June  2019, 11(2): 187-203. doi: 10.3934/jgm.2019010

Conservation laws in discrete geometry

1. 

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

2. 

Theoretical Design Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

* Corresponding author: L. G. Margolin

Received  April 2018 Revised  January 2019 Published  May 2019

The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry of computational grids. Historically, the simulations of flows with shocks and/or turbulence have relied on solving the Euler equations with dissipative regularization. In this paper, we begin by reviewing the regularization strategies used in shock wave calculations in both a Lagrangian and an Eulerian framework. We exhibit the essential similarities with Large Eddy Simulation models of turbulence, namely that almost all of these depend on the square of the size of the computational cell. In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. Those evolution equations, termed finite scale Navier-Stokes (FSNS), contain dissipative terms similar to the artificial viscosity first proposed by von Neumann and Richtmyer. We describe the properties of FSNS, provide a physical interpretation of the dissipative terms and show the connection to recent concepts in fluid dynamics, including inviscid dissipation and bi-velocity hydrodynamics.

Citation: Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
References:
[1]

A. Alexander, Duel at Dawn, Harvard University Press, Cambridge, MA, 2010.  Google Scholar

[2]

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.   Google Scholar

[3]

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

[4]

H. A. Bethe, On the theory of shock waves for an arbitrary equation of state, 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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

H. Brenner, Kinematics of volume transport, Physica A, 349 (2004), 11-59.   Google Scholar

[8]

H. Brenner, Steady-state heat conduction in quiescent fluids: Incompleteness of the Navier–Stokes–Fourier equations, Physica A, 390 (2011), 3216-3244.  doi: 10.1016/j.physa.2011.04.023.  Google Scholar

[9]

E. J. CaramanaM. 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.  Google Scholar

[10]

S. Y. ChenD. D. HolmL. G. Margolin and R. Zhang, Direct numerical simulations of the Navier–Stokes alpha model, Physica D, 133 (1999), 66-83.  doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar

[11]

S. Y. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

[12]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. Royal Soc. A, 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.  Google Scholar

[13]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, NY, 2010, third edition. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[14]

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

[15]

L. Euler, Principes généraux du mouvement des fluides, Mém. Acad. Sci. Berlin, 11, 274–315. See also an English translation by T.E. Burton, 1999: “General laws of the motion of fluids,” Fluid Dyn., 34 (1999), 801–822. Google Scholar

[16]

G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics, Physica D, 78 (1994), 222-240.  doi: 10.1016/0167-2789(94)90117-1.  Google Scholar

[17]

U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

L. S. García–ColínR. M. Velasco and F. J. Uribe, Beyond the Navier–Stokes equations: Burnett hydrodynamics, Phys. Reports, 465 (2008), 149-189.  doi: 10.1016/j.physrep.2008.04.010.  Google Scholar

[19]

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

[20]

B. J. Geurts and D. D. Holm, Leray and LANS-$\alpha$ modeling of turbulent mixing, J. Turbulence, 7 (2006), Paper 10, 33 pp. doi: 10.1080/14685240500501601.  Google Scholar

[21]

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

[22]

C. J. Greenshields and J. M. Reese, The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations, J. Fluid Mech., 580 (2007), 407-429.  doi: 10.1017/S0022112007005575.  Google Scholar

[23]

F. F. Grinstein, L. G. Margolin and W. J. Rider, Implicit Large Eddy Simulation, Cambridge University Press, NY, NY, 2007. doi: 10.1017/CBO9780511618604.  Google Scholar

[24]

J. L. GuermondJ. T. Oden and S. Prudhomme, An interpretation of the Navier–Stokes alpha model as a frame–indifferent Leray regularization, Physica D, 177 (2003), 23-30.  doi: 10.1016/S0167-2789(02)00748-0.  Google Scholar

[25]

J. L. GuermondR. Pasquetti and B. Popov, Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230 (2011), 4248-4267.  doi: 10.1016/j.jcp.2010.11.043.  Google Scholar

[26]

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

[27]

M. W. Hecht, D. D. Holm, M. R. Petersen and B. A. Wingate, The LANS-alpha and Leray turbulence parameterizations in primitive equation ocean modeling, J. Physics A, 41 (2008), 344009, 23 pp. doi: 10.1088/1751-8113/41/34/344009.  Google Scholar

