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doi: 10.3934/era.2021019
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Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747, USA

Received  October 2020 Revised  January 2021 Early access March 2021

The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.

Citation: Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, doi: 10.3934/era.2021019
References:
[1]

L. Abia and J. M. Sanz-Serna, The spectral accuracy of a fully-discrete scheme for a nonlinear third order equation, Computing, 44 (1990), 187-196.  doi: 10.1007/BF02262215.  Google Scholar

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J. B. BellP. Colella and H. M. Glaz, A second order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), 257-283.  doi: 10.1016/0021-9991(89)90151-4.  Google Scholar

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W. ChenS. CondeC. WangX. Wang and S. M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.  doi: 10.1007/s10915-011-9559-2.  Google Scholar

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W. ChenW. LiZ. LuoC. Wang and X. Wang, A stabilized second order ETD multistep method for thin film growth model without slope selection, ESAIM Math. Model. Numer. Anal., 54 (2020), 727-750.  doi: 10.1051/m2an/2019054.  Google Scholar

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W. Chen, W. Li, C. Wang, S. Wang and X. Wang, Energy stable higher order linear ETD multi-step methods for gradient flows: Application to thin film epitaxy, Res. Math. Sci., 7 (2020), Paper No. 13, 27 pp. doi: 10.1007/s40687-020-00212-9.  Google Scholar

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W. Chen, C. Wang, S. Wang, X. Wang and S. M. Wise, Energy stable numerical schemes for ternary Cahn-Hilliard system, J. Sci. Comput., 84 (2020), Paper No. 27, 36 pp. doi: 10.1007/s10915-020-01276-z.  Google Scholar

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W. ChenC. WangX. Wang and S. M. Wise, A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection., J. Sci. Comput., 59 (2014), 574-601.  doi: 10.1007/s10915-013-9774-0.  Google Scholar

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K. ChengW. FengS. Gottlieb and C. Wang, A Fourier pseudospectral method for the "Good" Boussinesq equation with second-order temporal accuracy, Numer. Methods Partial Differential Equations, 31 (2015), 202-224.  doi: 10.1002/num.21899.  Google Scholar

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K. ChengC. Wang and S. M. Wise, An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation, Commun. Comput. Phys., 26 (2019), 1335-1364.  doi: 10.4208/cicp.2019.js60.10.  Google Scholar

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K. ChengC. WangS. M. Wise and X. Yue, A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn-Hilliard equation and its solution by the homogeneous linear iteration method, J. Sci. Comput., 69 (2016), 1083-1114.  doi: 10.1007/s10915-016-0228-3.  Google Scholar

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M. Crouzeix, Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques, Numer. Math., 35 (1980), 257-276.  doi: 10.1007/BF01396412.  Google Scholar

[20]

J. De FrutosT. Ortega and J. M. Sanz-Serna, Pseudo-spectral method for the "Good" boussinesq equation, Math. Comp., 57 (1991), 109-122.  doi: 10.2307/2938665.  Google Scholar

[21]

W. E, Convergence of Fourier methods for Navier-Stokes equations, SIAM J. Numer. Anal., 30 (1993), 650-674.  doi: 10.1137/0730032.  Google Scholar

[22]

W. E, Convergence of spectral methods for the {Burgers'} equation, SIAM J. Numer. Anal., 29 (1992), 1520-1541.  doi: 10.1137/0729088.  Google Scholar

[23]

W. E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057.  doi: 10.1137/0732047.  Google Scholar

[24]

W. E and J.-G. Liu, Gauge finite element method for incompressible flows, Int. J. Num. Meth. Fluids, 34 (2000), 701-710.  doi: 10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B.  Google Scholar

[25]

W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Commu. Math. Sci., 1 (2003), 317-332.  doi: 10.4310/CMS.2003.v1.n2.a6.  Google Scholar

[26]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods, Theory and Applications, SIAM, Philadelphia, PA, 1977.  Google Scholar

[27]

S. GottliebF. ToneC. WangX. 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.  Google Scholar

[28]

S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers' equation, J. Sci. Comput., 53 (2012), 102-128.  doi: 10.1007/s10915-012-9621-8.  Google Scholar

[29]

B. Y. Guo, A spectral method for the vorticity equation on the surface, Math. Comp., 64 (1995), 1067-1079.  doi: 10.1090/S0025-5718-1995-1297463-5.  Google Scholar

[30]

B.-Y. Guo and W. Huang, Mixed Jacobi-Spherical harmonic spectral method for Navier-Stokes equations, Appl. Numer. Math., 57 (2007), 939-961.  doi: 10.1016/j.apnum.2006.09.003.  Google Scholar

[31]

