March 2019, 8(1): 117-137. doi: 10.3934/eect.2019007

Discrete regularization

Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

Received  January 2018 Revised  April 2018 Published  January 2019

In this paper we discuss discrete regularization, more specifically about how to add finite dissipation to the discretized Euler equations so as to ensure the stability and convergence of numerical solutions of high Reynolds number flows. We will briefly review regularization strategies widely used in Lagrangian shockwave simulations (artificial viscosity), in Eulerian nonoscillatory finite volume simulations, and in Eulerian simulations of turbulent flow (explicit and implicit large eddy simulations). We will describe an alternate strategy for regularization in which we introduce a finite length scale into the discrete model by volume averaging the equations over a computational cell. The new equations, which we term Finite Scale Navier-Stokes, contain explicit (inviscid) dissipation in a uniquely specified form and obey an entropy principle. We will describe features of the new equations including control of the small scales of motion by the larger resolved scales, a principle concerning the partition of total flux of conserved quantities into advective and diffusive components, and a physical basis for the inviscid dissipation.

Citation: Len Margolin, Catherine Plesko. Discrete regularization. Evolution Equations & Control Theory, 2019, 8 (1) : 117-137. doi: 10.3934/eect.2019007
References:
[1]

H. Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, J. Fluid Mech., 74 (1976), 497-513.

[2]

W. S. Baring-Gould, The Annotated Sherlock Holmes, Clarkson Potter; 2nd edition, NY, 1988.

[3]

R. Becker, Stoßbwelle und detonation, (In German), Zeitschrift für Physik, 8 (1922), 321-362.

[4]

H. A. Bethe, 1942: On the theory of shock waves for an arbitrary equation of state, reprinted in Classic Papers in Shock Compression Science, J.N. Johnson & R. Cheret, eds., Springer-Verlag, New York, 1998,421-492. doi: 10.1007/978-1-4612-2218-7_11.

[5]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342. doi: 10.4007/annals.2005.161.223.

[6]

J. P. Boris and D. L. Book, Flux-corrected transport, J. Comput. Phys., 11 (1973), 38-69. doi: 10.1006/jcph.1997.5756.

[7]

J. C. Campbell and M. J. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172 (2001), 739-765. doi: 10.1006/jcph.2001.6856.

[8]

E. J. Caramana, M. J. Shashkov and P. P. Whalen, Formulations of artificial viscosity for multi-dimensional shock wave computations, J. Comput. Phys., 144 (1998), 70-97. doi: 10.1006/jcph.1998.5989.

[9]

R. B. Christiansen, Godunov Methods on a Staggered Mesh: An Improved Artificial Viscosity, Lawrence Livermore National Laboratory Report, UCRL-JC-105269, 1991.

[10]

F. M. Denaro, What does finite volume-based implicit filtering really resolve in large-eddy simulations?, J. Comput. Phys., 230 (2011), 3849-3883. doi: 10.1016/j.jcp.2011.02.011.

[11]

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

[12]

T. D. Drivas and G. L. Eyink, An Onsager singularity theorem for turbulent solutions of compressible Euler equations, Commun. Math. Phys., 359 (2018), 733-763. doi: 10.1007/s00220-017-3078-4.

[13]

J. K. Dukowicz, A general non-iterative Riemann solver for Godunov's method, J. Comput. Phys., 61 (1985), 119-137. doi: 10.1016/0021-9991(85)90064-6.

[14]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.

[15]

K. O. Friedrichs and P. D. Lax, Systems of conservation equation with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.

[16] U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.
[17]

B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation, Phys. Fluids, 15 (2003), L13-L16. doi: 10.1063/1.1529180.

[18]

S. K. Godunov, (Ph.D. Dissertation) Different Methods for Shock Waves., Moscow State University, 1954.

[19] F. F. GrinsteinL. G. Margolin and W. J. Rider, Implicit Large Eddy Simulation, Cambridge University Press, NY, NY, 2007. doi: 10.1017/CBO9780511618604.
[20]

J.-L. Guermond and B. Popov, Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74 (2014), 284-305. doi: 10.1137/120903312.

[21]

A. Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49 (1983), 151-164. doi: 10.1016/0021-9991(83)90118-3.

[22]

A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), 357-393. doi: 10.1016/0021-9991(83)90136-5.

[23]

W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science, Prometheus Books, Amherst, NY, 1999.