[28]

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

[29]

D. D. Holm, Kármán–Howarth theorem for the Lagrangian–averaged Navier–Stokes–alpha model of turbulence, J. Fluid Mech., 467 (2002), 205-214.  doi: 10.1017/S002211200200160X.  Google Scholar

[30]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler–Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), 4173-4176.   Google Scholar

[31]

G.M. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases, Springer, NY, 2010. doi: 10.1007/978-3-642-11696-4.  Google Scholar

[32]

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

[33]

J. Leray, Sur les movements dun fluide visqueux remplaissant lespace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[34] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar
[35]

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.  Google Scholar

[36]

L. G. Margolin, The role of the observer in classical fluid flow, Mech. Res. Comm., 57 (2014), 10-17.   Google Scholar

[37]

L. G. Margolin and A. Hunter, Discrete thermodynamics, Mech. Res. Comm., 93 (2018), 103-107.  doi: 10.1016/j.mechrescom.2017.10.006.  Google Scholar

[38]

L. G. Margolin and C. S. Plesko, Discrete regularization, Evolution Equations and Control Theory, 8 (2019), 117-137.   Google Scholar

[39]

L. G. MargolinJ. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, Int. J. Nonlinear Mech., 95 (2017), 333-346.   Google Scholar

[40]

L. G. MargolinP. 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.  Google Scholar

[41]

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.  Google Scholar

[42]

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.  Google Scholar

[43]

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.  Google Scholar

[44]

I. Múller, On the entropy inequality, Archive for Rational Mechanics and Analysis, 26 (1967), 118-141.  doi: 10.1007/BF00285677.  Google Scholar

[45]

P. Névir, Ertel's vorticity theorems, the particle relabeling symmetry and the energy–vorticity theory of mechanics, Meteorologische Zeitschrift, 13 (2004), 485-498.   Google Scholar

[46]

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.   Google Scholar

[47]

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

[48]

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

[49]

P. Saugat, Large Eddy Simulation for Incompressible Flows, Scientific Computation. Springer-Verlag, Berlin, 2006.  Google Scholar

[50]

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

[51]

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

[52]

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.  Google Scholar

[53]

G. G. Stokes, On the theories of the internal friction of fluids in motion, Trans. Camb. Phil. Soc., 8 (1845), 287-305.   Google Scholar

[54]

P. A. Thompson, Compressible–Fluid Dynamics, McGraw–Hill, NY, 1972. Google Scholar

[55]

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

[56]

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.  Google Scholar

[57]

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.  Google Scholar

show all references

References:
[1]

A. Alexander, Duel at Dawn, Harvard University Press, Cambridge, MA, 2010.  Google Scholar

[2]

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.   Google Scholar

[3]

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

[4]

H. A. Bethe, On the theory of shock waves for an arbitrary equation of state, 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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

H. Brenner, Kinematics of volume transport, Physica A, 349 (2004), 11-59.   Google Scholar

[8]

H. Brenner, Steady-state heat conduction in quiescent fluids: Incompleteness of the Navier–Stokes–Fourier equations, Physica A, 390 (2011), 3216-3244.  doi: 10.1016/j.physa.2011.04.023.  Google Scholar

[9]

E. J. CaramanaM. 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.  Google Scholar

[10]

S. Y. ChenD. D. HolmL. G. Margolin and R. Zhang, Direct numerical simulations of the Navier–Stokes alpha model, Physica D, 133 (1999), 66-83.  doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar

[11]

S. Y. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

[12]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. Royal Soc. A, 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.  Google Scholar

[13]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, NY, 2010, third edition. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[14]

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

[15]

L. Euler, Principes généraux du mouvement des fluides, Mém. Acad. Sci. Berlin, 11, 274–315. See also an English translation by T.E. Burton, 1999: “General laws of the motion of fluids,” Fluid Dyn., 34 (1999), 801–822. Google Scholar

[16]

G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics, Physica D, 78 (1994), 222-240.  doi: 10.1016/0167-2789(94)90117-1.  Google Scholar

[17]

