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Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws
Numerical algorithms for stationary statistical properties of dissipative dynamical systems
1. | Department of Mathematics, The Florida State University, Tallahassee, FL 32306-4510 |
References:
[1] |
P. Billingsley, Weak Convergence of Measures: Applications in Probability, SIAM, Philadelphia, 1971. |
[2] |
E. Cancs, E. Legoll and G. Stoltz, Theoretical and numerical comparison of some sampling methods for molecular dynamics, ESAIM: Mathematical Modelling and Numerical Analysis, 41 (2007), 351-389.
doi: 10.1051/m2an:2007014. |
[3] |
W. Cheng and X. Wang, A uniformly dissipative scheme for stationary statistical properties of the infinite prandtl number model, Applied Mathematics Letters, 21 (2008), 1281-1285.
doi: 10.1016/j.aml.2007.07.036. |
[4] |
W. Cheng and X. Wang, A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model, SIAM J. Num. Anal., 47 (2008), 250-270.
doi: 10.1137/080713501. |
[5] |
C. Chiu, Q. Du and T. Y. Li, Error estimates of the Markov finite approximation of the Frobenius-Perron operator, Nonlinear Anal., 19 (1992), 291-308.
doi: 10.1016/0362-546X(92)90175-E. |
[6] |
A. Chorin, Vorticity and Turbulence, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8728-0. |
[7] |
P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988. |
[8] |
W. E and D. Li, The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136.
doi: 10.1002/cpa.20198. |
[9] |
C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613.
doi: 10.1088/0951-7715/4/3/001. |
[10] |
C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Phys. Lett. A, 186 (1994), 87-96.
doi: 10.1016/0375-9601(94)90926-1. |
[11] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511546754. |
[12] |
T. Geveci, On the convergence of a time discretization scheme for the Navier-Stokes equations, Math. Comp., 53 (1989), 43-53.
doi: 10.1090/S0025-5718-1989-0969488-5. |
[13] |
S. Gottlieb, F. Tone, C. Wang, X. Wang and D. Wirosoetisno, Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 50 (2012), 126-150.
doi: 10.1137/110834901. |
[14] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, R.I. : American Mathematical Society, 1988. |
[15] |
J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23 (1986), 750-777.
doi: 10.1137/0723049. |
[16] |
A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equation, IMA J. Numer. Anal., 20 (2000), 663-667.
doi: 10.1093/imanum/20.4.633. |
[17] |
D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, J. Math. Anal. Appl., 219 (1998), 479-502.
doi: 10.1006/jmaa.1997.5847. |
[18] |
N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597.
doi: 10.1093/imanum/22.4.577. |
[19] |
L. P. Kadanoff, Turbulent heat flow: Structures and scaling, Physics Today, 54 (2001), 34-39.
doi: 10.1063/1.1404847. |
[20] |
S. Larsson, The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems, SIAM J. Numer. Anal., 26 (1989), 348-365.
doi: 10.1137/0726019. |
[21] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd, ed. New York, Springer-Verlag, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[22] | |
[23] |
P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), 267-293.
doi: 10.1002/cpa.3160090206. |
[24] |
G. J. Lord, Attractors and inertial manifolds for finite-difference approximation of the complex Ginzburg-Landau equation, SIAM J. Numer. Anal., 34 (1997), 1483-1512.
doi: 10.1137/S003614299528554X. |
[25] |
G. J. Lord and A. M. Stuart, Discrete Gevrey regularity, attractors and upper-semicontinuity for a finite-difference approximation to the Ginzburg-Landau equation, Numer. Funct. Anal. Optim., 16 (1995), 1003-1047.
doi: 10.1080/01630569508816658. |
[26] |
A. J. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, England, 2002. |
[27] |
A. J. Majda and X. Wang, Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows, Cambridge University Press, Cambridge, England, 2006.
doi: 10.1017/CBO9780511616778. |
[28] |
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics; Mechanics of Turbulence, English ed. updated, augmented and rev. by the authors. MIT Press, Cambridge, Mass., 1975. |
[29] |
G. Raugel, Global attractors in partial differential equations, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 885-982.
doi: 10.1016/S1874-575X(02)80038-8. |
[30] |
S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570.
doi: 10.1137/S0036142997329797. |
[31] |
J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. and Optimiz., 10 (1989), 1213-1234.
doi: 10.1080/01630568908816354. |
[32] |
J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.
doi: 10.1080/00036819008839963. |
[33] |
H. Sigurgeirsson and A. M. Stuart, Statistics from computations, in Foundations of Computational Mathematics, Edited by R. Devore, A. Iserles and E. Suli, Cambridge University Press, 284 (2001), 323-344. |
[34] |
A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996. |
[35] |
D. Talay, Simulation of stochastic differential systems, in Probabilistic Methods in Applied Physics, P. Kree and W. Wedig (Eds), Lecture Notes in Physics, Springer-Verlag, 451 (1995), 54-96.