B. Y. Guo and J. Zou, Fourier spectral projection method and nonlinear convergence analysis for Navier-Stokes equation, J. Math. Anal. Appl., 282 (2003), 766-791.  doi: 10.1016/S0022-247X(03)00254-3.  Google Scholar

[32]

Y. HaoQ. Huang and C. Wang, A third order BDF energy stable linear scheme for the no-slope-selection thin film model, Commun. Comput. Phys., 29 (2021), 905-929.  doi: 10.4208/cicp.OA-2020-0074.  Google Scholar

[33]

H. Johnston and J.-G. Liu, A finite difference scheme for incompressible flow based on local pressure boundary conditions, J. Comput. Phys., 180 (2002), 120-154.  doi: 10.1006/jcph.2002.7079.  Google Scholar

[34]

H. Johnston and J.-G. Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys., 199 (2004), 221-259.  doi: 10.1016/j.jcp.2004.02.009.  Google Scholar

[35]

G. E. KarniadakisM. Israeli and S. A. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414-443.  doi: 10.1016/0021-9991(91)90007-8.  Google Scholar

[36]

J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.  doi: 10.1016/0021-9991(85)90148-2.  Google Scholar

[37]

Y. Maday and A. Quarteroni, Spectral and pseudospectral approximation of the Navier-Stokes equations, SIAM J. Numer. Anal., 19 (1982), 761-780.  doi: 10.1137/0719053.  Google Scholar

[38]

X. MengZ. QiaoC. Wang and Z. Zhang, Artificial regularization parameter analysis for the no-slope-selection epitaxial thin film model, CSIAM Trans. Appl. Math., 1 (2020), 441-462.  doi: 10.4208/csiam-am.2020-0015.  Google Scholar

[39]

S. A. OrszagM. Israeli and M. O. Deville, Boundary conditions for incompressible flows, J. Sci. Comput., 1 (1986), 75-111.  doi: 10.1007/BF01061454.  Google Scholar

[40]

R. Peyret, Spectral Methods for Incompressible Viscous Flow, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-6557-1.  Google Scholar

[41]

E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal., 23 (1986), 1-10.  doi: 10.1137/0723001.  Google Scholar

[42]

E. Tadmor, Convergence of spectral methods to nonlinear conservation laws, SIAM J. Numer. Anal., 26 (1989), 30-44.  doi: 10.1137/0726003.  Google Scholar

[43]

E. Tadmor, Shock capturing by the spectral viscosity method, Comput. Methods Appl. Mech. Engrg., 80 (1990), 197-208.  doi: 10.1016/0045-7825(90)90023-F.  Google Scholar

[44]

R. Témam, Sur l'approximation de la Solution Des équation de Navier-Stokes par la Méthode Des Fractionnarires II, Arch. Rational Mech. Anal., 33 (1969), 377-385.  doi: 10.1007/BF00247696.  Google Scholar

[45]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, Providence, Rhode Island, 2001. doi: 10.1090/chel/343.  Google Scholar

[46]

M. Wang, Q. Huang and C. Wang, A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation, J. Sci. Comput., Accepted and in press. Google Scholar

[47]

C. Wang and J.-G. Liu, Convergence of gauge method for incompressible flow, Math. Comp., 69 (2000), 1385-1407.  doi: 10.1090/S0025-5718-00-01248-5.  Google Scholar

[48]

C. ZhangH. WangJ. HuangC. Wang and X. Yue, A second order operator splitting numerical scheme for the "Good" Boussinesq equation, Appl. Numer. Math., 119 (2017), 179-193.  doi: 10.1016/j.apnum.2017.04.006.  Google Scholar

show all references

References:
[1]

L. Abia and J. M. Sanz-Serna, The spectral accuracy of a fully-discrete scheme for a nonlinear third order equation, Computing, 44 (1990), 187-196.  doi: 10.1007/BF02262215.  Google Scholar

[2]

J. B. BellP. Colella and H. M. Glaz, A second order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), 257-283.  doi: 10.1016/0021-9991(89)90151-4.  Google Scholar

[3]

O. Botella, On the solution of the Navier-Stokes equations using projection schemes with third- order accuracy in time, Comput. Fluids, 26 (1997), 107-116.  doi: 10.1016/S0045-7930(96)00032-1.  Google Scholar

[4]

N. Bressan and A. Quarteroni, An implicit/explcit spectral method for Burgers' equation, Calcolo, 23 (1986), 265-284.  doi: 10.1007/BF02576532.  Google Scholar

[5]

C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), 67-86.  doi: 10.1090/S0025-5718-1982-0637287-3.  Google Scholar

[6]

W. ChenS. CondeC. WangX. Wang and S. M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.  doi: 10.1007/s10915-011-9559-2.  Google Scholar

