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

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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

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

Abstract
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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.

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 - 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. 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 (2000), 33. Google Scholar
[3]	C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Springer, 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). doi: 10.1090/memo/0624. Google Scholar*
[5]	R. J. DiPerna, Measure valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[14]	P. D. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar
[15]	P. D. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis, (1971), 603. 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. doi: 10.1016/0167-2789(86)90185-5. Google Scholar
[18]	P. D. Lax, Mathematics and Physics,, Bull. AMS, 45 (2008), 135. 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, (2014). 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*, (1993), 1980. Google Scholar
[21]	P. D. Lax and B. Wendroff, Systems of conservation laws,, Comm. Pure Appl. Math., 13 (1960), 217. 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. doi: 10.1137/S0036142998345256. Google Scholar
[23]	M. S. Mock, Systems of conservation of mixed type,, J. Diff. Eqns., 37 (1980), 70. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0219891606000896. Google Scholar

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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. 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 (2000), 33. Google Scholar
[3]	C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Springer, 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). doi: 10.1090/memo/0624. Google Scholar*
[5]	R. J. DiPerna, Measure valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[14]	P. D. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar
[15]	P. D. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis, (1971), 603. 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. doi: 10.1016/0167-2789(86)90185-5. Google Scholar
[18]	P. D. Lax, Mathematics and Physics,, Bull. AMS, 45 (2008), 135. 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, (2014). 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*, (1993), 1980. Google Scholar
[21]	P. D. Lax and B. Wendroff, Systems of conservation laws,, Comm. Pure Appl. Math., 13 (1960), 217. 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. doi: 10.1137/S0036142998345256. Google Scholar
[23]	M. S. Mock, Systems of conservation of mixed type,, J. Diff. Eqns., 37 (1980), 70. 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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0219891606000896. Google Scholar

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