U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

L. S. García–ColínR. M. Velasco and F. J. Uribe, Beyond the Navier–Stokes equations: Burnett hydrodynamics, Phys. Reports, 465 (2008), 149-189.  doi: 10.1016/j.physrep.2008.04.010.  Google Scholar

[19]

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

[20]

B. J. Geurts and D. D. Holm, Leray and LANS-$\alpha$ modeling of turbulent mixing, J. Turbulence, 7 (2006), Paper 10, 33 pp. doi: 10.1080/14685240500501601.  Google Scholar

[21]

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

[22]

C. J. Greenshields and J. M. Reese, The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations, J. Fluid Mech., 580 (2007), 407-429.  doi: 10.1017/S0022112007005575.  Google Scholar

[23]

F. F. Grinstein, L. G. Margolin and W. J. Rider, Implicit Large Eddy Simulation, Cambridge University Press, NY, NY, 2007. doi: 10.1017/CBO9780511618604.  Google Scholar

[24]

J. L. GuermondJ. T. Oden and S. Prudhomme, An interpretation of the Navier–Stokes alpha model as a frame–indifferent Leray regularization, Physica D, 177 (2003), 23-30.  doi: 10.1016/S0167-2789(02)00748-0.  Google Scholar

[25]

J. L. GuermondR. Pasquetti and B. Popov, Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230 (2011), 4248-4267.  doi: 10.1016/j.jcp.2010.11.043.  Google Scholar

[26]

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

[27]

M. W. Hecht, D. D. Holm, M. R. Petersen and B. A. Wingate, The LANS-alpha and Leray turbulence parameterizations in primitive equation ocean modeling, J. Physics A, 41 (2008), 344009, 23 pp. doi: 10.1088/1751-8113/41/34/344009.  Google Scholar

[28]

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

[29]

D. D. Holm, Kármán–Howarth theorem for the Lagrangian–averaged Navier–Stokes–alpha model of turbulence, J. Fluid Mech., 467 (2002), 205-214.  doi: 10.1017/S002211200200160X.  Google Scholar

[30]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler–Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), 4173-4176.   Google Scholar

[31]

G.M. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases, Springer, NY, 2010. doi: 10.1007/978-3-642-11696-4.  Google Scholar

[32]

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

[33]

J. Leray, Sur les movements dun fluide visqueux remplaissant lespace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[34] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar
[35]

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.  Google Scholar

[36]

L. G. Margolin, The role of the observer in classical fluid flow, Mech. Res. Comm., 57 (2014), 10-17.   Google Scholar

[37]

L. G. Margolin and A. Hunter, Discrete thermodynamics, Mech. Res. Comm., 93 (2018), 103-107.  doi: 10.1016/j.mechrescom.2017.10.006.  Google Scholar

[38]

L. G. Margolin and C. S. Plesko, Discrete regularization, Evolution Equations and Control Theory, 8 (2019), 117-137.   Google Scholar

[39]

L. G. MargolinJ. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, Int. J. Nonlinear Mech., 95 (2017), 333-346.   Google Scholar

[40]

L. G. MargolinP. 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.  Google Scholar

[41]

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.  Google Scholar

[42]

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.  Google Scholar

[43]

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.  Google Scholar

[44]

I. Múller, On the entropy inequality, Archive for Rational Mechanics and Analysis, 26 (1967), 118-141.  doi: 10.1007/BF00285677.  Google Scholar

[45]

P. Névir, Ertel's vorticity theorems, the particle relabeling symmetry and the energy–vorticity theory of mechanics, Meteorologische Zeitschrift, 13 (2004), 485-498.   Google Scholar

[46]

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.   Google Scholar

[47]

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

[48]

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

[49]

P. Saugat, Large Eddy Simulation for Incompressible Flows, Scientific Computation. Springer-Verlag, Berlin, 2006.  Google Scholar

[50]

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

[51]

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

[52]

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.  Google Scholar

[53]

G. G. Stokes, On the theories of the internal friction of fluids in motion, Trans. Camb. Phil. Soc., 8 (1845), 287-305.   Google Scholar

[54]

P. A. Thompson, Compressible–Fluid Dynamics, McGraw–Hill, NY, 1972. Google Scholar

[55]

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

[56]

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.  Google Scholar

[57]

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.  Google Scholar

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