[7]

W. ChenW. LiZ. LuoC. Wang and X. Wang, A stabilized second order ETD multistep method for thin film growth model without slope selection, ESAIM Math. Model. Numer. Anal., 54 (2020), 727-750.  doi: 10.1051/m2an/2019054.  Google Scholar

[8]

W. Chen, W. Li, C. Wang, S. Wang and X. Wang, Energy stable higher order linear ETD multi-step methods for gradient flows: Application to thin film epitaxy, Res. Math. Sci., 7 (2020), Paper No. 13, 27 pp. doi: 10.1007/s40687-020-00212-9.  Google Scholar

[9]

W. Chen, C. Wang, S. Wang, X. Wang and S. M. Wise, Energy stable numerical schemes for ternary Cahn-Hilliard system, J. Sci. Comput., 84 (2020), Paper No. 27, 36 pp. doi: 10.1007/s10915-020-01276-z.  Google Scholar

[10]

W. ChenC. WangX. Wang and S. M. Wise, A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection., J. Sci. Comput., 59 (2014), 574-601.  doi: 10.1007/s10915-013-9774-0.  Google Scholar

[11]

K. ChengW. FengS. Gottlieb and C. Wang, A Fourier pseudospectral method for the "Good" Boussinesq equation with second-order temporal accuracy, Numer. Methods Partial Differential Equations, 31 (2015), 202-224.  doi: 10.1002/num.21899.  Google Scholar

[12]

K. ChengZ. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput., 81 (2019), 154-185.  doi: 10.1007/s10915-019-01008-y.  Google Scholar

[13]

K. Cheng and C. Wang, Long time stability of high order multi-step numerical schemes for two-dimensional incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 54 (2016), 3123-3144.  doi: 10.1137/16M1061588.  Google Scholar

[14]

Q. Cheng and C. Wang, Error estimate of a second order accurate scalar auxiliary variable (SAV) scheme for the thin film epitaxial equation, Adv. Appl. Math. Mech., Accepted and in press. Google Scholar

[15]

K. ChengC. Wang and S. M. Wise, An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation, Commun. Comput. Phys., 26 (2019), 1335-1364.  doi: 10.4208/cicp.2019.js60.10.  Google Scholar

[16]

K. Cheng, C. Wang and S. M. Wise, A weakly nonlinear energy stable scheme for the strongly anisotropic Cahn-Hilliard system and its convergence analysis, J. Comput. Phys., 405 (2020), 109109, 28 pp. doi: 10.1016/j.jcp.2019.109109.  Google Scholar

[17]

K. ChengC. WangS. M. Wise and X. Yue, A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn-Hilliard equation and its solution by the homogeneous linear iteration method, J. Sci. Comput., 69 (2016), 1083-1114.  doi: 10.1007/s10915-016-0228-3.  Google Scholar

[18]

A. J. Chorin, Numerical solution of Navier-Stokes equations, Math. Comp., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[19]

M. Crouzeix, Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques, Numer. Math., 35 (1980), 257-276.  doi: 10.1007/BF01396412.  Google Scholar

[20]

J. De FrutosT. Ortega and J. M. Sanz-Serna, Pseudo-spectral method for the "Good" boussinesq equation, Math. Comp., 57 (1991), 109-122.  doi: 10.2307/2938665.  Google Scholar

[21]

W. E, Convergence of Fourier methods for Navier-Stokes equations, SIAM J. Numer. Anal., 30 (1993), 650-674.  doi: 10.1137/0730032.  Google Scholar

[22]

W. E, Convergence of spectral methods for the {Burgers'} equation, SIAM J. Numer. Anal., 29 (1992), 1520-1541.  doi: 10.1137/0729088.  Google Scholar

[23]

W. E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057.  doi: 10.1137/0732047.  Google Scholar

[24]

W. E and J.-G. Liu, Gauge finite element method for incompressible flows, Int. J. Num. Meth. Fluids, 34 (2000), 701-710.  doi: 10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B.  Google Scholar

[25]

W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Commu. Math. Sci., 1 (2003), 317-332.  doi: 10.4310/CMS.2003.v1.n2.a6.  Google Scholar

[26]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods, Theory and Applications, SIAM, Philadelphia, PA, 1977.  Google Scholar

[27]

S. GottliebF. ToneC. WangX. 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.  Google Scholar

[28]

S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers' equation, J. Sci. Comput., 53 (2012), 102-128.  doi: 10.1007/s10915-012-9621-8.  Google Scholar

[29]

B. Y. Guo, A spectral method for the vorticity equation on the surface, Math. Comp., 64 (1995), 1067-1079.  doi: 10.1090/S0025-5718-1995-1297463-5.  Google Scholar

