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Telescoping method, summation formulas, and inversion pairs
Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations
1. | School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634, USA |
2. | Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA |
We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.
References:
[1] |
M. Akbas, S. Kaya and L. G. Rebholz,
On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems, Numer. Methods Partial Differential Equations, 33 (2017), 995-1017.
doi: 10.1002/num.22061. |
[2] |
M. Akbas, L. G. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations, Calcolo, 55 (2018), Paper No. 3, 29 pp. 1–29.
doi: 10.1007/s10092-018-0246-7. |
[3] |
D. A. F. Albanez, H. Nussenzveig Lopes and E. S. Titi,
Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model, Asymptotic Anal., 97 (2016), 139-164.
doi: 10.3233/ASY-151351. |
[4] |
R. A. Anthes,
Data assimilation and initialization of hurricane prediction models, J. Atmos. Sci., 31 (1974), 702-719.
doi: 10.1175/1520-0469(1974)031<0702:DAAIOH>2.0.CO;2. |
[5] |
A. Azouani, E. Olson and E. S. Titi,
Continuous data assimilation using general interpolant observables, Journal of Nonlinear Science, 24 (2014), 277-304.
doi: 10.1007/s00332-013-9189-y. |
[6] |
H. Bessaih, E. Olson and E. S. Titi,
Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753.
doi: 10.1088/0951-7715/28/3/729. |
[7] |
A. Biswas, C. Foias, C. F. Mondaini and E. S. Titi,
Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 295-326.
doi: 10.1016/j.anihpc.2018.05.004. |
[8] |
A. Biswas, J. Hudson, A. Larios and Y. Pei,
Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields, Asymptot. Anal., 108 (2018), 1-43.
|
[9] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[10] |
E. Carlson, J. Hudson and A. Larios, Parameter recovery for the 2 dimensional Navier-Stokes equations via continuous data assimilation, SIAM J. Sci. Comput., 42 (2020), A250–A270.
doi: 10.1137/19M1248583. |
[11] |
E. Celik, E. Olson and E. S. Titi,
Spectral filtering of interpolant observables for a discrete-in-time downscaling data assimilation algorithm, SIAM J. Appl. Dyn. Syst., 18 (2019), 1118-1142.
doi: 10.1137/18M1218480. |
[12] |
T. Charnyi, T. Heister, M. A. Olshanskii and L. G. Rebholz,
On conservation laws of Navier-Stokes Galerkin discretizations, Journal of Computational Physics, 337 (2017), 289-308.
doi: 10.1016/j.jcp.2017.02.039. |
[13] |
P. Clark Di Leoni, A. Mazzino and L. Biferale, Synchronization to big-data: Nudging the Navier-Stokes equations for data assimilation of turbulent flows, Physical Review X, 10 (2020), 1-15. Google Scholar |
[14] |
R. Daley, Atmospheric Data Analysis, Cambridge Atmospheric and Space Science Series, Cambridge University Press, 1993.
doi: 10.4267/2042/51948. |
[15] |
S. Desamsetti, I. Hoteit, O. Knio, E. Titi, S. Langodan and H. P. Dasari,
Efficient dynamical downscaling of general circulation models using continuous data assimilation, Quarterly Journal of the Royal Meteorological Society, 145 (2019), 3175-3194.
doi: 10.1002/qj.3612. |
[16] |
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
[17] |
A. Farhat, N. E. Glatt-Holtz, V. R. Martinez, S. A. McQuarrie and J. P. Whitehead,
Data assimilation in large Prandtl Rayleigh–Bénard convection from thermal measurements, SIAM J. Appl. Dyn. Syst., 19 (2020), 510-540.
doi: 10.1137/19M1248327. |
[18] |
A. Farhat, M. S. Jolly and E. S. Titi,
Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66.
doi: 10.1016/j.physd.2015.03.011. |
[19] |
A. Farhat, E. Lunasin and E. S. Titi, A data assimilation algorithm: The paradigm of the 3D Leray-α model of turbulence, Partial Differential Equations Arising from Physics and Geometry, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 450 (2019), 253-273. |
[20] |
C. Foias, C. F. Mondaini and E. S. Titi,
A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15 (2016), 2109-2142.
doi: 10.1137/16M1076526. |
[21] |
B. Garcia-Archilla, J. Novo and E. S. Titi,
Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations, SIAM Journal on Numerical Analysis, 58 (2020), 410-429.
