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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 11
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Research Article

Fully decoupled, linear and unconditional stability implicit/explicit scheme for the natural convection problem

Pages 3020-3042 | Received 20 Oct 2021, Accepted 23 Feb 2022, Published online: 07 Mar 2022

References

  • Luo ZD. The bases and applications of mixed finite element methods. Beijing: Chinese Science Press; 2006.
  • Manzari MT. An explicit finite element algorithm for convective heat transfer problems. Int J Numer Methods Heat Fluid Flow. 1999;9:860–877.
  • Massarotti N, Nithiarasu P, Zienkiewicz OC. Characteristic-Based-Split (CBS) algorithm for incompressible flow problems with heat transfer. Int J Numer Methods Heat Fluid Flow. 1998;8:969–990.
  • de Vahl Davis D. Natural convection of air in a square cavity: a benchmark solution. Int J Numer Methods Fluids. 1983;3:249–264.
  • Boland J, Layton W. Error analysis for finite element methods for steady natural convection problems. Numer Funct Anal Optim. 1990;11:449–483.
  • Cibik A, Kaya S. A projection-based stabilized finite element method for steady-state natural convection problem. J Math Anal Appl. 2011;381:469–484.
  • Liang HX, Zhang T. Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data. Comput Math Appl. 2019;77:2221–2241.
  • Shi DY, Ren JC. A least squares Galerkin-Petrov nonconforming mixed finite element method for the stationary conduction-convection problem. Nonlinear Anal Theory Methods Appl. 2010;72(3–4):1653–1667.
  • Zhang T, Yuan JY, Si ZY. Decoupled two grid finite element method for the time-dependent natural convection problem I: spatial discretization. Numer Method Partial Differ Equations. 2015;31:2135–2168.
  • Giraldo FX, Restelli M, Laeuter M. Semi-implicit formulations of the Navier-Stokes equations: application to nonhydrostatic atmospheric modeling. SIAM J Sci Comput. 2010;32:3394–3425.
  • He YN, Li J. A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J Comput Appl Math. 2010;235(3):708–725.
  • Marti J, Ryzhakov PB. An explicit-implicit finite element model for the numerical solution of incompressible Navier-Stokes equations on moving grids. Comput Methods Appl Mech Eng. 2019;350:750–765.
  • Su J, He YN. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete Continuous Dyn Syst Ser B. 2017;22(9):3421–3438.
  • Shi H, Li Y. Local discontinuous Galerkin methods with implicit-explicit multistep time-marching for solving the nonlinear Cahn-Hilliard equation. J Comput Phys. 2019;394:719–731.
  • Ak T, GaziKarakoc SB, Biswas A. Numerical scheme to dispersive shallow water waves. J Comput Theory Nanosci. 2016;13(10):7084–7092.
  • Ak T, GaziKarakoc SB. A numerical technique based on collocation method for solving modified Kawahara equation. J Ocean Eng Sci. 2018;3(1):67–75.
  • Ak T, GaziKarakoc SB, Triki H. Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation. Eur Phys J Plus. 2016;131:1–15.
  • GaziKarakoc SB, Bhowmik SK. Galerkin finite element solution for Benjamin-Bona-Mahony-Burgers equation with cubic B-Splines. Comput Math Appl. 2019;77(7):1917–1932.
  • GaziKarakoc SB, Neilan M. A C0 finite element method for the Biharmonic problem without extrinsic penalization. Numer Methods Partial Differ Equ. 2014;30(4):1254–1278.
  • Hansen E, Stillfjord T. Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations. Math Comput. 2013;82(284):1975–1985.
  • He YN. The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math Comput. 2008;77(264):2097–2124.
  • He YN, Huang PZ, Feng XL. H2-stability of first order fully discrete schemes for the time-dependent Navier-Stokes equations. J Sci Comput. 2015;62:230–264.
  • Jin JJ, Zhang T, Li J. H2 stability of the first order Galerkin method for the Boussinesq equations with smooth and nonsmooth initial data. Comput Math Appl. 2018;75(1):248–288.
  • Gerstenberger A, Wall WA. An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction. Comput Methods Appl Mech Eng. 2008;197(19–20):1699–1714.
  • Kim MY, Shin DW. A high order discontinuous Galerkin method with Lagrange multipliers for second-order elliptic problems. Appl Numer Math. 2019;135:47–68.
  • Zhang JP, Chen K, Yu B. An iterative Lagrange multiplier method for constrained total-variation-based image denoising. SIAM J Numer Anal. 2012;50(3):983–1003.