[30]

B.-Y. Guo and W. Huang, Mixed Jacobi-Spherical harmonic spectral method for Navier-Stokes equations, Appl. Numer. Math., 57 (2007), 939-961.  doi: 10.1016/j.apnum.2006.09.003.  Google Scholar

[31]

B. Y. Guo and J. Zou, Fourier spectral projection method and nonlinear convergence analysis for Navier-Stokes equation, J. Math. Anal. Appl., 282 (2003), 766-791.  doi: 10.1016/S0022-247X(03)00254-3.  Google Scholar

[32]

Y. HaoQ. Huang and C. Wang, A third order BDF energy stable linear scheme for the no-slope-selection thin film model, Commun. Comput. Phys., 29 (2021), 905-929.  doi: 10.4208/cicp.OA-2020-0074.  Google Scholar

[33]

H. Johnston and J.-G. Liu, A finite difference scheme for incompressible flow based on local pressure boundary conditions, J. Comput. Phys., 180 (2002), 120-154.  doi: 10.1006/jcph.2002.7079.  Google Scholar

[34]

H. Johnston and J.-G. Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys., 199 (2004), 221-259.  doi: 10.1016/j.jcp.2004.02.009.  Google Scholar

[35]

G. E. KarniadakisM. Israeli and S. A. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414-443.  doi: 10.1016/0021-9991(91)90007-8.  Google Scholar

[36]

J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.  doi: 10.1016/0021-9991(85)90148-2.  Google Scholar

[37]

Y. Maday and A. Quarteroni, Spectral and pseudospectral approximation of the Navier-Stokes equations, SIAM J. Numer. Anal., 19 (1982), 761-780.  doi: 10.1137/0719053.  Google Scholar

[38]

X. MengZ. QiaoC. Wang and Z. Zhang, Artificial regularization parameter analysis for the no-slope-selection epitaxial thin film model, CSIAM Trans. Appl. Math., 1 (2020), 441-462.  doi: 10.4208/csiam-am.2020-0015.  Google Scholar

[39]

S. A. OrszagM. Israeli and M. O. Deville, Boundary conditions for incompressible flows, J. Sci. Comput., 1 (1986), 75-111.  doi: 10.1007/BF01061454.  Google Scholar

[40]

R. Peyret, Spectral Methods for Incompressible Viscous Flow, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-6557-1.  Google Scholar

[41]

E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal., 23 (1986), 1-10.  doi: 10.1137/0723001.  Google Scholar

[42]

E. Tadmor, Convergence of spectral methods to nonlinear conservation laws, SIAM J. Numer. Anal., 26 (1989), 30-44.  doi: 10.1137/0726003.  Google Scholar

[43]

E. Tadmor, Shock capturing by the spectral viscosity method, Comput. Methods Appl. Mech. Engrg., 80 (1990), 197-208.  doi: 10.1016/0045-7825(90)90023-F.  Google Scholar

[44]

R. Témam, Sur l'approximation de la Solution Des équation de Navier-Stokes par la Méthode Des Fractionnarires II, Arch. Rational Mech. Anal., 33 (1969), 377-385.  doi: 10.1007/BF00247696.  Google Scholar

[45]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, Providence, Rhode Island, 2001. doi: 10.1090/chel/343.  Google Scholar

[46]

M. Wang, Q. Huang and C. Wang, A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation, J. Sci. Comput., Accepted and in press. Google Scholar

[47]

C. Wang and J.-G. Liu, Convergence of gauge method for incompressible flow, Math. Comp., 69 (2000), 1385-1407.  doi: 10.1090/S0025-5718-00-01248-5.  Google Scholar

[48]

C. ZhangH. WangJ. HuangC. Wang and X. Yue, A second order operator splitting numerical scheme for the "Good" Boussinesq equation, Appl. Numer. Math., 119 (2017), 179-193.  doi: 10.1016/j.apnum.2017.04.006.  Google Scholar

Figure 1.  The discrete $ \ell^2 $ and $ \ell^\infty $ numerical errors vs. temporal resolution $ N_T $ for $ N_T = 100:100:1000 $, with a spatial resolution $ N = 256 $. The numerical results are obtained by the computation using the proposed scheme (3.1), (5.2), (5.5), (5.6), in the first, second, third and fourth order temporal accuracy orders, respectively. The viscosity parameter is taken as $ \nu = 0.5 $. The numerical errors for the velocity variable $ u $ are displayed. The data lie roughly on curves $ CN_T^{-1} $, $ C N_T^{-2} $, $ C N_T^{-3} $ and $ C N_T^{-4} $, respectively, for appropriate choices of $ C $, confirming the full temporal accuracy orders of the proposed schemes
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