doi: 10.1137/19M1246845. |
[22] |
M. Gesho, E. Olson and E. S. Titi,
A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110.
doi: 10.4208/cicp.060515.161115a. |
[23] |
P. Gresho and R. Sani, Incompressible Flow and the Finite Element Method, Vol. 2, Wiley, 1998. Google Scholar |
[24] |
J. Guzman and L. R. Scott,
The Scott-Vogelius finite elements revisited, Math. Comp., 88 (2019), 515-529.
doi: 10.1090/mcom/3346. |
[25] |
T. Heister, M. A. Olshanskii and L. G. Rebholz,
Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135 (2017), 143-167.
doi: 10.1007/s00211-016-0794-1. |
[26] |
J. E. Hoke and R. A. Anthes,
The initialization of numerical models by a dynamic-initialization technique, Monthly Weather Review, 104 (1976), 1551-1556.
doi: 10.1175/1520-0493(1976)104<1551:TIONMB>2.0.CO;2. |
[27] |
H. A. Ibdah, C. F. Mondaini and E. S. Titi, Fully discrete numerical schemes of a data assimilation algorithm: Uniform-in-time error estimates, IMA Journal of Numerical Analysis, Drz043, (2019).
doi: 10.1093/imanum/drz043. |
[28] |
N. Jiang,
A second order ensemble method based on a blended BDF time-stepping scheme for time dependent Navier-Stokes equations, Numerical Methods for Partial Differential Equations, 33 (2017), 34-61.
doi: 10.1002/num.22070. |
[29] |
R. E. Kalman,
A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.
doi: 10.1115/1.3662552. |
[30] |
A. Larios, L. G. Rebholz and C. Zerfas,
Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 345 (2019), 1077-1093.
doi: 10.1016/j.cma.2018.09.004. |
[31] |
A. Larios and C. Victor, Continuous data assimilation with a moving cluster of data points for a reaction diffusion equation: A computational study, Commun. Comp. Phys., (accepted for publication). Google Scholar |
[32] |
K. Law, A. Stuart and K. Zygalakis, A Mathematical Introduction to Data Assimilation, Texts in Applied Mathematics, 62. Springer, Cham, 2015.
doi: 10.1007/978-3-319-20325-6. |
[33] |
W. Layton, C. C. Manica, M. Neda, M. Olshanskii and L. G. Rebholz,
On the accuracy of the rotation form in simulations of the Navier-Stokes equations, Journal of Computational Physics, 228 (2009), 3433-3447.
doi: 10.1016/j.jcp.2009.01.027. |
[34] |
H. K. Lee, M. A. Olshanskii and L. G. Rebholz,
On error analysis for the 3D Navier-Stokes equations in velocity-vorticity-helicity form, SIAM Journal on Numerical Analysis, 49 (2011), 711-732.
doi: 10.1137/10080124X. |
[35] |
C. F. Mondaini and E. S. Titi,
Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM J. Numer. Anal., 56 (2018), 78-110.
doi: 10.1137/16M110962X. |
[36] |
M. A. Olshanskii, T. Heister, L. G. Rebholz and K. J. Galvin,
Natural vorticity boundary conditions on solid walls, Computer Methods in Applied Mechanics and Engineering, 297 (2015), 18-37.
doi: 10.1016/j.cma.2015.08.011. |
[37] |
M. A. Olshanskii and L. G. Rebholz,
Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations, Journal of Computational Physics, 229 (2010), 4291-4303.
doi: 10.1016/j.jcp.2010.02.012. |
[38] |
M. A. Olshanskii, L. G. Rebholz and A. J. Salgado,
On well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions, Discrete Contin. Dyn. Syst., 38 (2018), 3459-3477.
doi: 10.3934/dcds.2018148. |
[39] |
M. A. Olshanskii and A. Reusken,
Grad-div stabilization for the Stokes equations, Math. Comp., 73 (2004), 1699-1718.
doi: 10.1090/S0025-5718-03-01629-6. |
[40] |
Y. Pei,
Continuous data assimilation for the 3D primitive equations of the ocean, Comm. Pure Appl. Math., 18 (2019), 643-661.