[24]

C. W. Hirt, Heuristic stability theory for finite difference equations, J. Comput. Phys., 2 (1968), 339-355.

[25]

V. P. Kolgan, Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics, J. Comput. Phys., 230 (2011), 2384-2390. doi: 10.1016/j.jcp.2010.12.033.

[26]

A. N. Kolmogorov, A refinement of previous hypothesis concerning the local structure of turbulence in viscous incompressible fluid at high Reynolds number, J. Fluid Mech., 13 (1962), 82-85. doi: 10.1017/S0022112062000518.

[27]

P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E.H. Zarantonello, Academic Press, NY, 1971, 603-634.

[28]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973.

[29]

P. D. Lax, Mathematics and physics, Bull. Amer. Math. Soc., 45 (2008), 135-152. doi: 10.1090/S0273-0979-07-01182-2.

[30] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[31]

D. K. Lily, The meteorological development of large eddy simulation, in IUTAM Symposium on Developments in Geophysical Turbulence, R. Kerr and Y. Kimura, eds., Kluwer Academic Publishers, Boulder, CO, 2000.

[32]

P. F. Linden, J. M. Redondo and D. L. Youngs, Molecular mixing in Rayleigh-Taylor instability, J. Fluid Mech., 265 (1994), 97-124.

[33]

J. L. Lumley, ed., Whither Turbulence? Turbulence at the Crossroads, Lecture Notes in Physics, Springer-Verlag Berlin Heidelberg, 1990. doi: 10.1007/3-540-52535-1.

[34]

L. G. Margolin, Finite-scale equations for compressible fluid flow, Phil. Trans. R. Soc. A, 367 (2009), 2861-2871. doi: 10.1098/rsta.2008.0290.

[35]

L. G. Margolin, Scale matters, Philos. Trans. Roy. Soc. A, 376 (2018), 20170235, 12 pp. doi: 10.1098/rsta.2017.0235.

[36]

L. G. Margolin, J. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, Int. J. Non-Linear Mech., 95 (2017), 333-346.

[37]

L. G. Margolin and W. J. Rider, A rationale for implicit turbulence modelling, Int. J. Num. Methods Fluids, 39 (2002), 821-841. doi: 10.1002/fld.331.

[38]

L. G. Margolin, W. J. Rider and F. F. Grinstein, Modeling turbulent flow with implicit LES, J. Turbulence, 7 (2006), Paper 15, 27 pp. doi: 10.1080/14685240500331595.

[39]

L. G. Margolin, H. M. Ruppel and R. B. Demuth, Gradient scaling for nonuniform meshes, Proc. 4th International Conference on Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, Swansea, UK, 1985, 1477-1488.

[40]

L. G. Margolin and M. Shashkov, Finite volume methods and the equations of finite scale: A mimetic approach, Int. J. Num. Methods Fluids, 56 (2008), 991-1002. doi: 10.1002/fld.1592.

[41]

L. G. Margolin, M. Shashkov and P. K. Smolarkiewicz, A discrete operator calculus for finite difference approximations, Comput. Methods Appl. Mech. Engr., 187 (2000), 365-383. doi: 10.1016/S0045-7825(00)80001-8.

[42]

L. G. Margolin, P. K. Smolarkiewicz and Z. Sorbjan, Large-eddy simulations of convective boundary layers using nonoscillatory differencing, Physica D., 133 (1999), 390-397. doi: 10.1016/S0167-2789(99)00083-4.

[43]

A. E. Mattsson and W. J. Rider, Artificial viscosity: Back to basics, Int. J. of Num. Methods in Fluids, 77 (2015), 400-417. doi: 10.1002/fld.3981.

[44]

C. Meneveau and J. Katz, Scale-invariance and turbulence models for large eddy simulation, Annu. Rev. Fluid Mech., 32 (2000), 1-32. doi: 10.1146/annurev.fluid.32.1.1.

[45]

M. L. Merriam, Smoothing and the second law, Comp. Meth. Appl. Mech. Eng., 64 (1987), 177-193. doi: 10.1016/0045-7825(87)90039-9.

[46]

K. Mohseni, Derivation of regularized Euler equations from basic principles, AIAA paper 2009-5695, 39th AIAA Fluid Dynamics Conference, San Antonio, TX, 2009.

