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SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION

Yıl 2021, Cilt: 7 Sayı: 2, 45 - 51, 31.12.2021

Gamze Yüksel Mustafa Hicret Yaman

https://doi.org/10.22531/muglajsci.932484
Cited By: 1

Öz

In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with nonlinear time relaxation term. The differential filter κ(|u-u ̅ |(u-u ̅ )) term is added to SMHD equations. Also SMHD Nonlinear Time Relaxation Model (SMHDNTRM) is introduced. The model is discretized by Backward-Euler (BE) method to obtain the finite element solutions. Moreover, the stability of the method is proved. The method is found unconditionally stable. The effectiveness of the method is exemplified by several cases with comparing different methods. FreeFem++ is used for all computations.

Anahtar Kelimeler

: MagnetoHydroDynamics Backward-Euler Method Nonlinear Time Relaxation Finite Element Method

Destekleyen Kurum

Mugla Sitki Kocman University Research Projects Coordination Office.

Proje Numarası

Project Grant Number: 17/225

Kaynakça

  • Davidson, P. A., An Introduction to Magnetohydrodynamics. Cambridge University Press, United Kingdom, 2001.
  • Peterson, J. S., “On the finite element approximation of incompressible flows of an electrically conducting fluid”, Numerical Methods for Partial Differential Equations, vol.4, no.1, pp. 57-68, 1988.
  • Layton, W., Tran, H. and Trenchea C., “Stability of partitioned methods for magnetohydrodynamics flows at small magnetic Reynolds numbers”, Contemporary Mathematics, vol. 586, pp. 231- 238, 2013.
  • Yuksel, G. and Ingram, R., “Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flow at small magnetic Reynolds number”, International Journal of Numerical Analysis and Modeling, vol.10, no.1, pp. 74-98, 2013.
  • Yuksel, G. and Isik, O. R., “Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows”, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 7, 1889-1898, 2015.
  • Layton, W., Tran, H. and Trenchea C., “Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows”, Numerical Methods for Partial Differential Equations, vol. 30, no. 4, pp. 1083-1102, 2014.
  • Rong, Y., Hou, Y. and Zhang, Y., “Numerical analysis of a second order algorithm for simplified magnetohydrodynamic flows”, Advances in Computational Mathematics, vol. 43, pp. 823-848, 2017.
  • Stolz, S., Adams, N. A. and Kleiser, L., “ The approximate deconvolution model for LES of compressible flows and its application to shock-turbulent-boundary layer interaction”, Physics of Fluids, vol. 13, no. 10, pp. 2985-3001, 2001.
  • Stolz, S., Adams, N. A. and Kleiser, L., “An approximate deconvolution model for large eddy simulation with application to wall-bounded flows”, Physics of Fluids, vol. 13, no.4, pp. 997-1014, 2001.
  • Adams, N. A., Stolz, S., Deconvolution methods for subgrid-scale approximation in LES, Modern Simulation Strategies for Turbulent Flow, R. T. Edwards, 2001.
  • Rosenau, P., “Extending hydrodynamics via the regularization of the Chapman--Enskog expansion”, Physical review. A, General physics, vol. 40, no. 12, pp. 7193-7196, 1989.
  • Schochet, S. and Tadmor E., The regularized Chapman--Enskog expansion for scalar conservation laws, Archive for Rational Mechanics and Analysis, vol. 119, pp. 95-107, 1992..
  • Breckling, S., Neda, M. and Hill, T., “A Review of Time Relaxation Methods”, Fluids, vol. 2, no. 3, pp. 40-59, 2017.
  • Pakzad, A., “On the long time behavior of time relaxation model of fluids”, Physica D: Nonlinear Phenomena, vol. 408, pp. 1-6, 2020.
  • Pruett, C.D., et al., The temporally filtered Navier--Stokes equations: properties of the residual-stress, Technical Report, National Science Foundation; United States, 2003.
  • Layton, W. and Neda, M., “Truncation of scales by time relaxation”, Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 788-807, 2007.
  • Isik, O. R., Yüksel, G. and Demir, B., “Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation”, Numerical Methods for Partial Differential Equations, vol. 34, pp. 2060-2078, 2018.
  • Yaman, M. H., Time Relaxation Models for Simplified MagnetoHydroDynamics Equations by Using Finite Element Method, M.Sc. Thesis, Grad. Sch. Nat. Appl. Sci. 2018.
  • Layton, W., Introduction to the Numerical Analysis of Incompressible Viscous Flows, SIAM, Philadelphia, 2008.
  • Yüksel, G. and Yaman, M. H., “Numerical Analysis of Backward-Euler Method for Simplified MagnetoHydroDynamics (SMHD) with Linear Time Relaxation”, Turkish Journal of Mathematics and Computer Science , vol. 13, no. 1, pp. 145-161, 2021.
  • Cibik, A., Eroglu, F.G. & Kaya, S. Analysis of Second Order Time Filtered Backward Euler Method for MHD Equations. J Sci Comput 82, 38, 2020.

