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

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.
Citation: Eitan Tadmor. Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 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. 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 (2000), 33.

[3]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Springer, 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). doi: 10.1090/memo/0624.

[5]

R. J. DiPerna, Measure valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. 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. 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. 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. 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. doi: 10.1073/pnas.68.8.1686.

[10]

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

[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. 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. 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.

[14]

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

[15]

P. D. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis, (1971), 603.

[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. doi: 10.1016/0167-2789(86)90185-5.

[18]

P. D. Lax, Mathematics and Physics,, Bull. AMS, 45 (2008), 135. 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, (2014).

[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, (1993), 1980.

[21]

P. D. Lax and B. Wendroff, Systems of conservation laws,, Comm. Pure Appl. Math., 13 (1960), 217. 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. doi: 10.1137/S0036142998345256.

[23]

M. S. Mock, Systems of conservation of mixed type,, J. Diff. Eqns., 37 (1980), 70. 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. doi: 10.1063/1.1699639.

[25]

P. L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes,, J. Comput. Phys., 43 (1981), 357. 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. 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. 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. 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. 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. doi: 10.1142/S0219891606000896.

show all references

References:
[1]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems,, Annals of Mathematics, 161 (2005), 223. 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 (2000), 33.

[3]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Springer, 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). doi: 10.1090/memo/0624.

[5]

R. J. DiPerna, Measure valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. 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. 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. 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. 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. doi: 10.1073/pnas.68.8.1686.

[10]

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

[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. 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. 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.

[14]

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

[15]

P. D. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis, (1971), 603.

[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. doi: 10.1016/0167-2789(86)90185-5.

[18]

P. D. Lax, Mathematics and Physics,, Bull. AMS, 45 (2008), 135. 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, (2014).

[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, (1993), 1980.

[21]

P. D. Lax and B. Wendroff, Systems of conservation laws,, Comm. Pure Appl. Math., 13 (1960), 217. 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. doi: 10.1137/S0036142998345256.

[23]

M. S. Mock, Systems of conservation of mixed type,, J. Diff. Eqns., 37 (1980), 70. 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. doi: 10.1063/1.1699639.

[25]

P. L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes,, J. Comput. Phys., 43 (1981), 357. 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. 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. 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. 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. 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. doi: 10.1142/S0219891606000896.

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