscillatory flux

  • Ritesh K. Dubey EMAIL logo
Veröffentlicht/Copyright: 16. August 2023

Abstract

This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux (F*) and a non-oscillatory flux (Fs). This novel approach paves a way to construct non-oscillatory entropy stable flux (Fˆ) as a simple combination of (F* and Fs) which inherently optimize the numerical diffusion in the entropy stable flux (Fˆ) such that it reduces to the underlying non-oscillatory flux (Fs) in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair (F*, Fs), (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that fully discrete schemes using these entropy stable fluxes do not exhibit nonphysical spurious oscillations in approximating the discontinuities and its non-oscillatory nature is comparable to the non-oscillatory schemes using underlying fluxes (Fs) only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair.

JEL Classification: 65M06; 35L65

Funding statement: Author acknowledges SERB India for fund through project EMR/2016/000394 to support authors research visit to Blockapps AI, Bangalore where initial work for this paper got carried out. Author also acknowledges SERB India for project CRG/2022/002659 which enabled author to complete this manuscript and further developments.

References

[1] M. Arora and P. L. Roe, A well-behaved TVD limiter for high-resolution calculations of unsteady flow. Journal of Computational Physics 132 (1997), 3–11.10.1007/s10444-017-9576-2Suche in Google Scholar

[2] B. Biswas and R. K. Dubey, Low dissipative entropy stable schemes using third order WENO and TVD reconstructions. Advances in Computational Mathematics 44 (2018), 1153–1181.10.1016/j.camwa.2020.10.005Suche in Google Scholar

[3] B. Biswas and R. K. Dubey, ENO and WENO schemes using arc-length based smoothness measurement. Computers & Mathematics with Applications 80 (2020), No. 12, 2780–2795.10.1090/S0025-5718-1985-0771028-7Suche in Google Scholar

[4] R. Borges, M. Carmona, B. Costa, and W. S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics 227 (2008), No. 6, 3191–3211.10.1016/j.jcp.2007.11.038Suche in Google Scholar

[5] M. Castro, B. Costa, and W. S. Don, High order weighted essentially non-oscillatory WENO-z schemes for hyperbolic conservation laws. Journal of Computational Physics 230 (2011), No. 5, 1766–1792.10.1016/j.jcp.2010.11.028Suche in Google Scholar

[6] P. Chandrashekar, Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier–Stokes equations. Communications in Computational Physics 14, (2013), No. 5, 1252–1286.10.4208/cicp.170712.010313aSuche in Google Scholar

[7] R. Chen and D.-K. Mao, Entropy-TVD scheme for nonlinear scalar conservation laws. Journal of Scientific Computing 47 (2011), No. 2, 150–169.10.1007/s10915-010-9431-9Suche in Google Scholar

[8] R. Chen, M. Zou, and L. Xiao, Entropy-TVD scheme for the shallow water equations in one dimension. Journal of Scientific Computing 71 (2017), No. 2, 822–838.10.1007/s10915-016-0322-6Suche in Google Scholar

[9] X. Cheng, A fourth order entropy stable scheme for hyperbolic conservation laws. Entropy 21 (2019), No. 5, 508.10.3390/e21050508Suche in Google Scholar PubMed PubMed Central

[10] X. Cheng and Y. Nie, A third-order entropy stable scheme for hyperbolic conservation laws. Journal of Hyperbolic Differential Equations 13 (2016), No. 1, 129–145.10.1142/S021989161650003XSuche in Google Scholar

[11] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Mathematics of Computation 34 (1980), 1–21.10.2307/2006218Suche in Google Scholar

[12] J. Duan and H. Tang, High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics. Journal of Computational Physics 431 (2021), 110136.10.1016/j.jcp.2021.110136Suche in Google Scholar

[13] R. K. Dubey and B. Biswas, Suitable diffusion for constructing non-oscillatory entropy stable schemes. Journal of Computational Physics 372 (2018), 912–930.10.1016/j.jcp.2018.04.037Suche in Google Scholar

[14] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 1998.Suche in Google Scholar

[15] P. Fan, Y. Shen, B. Tian, and C. Yang, A new smoothness indicator for improving the weighted essentially non-oscillatory scheme. Journal of Computational Physics 269 (2014), 329–354.10.1016/j.jcp.2014.03.032Suche in Google Scholar

[16] T. C. Fisher and M. H. Carpenter, High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics 252 (2013), 518–557.10.1016/j.jcp.2013.06.014Suche in Google Scholar

[17] U. S. Fjordholm, S. Mishra, and E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM Journal on Numerical Analysis 50 (2012), iNo. 2, 544–573.10.1137/110836961Suche in Google Scholar

[18] U. S. Fjordholm, S. Mishra, and E. Tadmor, Eno reconstruction and eno interpolation are stable. Foundations of Computational Mathematics 13, (2013), No. 2, 139–159.10.1007/s10208-012-9117-9Suche in Google Scholar