doi: 10.3934/cpaa.2019032. |
[41] |
L. Rebholz and C. Zerfas, Simple and efficient continuous data assimilation of evolution equations via algebraic nudging, Submitted. Google Scholar |
[42] |
P. W. Schroeder, C. Lehrenfeld, A. Linke and G. Lube,
Towards computable flows and robust estimates for inf-sup stable fem applied to the time dependent incompressible Navier-Stokes equations, SeMA J., 75 (2018), 629-653.
doi: 10.1007/s40324-018-0157-1. |
[43] |
C. Zerfas, Numerical Methods and Analysis for Continuous Data Assimilation in Fluid Models, PhD thesis, Clemson University, 2019,132 pp, https://tigerprints.clemson.edu/all_dissertations/2428. |
[44] |
C. Zerfas, L. G. Rebholz, M. Schneier and T. Iliescu, Continuous data assimilation reduced order models of fluid flow, Computer Methods in Applied Mechanics and Engineering, 357 (2019), 112596, 18 pp.
doi: 10.1016/j.cma.2019.112596. |
show all references
References:
[1] |
M. Akbas, S. Kaya and L. G. Rebholz,
On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems, Numer. Methods Partial Differential Equations, 33 (2017), 995-1017.
doi: 10.1002/num.22061. |
[2] |
M. Akbas, L. G. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations, Calcolo, 55 (2018), Paper No. 3, 29 pp. 1–29.
doi: 10.1007/s10092-018-0246-7. |
[3] |
D. A. F. Albanez, H. Nussenzveig Lopes and E. S. Titi,
Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model, Asymptotic Anal., 97 (2016), 139-164.
doi: 10.3233/ASY-151351. |
[4] |
R. A. Anthes,
Data assimilation and initialization of hurricane prediction models, J. Atmos. Sci., 31 (1974), 702-719.
doi: 10.1175/1520-0469(1974)031<0702:DAAIOH>2.0.CO;2. |
[5] |
A. Azouani, E. Olson and E. S. Titi,
Continuous data assimilation using general interpolant observables, Journal of Nonlinear Science, 24 (2014), 277-304.
doi: 10.1007/s00332-013-9189-y. |
[6] |
H. Bessaih, E. Olson and E. S. Titi,
Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753.
doi: 10.1088/0951-7715/28/3/729. |
[7] |
A. Biswas, C. Foias, C. F. Mondaini and E. S. Titi,
Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 295-326.
doi: 10.1016/j.anihpc.2018.05.004. |
[8] |
A. Biswas, J. Hudson, A. Larios and Y. Pei,
Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields, Asymptot. Anal., 108 (2018), 1-43.
|
[9] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[10] |
E. Carlson, J. Hudson and A. Larios, Parameter recovery for the 2 dimensional Navier-Stokes equations via continuous data assimilation, SIAM J. Sci. Comput., 42 (2020), A250–A270.
doi: 10.1137/19M1248583. |
[11] |
E. Celik, E. Olson and E. S. Titi,
Spectral filtering of interpolant observables for a discrete-in-time downscaling data assimilation algorithm, SIAM J. Appl. Dyn. Syst., 18 (2019), 1118-1142.
doi: 10.1137/18M1218480. |
[12] |
T. Charnyi, T. Heister, M. A. Olshanskii and L. G. Rebholz,
On conservation laws of Navier-Stokes Galerkin discretizations, Journal of Computational Physics, 337 (2017), 289-308.
doi: 10.1016/j.jcp.2017.02.039. |
[13] |
P. Clark Di Leoni, A. Mazzino and L. Biferale, Synchronization to big-data: Nudging the Navier-Stokes equations for data assimilation of turbulent flows, Physical Review X, 10 (2020), 1-15. Google Scholar |
[14] |
R. Daley, Atmospheric Data Analysis, Cambridge Atmospheric and Space Science Series, Cambridge University Press, 1993.
doi: 10.4267/2042/51948. |
[15] |
S. Desamsetti, I. Hoteit, O. Knio, E. Titi, S. Langodan and H. P. Dasari,
Efficient dynamical downscaling of general circulation models using continuous data assimilation, Quarterly Journal of the Royal Meteorological Society, 145 (2019), 3175-3194.
