# American Institute of Mathematical Sciences

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 and 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. [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.

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. [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.
 [1] Yinnian He, Pengzhan Huang, Jian Li. H2-stability of some second order fully discrete schemes for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2745-2780. doi: 10.3934/dcdsb.2018273 [2] Cleopatra Christoforou, Myrto Galanopoulou, Athanasios E. Tzavaras. Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6175-6206. doi: 10.3934/dcds.2019269 [3] Maria Michaela Porzio, Flavia Smarrazzo, Alberto Tesei. Radon measure-valued solutions of unsteady filtration equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022040 [4] Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks and Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145 [5] Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153 [6] Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29 (5) : 2915-2944. doi: 10.3934/era.2021019 [7] Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361 [8] Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101 [9] Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097 [10] Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353 [11] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic and Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 [12] Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041 [13] Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056 [14] Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure and Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459 [15] Casimir Emako, Farah Kanbar, Christian Klingenberg, Min Tang. A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving. Kinetic and Related Models, 2021, 14 (5) : 847-866. doi: 10.3934/krm.2021026 [16] Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234 [17] Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715 [18] Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829 [19] Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557 [20] Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230

2021 Impact Factor: 1.588