[19] K. O. Friedrichs, Symmetric hyperbolic linear differential equations. Communications on Pure and Applied Mathematics 7, (1954), No. 2, 345–392.10.1002/cpa.3160070206Suche in Google Scholar

[20] J. B. Goodman and R. J. LeVeque, A geometric approach to high resolution TVD schemes. SIAM Journal on Numerical Analysis 25, (1988), No. 2, 268–284.10.1137/0725019Suche in Google Scholar

[21] S. Gottlieb and C.-W. Shu, Total variation diminishing Runge–Kutta schemes. Math. Comput. 67 (1998), 221, 73–85.10.1090/S0025-5718-98-00913-2Suche in Google Scholar

[22] Y. Ha, C. H. Kim, Y. J. Lee, and J. Yoon, An improved weighted essentially non-oscillatory scheme with a new smoothness indicator. Journal of Computational Physics 232, (2013), No. 1, 68–86.10.1016/j.jcp.2012.06.016Suche in Google Scholar

[23] A. Harten, High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics 49 (1983), No. 3, 357–393.10.1016/0021-9991(83)90136-5Suche in Google Scholar

[24] A. Harten and P. D. Lax, On a class of high resolution total-variation-stable finite-difference schemes. SIAM Journal on Numerical Analysis 21 (1984), No. 1, 1–23.10.1137/0721001Suche in Google Scholar

[25] A. Harten and S. Osher, Uniformly high-order accurate nonoscillatory schemes, I. SIAM Journal on Numerical Analysis 24 (1987), No. 2, 279–309.10.1137/0724022Suche in Google Scholar

[26] A. K. Henrick, T. D. Aslam, and J. M. Powers, Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. Journal of Computational Physics 207 (2005), No. 2, 542–567.10.1016/j.jcp.2005.01.023Suche in Google Scholar

[27] W. Hui, P. Li, and Z. Li, A unified coordinate system for solving the two-dimensional Euler equations. Journal of Computational Physics 153 (1999), No. 2, 596–637.10.1006/jcph.1999.6295Suche in Google Scholar

[28] F. Ismail and P. L. Roe, Affordable, entropy-consistent euler flux functions, II: Entropy production at shocks. Journal of Computational Physics 228 (2009), No. 15, 5410–5436.10.1016/j.jcp.2009.04.021Suche in Google Scholar

[29] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes. Journal of Computational Physics 126 (1996), No. 1, 202–228.10.1006/jcph.1996.0130Suche in Google Scholar

[30] C. H. Kim, Y. Ha, and J. Yoon, Modified non-linear weights for fifth-order weighted essentially non-oscillatory schemes. Journal of Scientific Computing 67 (2016), No. 1, 299–323.10.1007/s10915-015-0079-3Suche in Google Scholar

[31] R. Kumar and M. Kadalbajoo, Efficient high-resolution relaxation schemes for hyperbolic systems of conservation laws. Int. Journal for Numerical Methods in Fluids 55 (2007), No. 5, 483–507.10.1002/fld.1479Suche in Google Scholar

[32] A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numerical Methods for Partial Differential Equations 18, (2002), No. 5, 584–608.10.1002/num.10025Suche in Google Scholar

[33] C. B. Laney, Computational Gas Dynamics, Cambridge University Press, 1998.10.1002/cpa.3160070112Suche in Google Scholar

[34] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation. Communications on Pure and Applied Mathematics 7 (1954), No. 1, 159–193.10.1137/S003614290240069XSuche in Google Scholar

[35] P. G. LeFloch, J. M. Mercier, and C. Rohde, Fully discrete, entropy conservative schemes of arbitrary order. SIAM Journal on Numerical Analysis 40 (2002), No. 5, 1968–1992.10.1017/CBO9780511791253Suche in Google Scholar

[36] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Vol. 31, Cambridge University Press, 2002.10.1007/978-3-642-05146-3Suche in Google Scholar

[37] R. Liska and B.Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM Journal on Scientific Computing 25 (2003), No. 3, 995–1017.10.1137/S1064827502402120Suche in Google Scholar

[38] Q. Liu, Y. Liu, and J. Feng, The scaled entropy variables reconstruction for entropy stable schemes with application to shallow water equations. Computers & Fluids 192 (2019), 104266.10.1016/j.compfluid.2019.104266Suche in Google Scholar

[39] X.-D. Liu, S. Osher, T. Chan, et al., Weighted essentially non-oscillatory schemes. Journal of Computational Physics 115 (1994), No. 1, 200–212.10.1006/jcph.1994.1187Suche in Google Scholar

[40] J. R. Magnus, On the concept of matrix derivative. Journal of Multivariate Analysis 101 (2010), No. 9, 2200–2206.10.1016/j.jmva.2010.05.005Suche in Google Scholar

[41] S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition. SIAM Journal on Numerical Analysis 21 (1984), No. 5, 955–984.10.1137/0721060Suche in Google Scholar