doi: 10.1002/qj.3612. |
[16] |
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
[17] |
A. Farhat, N. E. Glatt-Holtz, V. R. Martinez, S. A. McQuarrie and J. P. Whitehead,
Data assimilation in large Prandtl Rayleigh–Bénard convection from thermal measurements, SIAM J. Appl. Dyn. Syst., 19 (2020), 510-540.
doi: 10.1137/19M1248327. |
[18] |
A. Farhat, M. S. Jolly and E. S. Titi,
Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66.
doi: 10.1016/j.physd.2015.03.011. |
[19] |
A. Farhat, E. Lunasin and E. S. Titi, A data assimilation algorithm: The paradigm of the 3D Leray-α model of turbulence, Partial Differential Equations Arising from Physics and Geometry, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 450 (2019), 253-273. |
[20] |
C. Foias, C. F. Mondaini and E. S. Titi,
A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15 (2016), 2109-2142.
doi: 10.1137/16M1076526. |
[21] |
B. Garcia-Archilla, J. Novo and E. S. Titi,
Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations, SIAM Journal on Numerical Analysis, 58 (2020), 410-429.
doi: 10.1137/19M1246845. |
[22] |
M. Gesho, E. Olson and E. S. Titi,
A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110.
doi: 10.4208/cicp.060515.161115a. |
[23] |
P. Gresho and R. Sani, Incompressible Flow and the Finite Element Method, Vol. 2, Wiley, 1998. Google Scholar |
[24] |
J. Guzman and L. R. Scott,
The Scott-Vogelius finite elements revisited, Math. Comp., 88 (2019), 515-529.
doi: 10.1090/mcom/3346. |
[25] |
T. Heister, M. A. Olshanskii and L. G. Rebholz,
Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135 (2017), 143-167.
doi: 10.1007/s00211-016-0794-1. |
[26] |
J. E. Hoke and R. A. Anthes,
The initialization of numerical models by a dynamic-initialization technique, Monthly Weather Review, 104 (1976), 1551-1556.
doi: 10.1175/1520-0493(1976)104<1551:TIONMB>2.0.CO;2. |
[27] |
H. A. Ibdah, C. F. Mondaini and E. S. Titi, Fully discrete numerical schemes of a data assimilation algorithm: Uniform-in-time error estimates, IMA Journal of Numerical Analysis, Drz043, (2019).
doi: 10.1093/imanum/drz043. |
[28] |
N. Jiang,
A second order ensemble method based on a blended BDF time-stepping scheme for time dependent Navier-Stokes equations, Numerical Methods for Partial Differential Equations, 33 (2017), 34-61.
doi: 10.1002/num.22070. |
[29] |
R. E. Kalman,
A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.
doi: 10.1115/1.3662552. |
[30] |
A. Larios, L. G. Rebholz and C. Zerfas,
Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 345 (2019), 1077-1093.
doi: 10.1016/j.cma.2018.09.004. |
[31] |
A. Larios and C. Victor, Continuous data assimilation with a moving cluster of data points for a reaction diffusion equation: A computational study, Commun. Comp. Phys., (accepted for publication). Google Scholar |
[32] |
K. Law, A. Stuart and K. Zygalakis, A Mathematical Introduction to Data Assimilation, Texts in Applied Mathematics, 62. Springer, Cham, 2015.
doi: 10.1007/978-3-319-20325-6. |
[33] |
W. Layton, C. C. Manica, M. Neda, M. Olshanskii and L. G. Rebholz,
On the accuracy of the rotation form in simulations of the Navier-Stokes equations, Journal of Computational Physics, 228 (2009), 3433-3447.
doi: 10.1016/j.jcp.2009.01.027. |
[34] |
H. K. Lee, M. A. Olshanskii and L. G. Rebholz,
On error analysis for the 3D Navier-Stokes equations in velocity-vorticity-helicity form, SIAM Journal on Numerical Analysis, 49 (2011), 711-732.
doi: 10.1137/10080124X. |
[35] |
C. F. Mondaini and E. S. Titi,
Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM J. Numer. Anal., 56 (2018), 78-110.
doi: 10.1137/16M110962X. |
[36] |
M. A. Olshanskii, T. Heister, L. G. Rebholz and K. J. Galvin,
Natural vorticity boundary conditions on solid walls, Computer Methods in Applied Mechanics and Engineering, 297 (2015), 18-37.
doi: 10.1016/j.cma.2015.08.011. |
[37] |
M. A. Olshanskii and L. G. Rebholz,
Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations, Journal of Computational Physics, 229 (2010), 4291-4303.