doi: 10.1007/3-540-60214-3_51. |
[36] |
R. M. Temam, Sur l'approximation des solutions des équations de Navier-Stokes, C.R. Acad. Sci., Paris, Serie A, 262 (1966), 219-221. |
[37] |
R. M. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, CBMS-SIAM, SIAM, 1995.
doi: 10.1137/1.9781611970050. |
[38] |
R. M. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[39] |
F. Tone and X. Wang, Approximation of the stationary statistical properties of the dynamical systems generated by the two-dimensional Rayleigh-Benard convection problem, Analysis and Applications, 9 (2011), 421-446.
doi: 10.1142/S0219530511001935. |
[40] |
F. Tone, X. Wang and D. Wirosoetisno, Long-time dynamics of 2d double-diffusive convection: Analysis and/of numerics, Numer. Math., 130 (2015), 541-566.
doi: 10.1007/s00211-014-0670-9. |
[41] |
F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations, SIAM J. Num. Anal., 44 (2006), 29-40.
doi: 10.1137/040618527. |
[42] |
P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems, SIAM J. Applied Dynamical Systems, 4 (2005), 563-587.
doi: 10.1137/040603802. |
[43] |
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Acad. Publishers, Dordrecht/Boston/London, 1988.
doi: 10.1007/978-94-009-1423-0. |
[44] |
P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. |
[45] |
X. Wang, Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure and Appl. Math., 57 (2004), 1265-1282.
doi: 10.1002/cpa.3047. |
[46] |
X. Wang, Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number, Comm. Pure and Appl. Math., 61 (2008), 789-815.
doi: 10.1002/cpa.20214. |
[47] |
X. Wang, Upper Semi-Continuity of Stationary Statistical Properties of Dissipative Systems, Dedicated to Prof. Li Ta-Tsien on the occasion of his 70th birthday, Discrete and Continuous Dynamical Systems, A, 23 (2009), 521-540.
doi: 10.3934/dcds.2009.23.521. |
[48] |
X. Wang, Approximating stationary statistical properties, Chinese Ann. Math. Series B, (an invited article in a special issue dedicated to Andy Majda), 30 (2009), 831-844.
doi: 10.1007/s11401-009-0178-2. |
[49] |
X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp., 79 (2010), 259-280.
doi: 10.1090/S0025-5718-09-02256-X. |
[50] |
X. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations, Numer. Math., 121 (2012), 753-779.
doi: 10.1007/s00211-012-0450-3. |
[51] |
Y. Yan, Attractors and error estimates for discretizations of incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 33 (1996), 1451-1472.
doi: 10.1137/S0036142993248092. |
show all references
References:
[1] |
P. Billingsley, Weak Convergence of Measures: Applications in Probability, SIAM, Philadelphia, 1971. |
[2] |
E. Cancs, E. Legoll and G. Stoltz, Theoretical and numerical comparison of some sampling methods for molecular dynamics, ESAIM: Mathematical Modelling and Numerical Analysis, 41 (2007), 351-389.
doi: 10.1051/m2an:2007014. |
[3] |
W. Cheng and X. Wang, A uniformly dissipative scheme for stationary statistical properties of the infinite prandtl number model, Applied Mathematics Letters, 21 (2008), 1281-1285.
doi: 10.1016/j.aml.2007.07.036. |
[4] |
W. Cheng and X. Wang, A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model, SIAM J. Num. Anal., 47 (2008), 250-270.
doi: 10.1137/080713501. |
[5] |
C. Chiu, Q. Du and T. Y. Li, Error estimates of the Markov finite approximation of the Frobenius-Perron operator, Nonlinear Anal., 19 (1992), 291-308.
doi: 10.1016/0362-546X(92)90175-E. |
[6] |
A. Chorin, Vorticity and Turbulence, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8728-0. |
[7] |
P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988. |
[8] |
W. E and D. Li, The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136.
doi: 10.1002/cpa.20198. |
[9] |
C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613.
doi: 10.1088/0951-7715/4/3/001. |
[10] |
C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Phys. Lett. A, 186 (1994), 87-96.
doi: 10.1016/0375-9601(94)90926-1. |
[11] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511546754. |
[12] |
T. Geveci, On the convergence of a time discretization scheme for the Navier-Stokes equations, Math. Comp., 53 (1989), 43-53.
doi: 10.1090/S0025-5718-1989-0969488-5. |
[13] |
S. Gottlieb, F. Tone, C. Wang, X. Wang and D. Wirosoetisno, Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 50 (2012), 126-150.
doi: 10.1137/110834901. |
[14] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, R.I. : American Mathematical Society, 1988. |
[15] |
J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23 (1986), 750-777.
doi: 10.1137/0723049. |
[16] |
A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equation, IMA J. Numer. Anal., 20 (2000), 663-667.
doi: 10.1093/imanum/20.4.633. |
[17] |
D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, J. Math. Anal. Appl., 219 (1998), 479-502.
doi: 10.1006/jmaa.1997.5847. |
[18] |
N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597.
