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August  2016, 36(8): 4579-4598. doi: 10.3934/dcds.2016.36.4579

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

Eitan Tadmor 1,

1. 

Department of Mathematics, Institute for Physical Science & Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742

Received  May 2015 Revised  January 2016 Published  March 2016

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.
Keywords: Entropy conservative schemes, measure-valued solutions., Euler and Navier-Stokes equations, shallow water equations, entropy stability, energy preserving schemes, numerical viscosity.
Mathematics Subject Classification: Primary: 65M12, 35L65; Secondary: 65M06, 35R0.
Citation: Eitan Tadmor. Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4579-4598. doi: 10.3934/dcds.2016.36.4579
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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