[42] S. Osher and S. Chakravarthy, Very high order accurate TVD schemes. In: Oscillation Theory, Computation, and Methods of Compensated Compactness. Springer, 1986, pp. 229–274.10.1007/978-1-4613-8689-6_9Suche in Google Scholar

[43] S. Parvin and R. K. Dubey, A new framework to construct third-order weighted essentially nonoscillatory weights using weight limiter functions. Int. Journal for Numerical Methods in Fluids 93 (2021), No. 4, 1213–1234.10.1002/fld.4926Suche in Google Scholar

[44] H. Ranocha, Comparison of some entropy conservative numerical fluxes for the Euler equations. Journal of Scientific Computing 17 (2018), 216–242.10.1007/s10915-017-0618-1Suche in Google Scholar

[45] S. Rathan and G. N. Raju, A modified fifth-order WENO scheme for hyperbolic conservation laws. Computers & Mathematics with Applications 75 (2018), No. 5, 1531–1549.10.1016/j.camwa.2017.11.020Suche in Google Scholar

[46] V. V. Rusanov, Calculation of interaction of non-steady shock-waves with obstacles. USSR Comput. Math. & Math. Phys. 1 (1961), 267–279.Suche in Google Scholar

[47] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing. Mathematics of Computation 40 (1983), No. 161, 91–106.10.1090/S0025-5718-1983-0679435-6Suche in Google Scholar

[48] M. A. Schonbek, Second-order conservative schemes and the entropy condition. Mathematics of Computation 44 (1985), 31–38.10.1137/0914082Suche in Google Scholar

[49] C.W. Schulz-Rinne, J. P. Collins, and H. M. Glaz, Numerical solution of the riemann problem for two-dimensional gas dynamics. SIAM Journal on Scientific Computing 14 (1993), No. 6, 1394–1414.10.1016/j.jcp.2003.09.017Suche in Google Scholar

[50] S. Serna and A. Marquina, Power ENO methods: A fifth-order accurate weighted power ENO method. Journal of Computational Physics 194 (2004), No. 2, 632–658.10.1137/0909073Suche in Google Scholar

[51] C.-W. Shu, Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9 (1988), No. 6, 1073–1084.Suche in Google Scholar

[52] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Technical Report NASA/CR-97-206253, NAS 1.26:206253, ICASE-97-65, Institute for Computer Applications in Science and Engineering, Hampton, VA, 1997.10.1137/070679065Suche in Google Scholar

[53] C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Review 51 (2009), No. 1, 82–126.10.1016/0021-9991(88)90177-5Suche in Google Scholar

[54] C-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77 (1988), No. 2, 439–471.10.1016/0021-9991(78)90023-2Suche in Google Scholar

[55] G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics 27 (1978), No. 1, 1–31.10.1137/0721062Suche in Google Scholar

[56] P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal on Numerical Analysis 21 (1984), No. 5, 995–1011.10.1090/S0025-5718-1987-0890255-3Suche in Google Scholar

[57] E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comp 49 (1987), 91–103.10.1137/0725057Suche in Google Scholar

[58] E. Tadmor, Convenient total variation diminishing conditions for nonlinear difference schemes. SIAM Journal on Numerical Analysis 25 (1988), No. 5, 1002–1014.10.1017/S0962492902000156Suche in Google Scholar

[59] E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica 12 (2003), 451–512.10.3934/dcds.2016.36.4579Suche in Google Scholar

[60] E. Tadmor, Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete and Continuous Dynamical Systems 36 (2016), No. 8, 4579–4598.10.1093/imanum/20.1.47Suche in Google Scholar

[61] E. Toro and S. J. Billett, Centred TVD schemes for hyperbolic conservation laws. IMA Journal of Numerical Analysis 20 (2000), No. 1, 47–79.Suche in Google Scholar

[62] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer Science & Business Media, 2013.10.1016/j.jcp.2015.09.055Suche in Google Scholar

[63] P. Wesseling, Principles of Computational Fluid Dynamics, Springer-Verlag, Berlin, 2001.10.1016/0021-9991(84)90142-6Suche in Google Scholar

[64] A. R. Winters and G. J. Gassner, Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations. Journal of Computational Physics 304 (2016), 72–108.10.1093/imanum/drv020Suche in Google Scholar

[65] P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics 54 (1984), No. 1, 115–173.10.1137/090764384Suche in Google Scholar

[66] H. Zakerzadeh and U. S. Fjordholm, High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA Journal of Numerical Analysis 32 (2015), No. 2, 633–654.10.1006/jcph.1996.5514Suche in Google Scholar

[67] X. Zhang and C.-W. Shu, A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws. SIAM Journal on Numerical Analysis 48 (2010), No. 2, 772–795.10.1017/CBO9780511605604Suche in Google Scholar

[68] X. Zhang and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: Survey and new developments. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467 (2011), No. 2134, 2752–2776.10.1098/rspa.2011.0153Suche in Google Scholar

Received: 2022-08-23
Revised: 2023-01-14
Accepted: 2023-05-10
Published Online: 2023-08-16
Published in Print: 2024-03-25

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