  • Yang XF, Zhang GD. Convergence analysis for the invariant energy quadratization (IEQ) schemes for solving the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential. J Sci Comput. 2020;82(3):5. DOI:10.1007/s10915-020-01151-x
  • Yang XF, Zhao J, Wang Q. Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J Comput Phys. 2017;333:104–127.
  • Yang XF, Zhao J, He XM. Linear, second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method. J Comput Appl Math. 2018;343:80–97.
  • Zhao J, Wang Q, Yang XF. Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int J Numer Methods Eng. 2017;110(3):279–300.
  • Li XL, Shen J. Error analysis of the SAV-MAC scheme for the Navier-Stokes equations. SIAM J Numer Anal. 2020;58(5):2465–2491.
  • Li XL, Shen J. On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes phase-field model and its error analysis for the corresponding Cahn-Hilliard-Stokes case. Math Models Methods Appl Sci. 2020;30(12):2263–2297.
  • Li XL, Shen J, Liu ZG. New SAV-pressure correction methods for the Navier-Stokes equations: stability and error analysis; 21 Feb 2020. arXiv: 2002.09090v1, [math.NA].
  • Li XL, Shen J, Rui HX. Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math Comput. 2019;88(319):2047–2068.
  • Lin LL, Yang ZG, Dong SC. Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable. J Comput Phys. 2019;388:1–22.
  • Shen J, Xu J. Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J Numer Anal. 2018;56:2895–2912.
  • Shen J, Xu J, Yang J. The scalar auxiliary variable (SAV) approach for gradient flows. J Comput Phys. 2018;353:407–416.
  • Xu Z, Yang XF, Zhang H, et al. Efficient and linear schemes for anisotropic Cahn-Hilliard model using the stabilized-invariant energy quadratization (S-IEQ) approach. Comput Phys Commun. 2019;238:36–49.
  • Zhang J, Chen CJ, Yang XF. A novel decoupled and stable scheme for an anisotropic phase-field dendritic crystal growth model. Appl Math Lett. 2019;95:122–129.
  • Hou DM, Azaiez M, Xu CJ. A variant of scalar auxiliary variable approaches for gradient flows. J Comput Phys. 2019;395:307–332.
  • Yang JX, Kim J. A variant of stabilized-scalar auxiliary variable (S-SAV) approach for a modified phase-field surfactant model. Comput Phys Commun. 2021;261:107825.
  • Cheng Q, Shen J. Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model. SIAM J Sci Comput. 2018;40(6):A3982–A4006.
  • Huang FK, Shen J, Yang ZG. A highly efficient and accurate new scalar auxiliary variable approach for gradient flows. SIAM J Sci Comput. 2020;42(4):A2514–A2536.
  • Wang N, Shi DY. Two efficient spectral methods for the nonlinear fractional wave equation in unbounded domain. Math Computer Simul. 2021;185:696–718.
  • Shen J. On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations. Numer Math. 1992;62:49–73.
  • Shen J. On error estimates of the projection, methods for the Navier-Stokes equations: second-order schemes. Math Comput. 1996;65:1039–1066.
  • Liang HX, Zhang T. Stability and convergence of two-grid Crank-Nicolson extrapolation scheme for the time-dependent natural convection equations. Math Methods Appl Sci. 2019;42:6165–6191.
  • Temam R. Navier-Stokes equations, theory and numerical analysis. 3rd ed. Amsterdam: North-Holland; 1984.
  • Larsson S. The long time behavior of finite element approximations of solutions to semilinear parabolic problems. SIAM J Numer Anal. 1989;26:348–365.
  • Brezzi F, Fortin M. Mixed and hybrid finite element methods. New York (NY): Springer-Verlag; 1991.
  • Heywood J, Rannacher R. Finite element approximation of the nonstationary Navier-Stokes problem I; regularity of solutions and second-order error estimates for spatial discretization. SIAM J Numer Anal. 1982;19:275–311.
  • Wan DC, Patnaik BSV, Wei GW. A new benchmark quality solution for the buoyancy driven cavity by discrete singular convolution. Numer. Heat Trans B Fundam. 2001;40:199–228.
  • Mayne DA, Usmani AS, Crapper M. H-adaptive finite element solution of high Rayleigh number thermally driven cavity problem. Int J Numer Methods Heat Fluid Flow. 2000;10:598–615.
  • Liu ZG, Li XL. Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation. Numer Algorithm. 2020;85(1):107–132.

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