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Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws

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  • Entropy stability plays an important role in the dynamics of nonlinear systems of hyperbolic conservation laws and related convection-diffusion equations. Here we are concerned with the corresponding question of numerical entropy stability --- we review a general framework for designing entropy stable approximations of such systems. The framework, developed in [28,29] and in an ongoing series of works [30,6,7], is based on comparing numerical viscosities to certain entropy-conservative schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks.
        We demonstrate this approach with a host of second-- and higher--order accurate schemes, ranging from scalar examples to the systems of shallow-water, Euler and Navier-Stokes equations. We present a family of energy conservative schemes for the shallow-water equations with a well-balanced description of their steady-states. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in Euler equations, and we conclude with the computation of entropic measure-valued solutions based on the class of so-called TeCNO schemes --- arbitrarily high-order accurate, non-oscillatory and entropy stable schemes for systems of conservation laws.
    Mathematics Subject Classification: Primary: 65M12, 35L65; Secondary: 65M06, 35R06.

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  • [1]

    S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Annals of Mathematics, 161 (2005), 223-342.doi: 10.4007/annals.2005.161.223.

    [2]

    G.-Q. Chen, Compactness methods and nonlinear hyperbolic conservation laws: Some current topics on nonlinear conservation laws, in AMS/IP Stud. Adv. Math., 15, Amer. Math. Soc., Providence, RI, (2000), 33-75.

    [3]

    C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 325, 2000.doi: 10.1007/978-3-642-04048-1.

    [4]

    P. Deift and K. T. R. McLaughlin, A continuum limit of the Toda lattice, Mem. Amer. Math. Soc., 131 (1998), x+216 pp.doi: 10.1090/memo/0624.

    [5]

    R. J. DiPerna, Measure valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270.doi: 10.1007/BF00752112.

    [6]

    U. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, J. Computational Physics, 230 (2011), 5587-5609.doi: 10.1016/j.jcp.2011.03.042.

    [7]

    U. Fjordholm, S. Mishra and E. Tadmor, Arbitrarily high order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws, SIAM J. on Numerical Analysis, 50 (2012), 544-573.doi: 10.1137/110836961.

    [8]

    U. Fjordholm, R. Kappeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws, Foundations Comp. Math., (2015), 1-65.doi: 10.1007/s10208-015-9299-z.

    [9]

    K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688.doi: 10.1073/pnas.68.8.1686.

    [10]

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

    [11]

    A. Harten, B. Engquist, S. Osher and S. R. Chakravarty, Uniformly high order accurate essentially non-oscillatory schemes, J. Comput. Phys., 71 (1987), 231-303.doi: 10.1016/0021-9991(87)90031-3.

    [12]

    F. Ismail and P. L. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks, Journal of Computational Physics, 228 (2009), 5410-5436.doi: 10.1016/j.jcp.2009.04.021.

    [13]

    S. N. Kruzkhov, First order quasilinear equations in several independent variables, USSR Math. Sbornik, 10 (1970), 217-243.

    [14]

    P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.doi: 10.1002/cpa.3160100406.

    [15]

    P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, (E. A. Zarantonello, ed.), Academic Press, New York, (1971), 603-634.

    [16]

    P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Lectures in Applied Mathematics, 11, 1973.

    [17]

    P. D. Lax, On dispersive difference schemes, Physica D, 18 (1986), 250-254.doi: 10.1016/0167-2789(86)90185-5.

    [18]

    P. D. Lax, Mathematics and Physics, Bull. AMS, 45 (2008), 135-152.doi: 10.1090/S0273-0979-07-01182-2.

    [19]

    P. D. Lax, John von Neumann: The Early Years, The Years at Los Alamos and the Road to Computing, in "Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs'', 2014. Available from: http://www.ki-net.umd.edu/tn60/2014_04_30_Lax_Banquet_talk.pdf.

    [20]

    P. D. Lax, D. Levermore and S. Venakidis, The generation and propagation of oscillations in dispersive IVPs and their limiting behavior, in Important Developments in Soliton Theory 1980-1990 (T. Fokas and V. E. Zakharov, eds), Springer, Berlin, 1993.

    [21]

    P. D. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237.doi: 10.1002/cpa.3160130205.

    [22]

    P. LeFloch and C. Rohde, High-order schemes, entropy inequalities and non-classical shocks, SIAM J. Numer. Analm., 37 (2000), 2023-2060.doi: 10.1137/S0036142998345256.

    [23]

    M. S. Mock, Systems of conservation of mixed type, J. Diff. Eqns., 37 (1980), 70-88.doi: 10.1016/0022-0396(80)90089-3.

    [24]

    J. von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21 (1950), 232-237.doi: 10.1063/1.1699639.

    [25]

    P. L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), 357-372.doi: 10.1016/0021-9991(81)90128-5.

    [26]

    C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory schemes - II, J. Comput. Phys., 83 (1989), 32-78.doi: 10.1016/0021-9991(89)90222-2.

    [27]

    E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp., 43 (1984), 369-381.doi: 10.1090/S0025-5718-1984-0758189-X.

    [28]

    E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I, Math. Comp., 49 (1987), 91-103.doi: 10.1090/S0025-5718-1987-0890255-3.

    [29]

    E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems, Acta Numerica, 42 (2003), 451-512.doi: 10.1017/S0962492902000156.

    [30]

    E. Tadmor and W. Zhong, Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity, J. Hyperbolic DEs, 3 (2006), 529-559.doi: 10.1142/S0219891606000896.

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