doi: 10.1093/imanum/22.4.577. |
[19] |
L. P. Kadanoff, Turbulent heat flow: Structures and scaling, Physics Today, 54 (2001), 34-39.
doi: 10.1063/1.1404847. |
[20] |
S. Larsson, The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems, SIAM J. Numer. Anal., 26 (1989), 348-365.
doi: 10.1137/0726019. |
[21] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd, ed. New York, Springer-Verlag, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[22] | |
[23] |
P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), 267-293.
doi: 10.1002/cpa.3160090206. |
[24] |
G. J. Lord, Attractors and inertial manifolds for finite-difference approximation of the complex Ginzburg-Landau equation, SIAM J. Numer. Anal., 34 (1997), 1483-1512.
doi: 10.1137/S003614299528554X. |
[25] |
G. J. Lord and A. M. Stuart, Discrete Gevrey regularity, attractors and upper-semicontinuity for a finite-difference approximation to the Ginzburg-Landau equation, Numer. Funct. Anal. Optim., 16 (1995), 1003-1047.
doi: 10.1080/01630569508816658. |
[26] |
A. J. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, England, 2002. |
[27] |
A. J. Majda and X. Wang, Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows, Cambridge University Press, Cambridge, England, 2006.
doi: 10.1017/CBO9780511616778. |
[28] |
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics; Mechanics of Turbulence, English ed. updated, augmented and rev. by the authors. MIT Press, Cambridge, Mass., 1975. |
[29] |
G. Raugel, Global attractors in partial differential equations, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 885-982.
doi: 10.1016/S1874-575X(02)80038-8. |
[30] |
S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570.
doi: 10.1137/S0036142997329797. |
[31] |
J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. and Optimiz., 10 (1989), 1213-1234.
doi: 10.1080/01630568908816354. |
[32] |
J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.
doi: 10.1080/00036819008839963. |
[33] |
H. Sigurgeirsson and A. M. Stuart, Statistics from computations, in Foundations of Computational Mathematics, Edited by R. Devore, A. Iserles and E. Suli, Cambridge University Press, 284 (2001), 323-344. |
[34] |
A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996. |
[35] |
D. Talay, Simulation of stochastic differential systems, in Probabilistic Methods in Applied Physics, P. Kree and W. Wedig (Eds), Lecture Notes in Physics, Springer-Verlag, 451 (1995), 54-96.
doi: 10.1007/3-540-60214-3_51. |
[36] |
R. M. Temam, Sur l'approximation des solutions des équations de Navier-Stokes, C.R. Acad. Sci., Paris, Serie A, 262 (1966), 219-221. |
[37] |
R. M. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, CBMS-SIAM, SIAM, 1995.
doi: 10.1137/1.9781611970050. |
[38] |
R. M. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[39] |
F. Tone and X. Wang, Approximation of the stationary statistical properties of the dynamical systems generated by the two-dimensional Rayleigh-Benard convection problem, Analysis and Applications, 9 (2011), 421-446.
doi: 10.1142/S0219530511001935. |
[40] |
F. Tone, X. Wang and D. Wirosoetisno, Long-time dynamics of 2d double-diffusive convection: Analysis and/of numerics, Numer. Math., 130 (2015), 541-566.
doi: 10.1007/s00211-014-0670-9. |
[41] |
F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations, SIAM J. Num. Anal., 44 (2006), 29-40.
doi: 10.1137/040618527. |
[42] |
P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems, SIAM J. Applied Dynamical Systems, 4 (2005), 563-587.
doi: 10.1137/040603802. |
[43] |
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Acad. Publishers, Dordrecht/Boston/London, 1988.
doi: 10.1007/978-94-009-1423-0. |
[44] |
P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. |
[45] |
X. Wang, Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure and Appl. Math., 57 (2004), 1265-1282.
doi: 10.1002/cpa.3047. |
[46] |
X. Wang, Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number, Comm. Pure and Appl. Math., 61 (2008), 789-815.
doi: 10.1002/cpa.20214. |
[47] |
X. Wang, Upper Semi-Continuity of Stationary Statistical Properties of Dissipative Systems, Dedicated to Prof. Li Ta-Tsien on the occasion of his 70th birthday, Discrete and Continuous Dynamical Systems, A, 23 (2009), 521-540.
doi: 10.3934/dcds.2009.23.521. |
[48] |
X. Wang, Approximating stationary statistical properties, Chinese Ann. Math. Series B, (an invited article in a special issue dedicated to Andy Majda), 30 (2009), 831-844.
doi: 10.1007/s11401-009-0178-2. |
[49] |
X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp., 79 (2010), 259-280.
doi: 10.1090/S0025-5718-09-02256-X. |
[50] |
X. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations, Numer. Math., 121 (2012), 753-779.
doi: 10.1007/s00211-012-0450-3. |
[51] |
Y. Yan, Attractors and error estimates for discretizations of incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 33 (1996), 1451-1472.
doi: 10.1137/S0036142993248092. |
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