[47]

M. Morduchow and P. A. Libby, On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas, J. Aeronautical Sciences, 16 (1949), 674-684,704. doi: 10.2514/8.11882.

[48]

W. F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat conduction, J. Comput. Phys., 72 (1978), 78-120.

[49]

E. S. Oran and J. P. Boris, Computing turbulent shear flows-a convenient conspiracy, Computers in Physics, 7 (1993), 523-533.

[50]

A. Petersen, The philosophy of Niels Bohr, Bulletin of the Atomic Scientists, 19 (1963), 8-14.

[51]

D. H. Porter, A. Pouquet and P. R. Woodward, Kolmogorov-like spectra in decaying three-dimensional supersonic flows, Phys. Fluids, 6 (1994), 2133-2142.

[52]

R. D. Richtmyer, Proposed numerical method for calculation of shocks, Los Alamos Scientific Laboratory Report LA-671, 1948.

[53]

W. J. Rider, Revisiting wall heating, J. Comput. Phys., 162 (2000), 395-410.

[54]

B. Schmidt, Electron beam density measurements in shock waves in argon, J. Fluid Mech., 39 (1969), 361-373.

[55]

J. Smagorinsky, General circulation experiments with the primitive equations I. The basic experiment, Mon. Wea. Rev., 91 (1963), 99-164.

[56]

J. Smagorinsky, The beginnings of numerical weather prediction and general circulation modeling: Early recollections, Advances in Geophysics, 25 (1983), 3-37.

[57]

P. K. Smolarkiewicz and L. G. Margolin, MPDATA: A finite-difference solver for geophysical flows, J. Comput. Phys., 140 (1998), 459-480. doi: 10.1006/jcph.1998.5901.

[58]

P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Num. Anal., 21 (1984), 995-1011. doi: 10.1137/0721062.

[59]

E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numerica, 12 (2003), 451-512. doi: 10.1017/S0962492902000156.

[60]

E. Tadmor and W. Zhong, Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity, J. Hyperbolic Differential Equations, 3 (2006), 529-559. doi: 10.1142/S0219891606000896.

[61]

L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115A (1990), 193-230. doi: 10.1017/S0308210500020606.

[62]

P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, NY, 1972.

[63]

B. van Leer, Toward the ultimate conservative difference scheme V, J. Comput. Phys., 32 (1979), 101-136. doi: 10.1006/jcph.1997.5757.

[64]

B. van Leer, A historical oversight: Vladimir P. Kolgan and his high-resolution scheme, J. Comput. Phys., 230 (2011), 2378-2383. doi: 10.1016/j.jcp.2010.12.032.

[65]

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.

[66]

M. L. Wilkins, Use of artificial viscosity in multidimensional fluid dynamic calculations, J. Comput. Phys., 36 (1980), 281-303. doi: 10.1016/0021-9991(80)90161-8.

show all references

References:
[1]

H. Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, J. Fluid Mech., 74 (1976), 497-513.

[2]

W. S. Baring-Gould, The Annotated Sherlock Holmes, Clarkson Potter; 2nd edition, NY, 1988.

[3]

R. Becker, Stoßbwelle und detonation, (In German), Zeitschrift für Physik, 8 (1922), 321-362.

[4]

H. A. Bethe, 1942: On the theory of shock waves for an arbitrary equation of state, reprinted in Classic Papers in Shock Compression Science, J.N. Johnson & R. Cheret, eds., Springer-Verlag, New York, 1998,421-492. doi: 10.1007/978-1-4612-2218-7_11.

[5]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342. doi: 10.4007/annals.2005.161.223.

[6]

J. P. Boris and D. L. Book, Flux-corrected transport, J. Comput. Phys., 11 (1973), 38-69. doi: 10.1006/jcph.1997.5756.

[7]

J. C. Campbell and M. J. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172 (2001), 739-765. doi: 10.1006/jcph.2001.6856.

[8]

E. J. Caramana, M. J. Shashkov and P. P. Whalen, Formulations of artificial viscosity for multi-dimensional shock wave computations, J. Comput. Phys., 144 (1998), 70-97. doi: 10.1006/jcph.1998.5989.

[9]

R. B. Christiansen, Godunov Methods on a Staggered Mesh: An Improved Artificial Viscosity, Lawrence Livermore National Laboratory Report, UCRL-JC-105269, 1991.

