-
Previous Article
Numerical algorithms for stationary statistical properties of dissipative dynamical systems
- DCDS Home
- This Issue
-
Next Article
The relative entropy method for the stability of intermediate shock waves; the rich case
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 |
  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.
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. Google Scholar |
[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). 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.
|
[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. Google Scholar |
[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). 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.
|
[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. |
[1] |
Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 |
[2] |
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 & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230 |
[3] |
Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 |
[4] |
Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 |
[5] |
Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001 |
[6] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[7] |
Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041 |
[8] |
Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348 |
[9] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[10] |
Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020408 |
[11] |
Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 |
[12] |
Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 |
[13] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[14] |
Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021002 |
[15] |
Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 |
[16] |
Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039 |
[17] |
Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141 |
[18] |
Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319 |
[19] |
Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021017 |
[20] |
Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]