Yıl 2021, Cilt: 7 Sayı: 2, 45 - 51, 31.12.2021

Gamze Yüksel Mustafa Hicret Yaman

https://doi.org/10.22531/muglajsci.932484
Cited By: 1

Öz

Proje Numarası

Project Grant Number: 17/225

Kaynakça

  • Davidson, P. A., An Introduction to Magnetohydrodynamics. Cambridge University Press, United Kingdom, 2001.
  • Peterson, J. S., “On the finite element approximation of incompressible flows of an electrically conducting fluid”, Numerical Methods for Partial Differential Equations, vol.4, no.1, pp. 57-68, 1988.
  • Layton, W., Tran, H. and Trenchea C., “Stability of partitioned methods for magnetohydrodynamics flows at small magnetic Reynolds numbers”, Contemporary Mathematics, vol. 586, pp. 231- 238, 2013.
  • Yuksel, G. and Ingram, R., “Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flow at small magnetic Reynolds number”, International Journal of Numerical Analysis and Modeling, vol.10, no.1, pp. 74-98, 2013.
  • Yuksel, G. and Isik, O. R., “Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows”, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 7, 1889-1898, 2015.
  • Layton, W., Tran, H. and Trenchea C., “Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows”, Numerical Methods for Partial Differential Equations, vol. 30, no. 4, pp. 1083-1102, 2014.
  • Rong, Y., Hou, Y. and Zhang, Y., “Numerical analysis of a second order algorithm for simplified magnetohydrodynamic flows”, Advances in Computational Mathematics, vol. 43, pp. 823-848, 2017.
  • Stolz, S., Adams, N. A. and Kleiser, L., “ The approximate deconvolution model for LES of compressible flows and its application to shock-turbulent-boundary layer interaction”, Physics of Fluids, vol. 13, no. 10, pp. 2985-3001, 2001.
  • Stolz, S., Adams, N. A. and Kleiser, L., “An approximate deconvolution model for large eddy simulation with application to wall-bounded flows”, Physics of Fluids, vol. 13, no.4, pp. 997-1014, 2001.
  • Adams, N. A., Stolz, S., Deconvolution methods for subgrid-scale approximation in LES, Modern Simulation Strategies for Turbulent Flow, R. T. Edwards, 2001.
  • Rosenau, P., “Extending hydrodynamics via the regularization of the Chapman--Enskog expansion”, Physical review. A, General physics, vol. 40, no. 12, pp. 7193-7196, 1989.
  • Schochet, S. and Tadmor E., The regularized Chapman--Enskog expansion for scalar conservation laws, Archive for Rational Mechanics and Analysis, vol. 119, pp. 95-107, 1992..
  • Breckling, S., Neda, M. and Hill, T., “A Review of Time Relaxation Methods”, Fluids, vol. 2, no. 3, pp. 40-59, 2017.
  • Pakzad, A., “On the long time behavior of time relaxation model of fluids”, Physica D: Nonlinear Phenomena, vol. 408, pp. 1-6, 2020.
  • Pruett, C.D., et al., The temporally filtered Navier--Stokes equations: properties of the residual-stress, Technical Report, National Science Foundation; United States, 2003.
  • Layton, W. and Neda, M., “Truncation of scales by time relaxation”, Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 788-807, 2007.
  • Isik, O. R., Yüksel, G. and Demir, B., “Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation”, Numerical Methods for Partial Differential Equations, vol. 34, pp. 2060-2078, 2018.
  • Yaman, M. H., Time Relaxation Models for Simplified MagnetoHydroDynamics Equations by Using Finite Element Method, M.Sc. Thesis, Grad. Sch. Nat. Appl. Sci. 2018.
  • Layton, W., Introduction to the Numerical Analysis of Incompressible Viscous Flows, SIAM, Philadelphia, 2008.
  • Yüksel, G. and Yaman, M. H., “Numerical Analysis of Backward-Euler Method for Simplified MagnetoHydroDynamics (SMHD) with Linear Time Relaxation”, Turkish Journal of Mathematics and Computer Science , vol. 13, no. 1, pp. 145-161, 2021.
  • Cibik, A., Eroglu, F.G. & Kaya, S. Analysis of Second Order Time Filtered Backward Euler Method for MHD Equations. J Sci Comput 82, 38, 2020.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Makaleler
Yazarlar

Gamze Yüksel 0000-0003-3578-2762

Mustafa Hicret Yaman 0000-0001-6483-0140

Proje Numarası Project Grant Number: 17/225
Erken Görünüm Tarihi 22 Ekim 2021
Yayımlanma Tarihi 31 Aralık 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 7 Sayı: 2

Kaynak Göster

APA Yüksel, G., & Yaman, M. H. (2021). SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION. Mugla Journal of Science and Technology, 7(2), 45-51. https://doi.org/10.22531/muglajsci.932484
AMA Yüksel G, Yaman MH. SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION. Mugla Journal of Science and Technology. Aralık 2021;7(2):45-51. doi:10.22531/muglajsci.932484
Chicago Yüksel, Gamze, ve Mustafa Hicret Yaman. “SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION”. Mugla Journal of Science and Technology 7, sy. 2 (Aralık 2021): 45-51. https://doi.org/10.22531/muglajsci.932484.
EndNote Yüksel G, Yaman MH (01 Aralık 2021) SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION. Mugla Journal of Science and Technology 7 2 45–51.
IEEE G. Yüksel ve M. H. Yaman, “SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION”, Mugla Journal of Science and Technology, c. 7, sy. 2, ss. 45–51, 2021, doi: 10.22531/muglajsci.932484.
ISNAD Yüksel, Gamze - Yaman, Mustafa Hicret. “SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION”. Mugla Journal of Science and Technology 7/2 (Aralık 2021), 45-51. https://doi.org/10.22531/muglajsci.932484.
JAMA Yüksel G, Yaman MH. SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION. Mugla Journal of Science and Technology. 2021;7:45–51.
MLA Yüksel, Gamze ve Mustafa Hicret Yaman. “SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION”. Mugla Journal of Science and Technology, c. 7, sy. 2, 2021, ss. 45-51, doi:10.22531/muglajsci.932484.
Vancouver Yüksel G, Yaman MH. SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION. Mugla Journal of Science and Technology. 2021;7(2):45-51.

Cited By

SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION
Mugla Journal of Science and Technology
https://doi.org/10.22531/muglajsci.932484
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