[10]

F. M. Denaro, What does finite volume-based implicit filtering really resolve in large-eddy simulations?, J. Comput. Phys., 230 (2011), 3849-3883. doi: 10.1016/j.jcp.2011.02.011.

[11]

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

[12]

T. D. Drivas and G. L. Eyink, An Onsager singularity theorem for turbulent solutions of compressible Euler equations, Commun. Math. Phys., 359 (2018), 733-763. doi: 10.1007/s00220-017-3078-4.

[13]

J. K. Dukowicz, A general non-iterative Riemann solver for Godunov's method, J. Comput. Phys., 61 (1985), 119-137. doi: 10.1016/0021-9991(85)90064-6.

[14]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.

[15]

K. O. Friedrichs and P. D. Lax, Systems of conservation equation with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.

[16] U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.
[17]

B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation, Phys. Fluids, 15 (2003), L13-L16. doi: 10.1063/1.1529180.

[18]

S. K. Godunov, (Ph.D. Dissertation) Different Methods for Shock Waves., Moscow State University, 1954.

[19] F. F. GrinsteinL. G. Margolin and W. J. Rider, Implicit Large Eddy Simulation, Cambridge University Press, NY, NY, 2007. doi: 10.1017/CBO9780511618604.
[20]

J.-L. Guermond and B. Popov, Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74 (2014), 284-305. doi: 10.1137/120903312.

[21]

A. Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49 (1983), 151-164. doi: 10.1016/0021-9991(83)90118-3.

[22]

A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), 357-393. doi: 10.1016/0021-9991(83)90136-5.

[23]

W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science, Prometheus Books, Amherst, NY, 1999.

[24]

C. W. Hirt, Heuristic stability theory for finite difference equations, J. Comput. Phys., 2 (1968), 339-355.

[25]

V. P. Kolgan, Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics, J. Comput. Phys., 230 (2011), 2384-2390. doi: 10.1016/j.jcp.2010.12.033.

[26]

A. N. Kolmogorov, A refinement of previous hypothesis concerning the local structure of turbulence in viscous incompressible fluid at high Reynolds number, J. Fluid Mech., 13 (1962), 82-85. doi: 10.1017/S0022112062000518.

[27]

P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E.H. Zarantonello, Academic Press, NY, 1971, 603-634.

[28]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1973.

[29]

P. D. Lax, Mathematics and physics, Bull. Amer. Math. Soc., 45 (2008), 135-152. doi: 10.1090/S0273-0979-07-01182-2.

[30] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[31]

D. K. Lily, The meteorological development of large eddy simulation, in IUTAM Symposium on Developments in Geophysical Turbulence, R. Kerr and Y. Kimura, eds., Kluwer Academic Publishers, Boulder, CO, 2000.

[32]

P. F. Linden, J. M. Redondo and D. L. Youngs, Molecular mixing in Rayleigh-Taylor instability, J. Fluid Mech., 265 (1994), 97-124.

[33]

J. L. Lumley, ed., Whither Turbulence? Turbulence at the Crossroads, Lecture Notes in Physics, Springer-Verlag Berlin Heidelberg, 1990. doi: 10.1007/3-540-52535-1.

[34]

L. G. Margolin, Finite-scale equations for compressible fluid flow, Phil. Trans. R. Soc. A, 367 (2009), 2861-2871. doi: 10.1098/rsta.2008.0290.

[35]

L. G. Margolin, Scale matters, Philos. Trans. Roy. Soc. A, 376 (2018), 20170235, 12 pp. doi: 10.1098/rsta.2017.0235.

[36]

L. G. Margolin, J. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, Int. J. Non-Linear Mech., 95 (2017), 333-346.

[37]

L. G. Margolin and W. J. Rider, A rationale for implicit turbulence modelling, Int. J. Num. Methods Fluids, 39 (2002), 821-841. doi: 10.1002/fld.331.

[38]

L. G. Margolin, W. J. Rider and F. F. Grinstein, Modeling turbulent flow with implicit LES, J. Turbulence, 7 (2006), Paper 15, 27 pp. doi: 10.1080/14685240500331595.

[39]

L. G. Margolin, H. M. Ruppel and R. B. Demuth, Gradient scaling for nonuniform meshes, Proc. 4th International Conference on Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, Swansea, UK, 1985, 1477-1488.