doi: 10.1016/j.jcp.2010.02.012. |
[38] |
M. A. Olshanskii, L. G. Rebholz and A. J. Salgado,
On well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions, Discrete Contin. Dyn. Syst., 38 (2018), 3459-3477.
doi: 10.3934/dcds.2018148. |
[39] |
M. A. Olshanskii and A. Reusken,
Grad-div stabilization for the Stokes equations, Math. Comp., 73 (2004), 1699-1718.
doi: 10.1090/S0025-5718-03-01629-6. |
[40] |
Y. Pei,
Continuous data assimilation for the 3D primitive equations of the ocean, Comm. Pure Appl. Math., 18 (2019), 643-661.
doi: 10.3934/cpaa.2019032. |
[41] |
L. Rebholz and C. Zerfas, Simple and efficient continuous data assimilation of evolution equations via algebraic nudging, Submitted. Google Scholar |
[42] |
P. W. Schroeder, C. Lehrenfeld, A. Linke and G. Lube,
Towards computable flows and robust estimates for inf-sup stable fem applied to the time dependent incompressible Navier-Stokes equations, SeMA J., 75 (2018), 629-653.
doi: 10.1007/s40324-018-0157-1. |
[43] |
C. Zerfas, Numerical Methods and Analysis for Continuous Data Assimilation in Fluid Models, PhD thesis, Clemson University, 2019,132 pp, https://tigerprints.clemson.edu/all_dissertations/2428. |
[44] |
C. Zerfas, L. G. Rebholz, M. Schneier and T. Iliescu, Continuous data assimilation reduced order models of fluid flow, Computer Methods in Applied Mechanics and Engineering, 357 (2019), 112596, 18 pp.
doi: 10.1016/j.cma.2019.112596. |









h | rate | rate | ||
1/4 | 2.62008e-03 | - | 7.70647e-03 | - |
1/8 | 3.20467e-04 | 3.0314 | 9.68456e-04 | 2.9923 |
1/16 | 3.97307e-05 | 3.0146 | 1.20888e-04 | 3.0041 |
1/32 | 4.94529e-06 | 3.0061 | 1.50809e-05 | 3.0029 |
1/64 | 6.19332e-07 | 2.9973 | 1.99325e-06 | 2.9195 |
1/128 | 8.13141e-08 | 2.9247 | 3.15236e-07 | 2.5855 |
h | rate | rate | ||
1/4 | 2.62008e-03 | - | 7.70647e-03 | - |
1/8 | 3.20467e-04 | 3.0314 | 9.68456e-04 | 2.9923 |
1/16 | 3.97307e-05 | 3.0146 | 1.20888e-04 | 3.0041 |
1/32 | 4.94529e-06 | 3.0061 | 1.50809e-05 | 3.0029 |
1/64 | 6.19332e-07 | 2.9973 | 1.99325e-06 | 2.9195 |
1/128 | 8.13141e-08 | 2.9247 | 3.15236e-07 | 2.5855 |
h | rate | rate | ||
1/4 | 2.62003e-03 | - | 7.79431e-03 | - |
1/8 | 3.20466e-04 | 3.0313 | 9.70492e-04 | 3.0056 |
1/16 | 3.97175e-05 | 3.0123 | 1.20897e-04 | 3.0049 |
1/32 | 4.94501e-06 | 3.0057 | 1.50883e-05 | 3.0023 |
1/64 | 6.17406e-07 | 3.0017 | 2.08215e-06 | 2.8573 |
1/128 | 8.11244e-08 | 2.9280 | 9.37122e-07 | 1.1518 |
h | rate | rate | ||
1/4 | 2.62003e-03 | - | 7.79431e-03 | - |
1/8 | 3.20466e-04 | 3.0313 | 9.70492e-04 | 3.0056 |
1/16 | 3.97175e-05 | 3.0123 | 1.20897e-04 | 3.0049 |
1/32 | 4.94501e-06 | 3.0057 | 1.50883e-05 | 3.0023 |
1/64 | 6.17406e-07 | 3.0017 | 2.08215e-06 | 2.8573 |
1/128 | 8.11244e-08 | 2.9280 | 9.37122e-07 | 1.1518 |
[1] |
Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 |
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