[40]

L. G. Margolin and M. Shashkov, Finite volume methods and the equations of finite scale: A mimetic approach, Int. J. Num. Methods Fluids, 56 (2008), 991-1002. doi: 10.1002/fld.1592.

[41]

L. G. Margolin, M. Shashkov and P. K. Smolarkiewicz, A discrete operator calculus for finite difference approximations, Comput. Methods Appl. Mech. Engr., 187 (2000), 365-383. doi: 10.1016/S0045-7825(00)80001-8.

[42]

L. G. Margolin, P. K. Smolarkiewicz and Z. Sorbjan, Large-eddy simulations of convective boundary layers using nonoscillatory differencing, Physica D., 133 (1999), 390-397. doi: 10.1016/S0167-2789(99)00083-4.

[43]

A. E. Mattsson and W. J. Rider, Artificial viscosity: Back to basics, Int. J. of Num. Methods in Fluids, 77 (2015), 400-417. doi: 10.1002/fld.3981.

[44]

C. Meneveau and J. Katz, Scale-invariance and turbulence models for large eddy simulation, Annu. Rev. Fluid Mech., 32 (2000), 1-32. doi: 10.1146/annurev.fluid.32.1.1.

[45]

M. L. Merriam, Smoothing and the second law, Comp. Meth. Appl. Mech. Eng., 64 (1987), 177-193. doi: 10.1016/0045-7825(87)90039-9.

[46]

K. Mohseni, Derivation of regularized Euler equations from basic principles, AIAA paper 2009-5695, 39th AIAA Fluid Dynamics Conference, San Antonio, TX, 2009.

[47]

M. Morduchow and P. A. Libby, On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas, J. Aeronautical Sciences, 16 (1949), 674-684,704. doi: 10.2514/8.11882.

[48]

W. F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat conduction, J. Comput. Phys., 72 (1978), 78-120.

[49]

E. S. Oran and J. P. Boris, Computing turbulent shear flows-a convenient conspiracy, Computers in Physics, 7 (1993), 523-533.

[50]

A. Petersen, The philosophy of Niels Bohr, Bulletin of the Atomic Scientists, 19 (1963), 8-14.

[51]

D. H. Porter, A. Pouquet and P. R. Woodward, Kolmogorov-like spectra in decaying three-dimensional supersonic flows, Phys. Fluids, 6 (1994), 2133-2142.

[52]

R. D. Richtmyer, Proposed numerical method for calculation of shocks, Los Alamos Scientific Laboratory Report LA-671, 1948.

[53]

W. J. Rider, Revisiting wall heating, J. Comput. Phys., 162 (2000), 395-410.

[54]

B. Schmidt, Electron beam density measurements in shock waves in argon, J. Fluid Mech., 39 (1969), 361-373.

[55]

J. Smagorinsky, General circulation experiments with the primitive equations I. The basic experiment, Mon. Wea. Rev., 91 (1963), 99-164.

[56]

J. Smagorinsky, The beginnings of numerical weather prediction and general circulation modeling: Early recollections, Advances in Geophysics, 25 (1983), 3-37.

[57]

P. K. Smolarkiewicz and L. G. Margolin, MPDATA: A finite-difference solver for geophysical flows, J. Comput. Phys., 140 (1998), 459-480. doi: 10.1006/jcph.1998.5901.

[58]

P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Num. Anal., 21 (1984), 995-1011. doi: 10.1137/0721062.

[59]

E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numerica, 12 (2003), 451-512. doi: 10.1017/S0962492902000156.

[60]

E. Tadmor and W. Zhong, Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity, J. Hyperbolic Differential Equations, 3 (2006), 529-559. doi: 10.1142/S0219891606000896.

[61]

L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115A (1990), 193-230. doi: 10.1017/S0308210500020606.

[62]

P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, NY, 1972.

[63]

B. van Leer, Toward the ultimate conservative difference scheme V, J. Comput. Phys., 32 (1979), 101-136. doi: 10.1006/jcph.1997.5757.

[64]

B. van Leer, A historical oversight: Vladimir P. Kolgan and his high-resolution scheme, J. Comput. Phys., 230 (2011), 2378-2383. doi: 10.1016/j.jcp.2010.12.032.

[65]

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.

[66]

M. L. Wilkins, Use of artificial viscosity in multidimensional fluid dynamic calculations, J. Comput. Phys., 36 (1980), 281-303. doi: 10.1016/0021-9991(80)90161-8.

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