Research Article
SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION
Abstract
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.
Keywords
: MagnetoHydroDynamics Backward-Euler Method Nonlinear Time Relaxation Finite Element Method
Supporting Institution
Mugla Sitki Kocman University Research Projects Coordination Office.
Project Number
Project Grant Number: 17/225
References
- 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.
Year 2021,
Volume: 7 Issue: 2, 45 - 51, 31.12.2021
Abstract
Project Number
Project Grant Number: 17/225
References
- 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.
There are 21 citations in total.
Details
| Primary Language | English |
|---|---|
| Journal Section | Journals |
| Authors | |
| Project Number | Project Grant Number: 17/225 |
| Early Pub Date | October 22, 2021 |
| Publication Date | December 31, 2021 |
| Published in Issue | Year 2021 Volume: 7 Issue: 2 |
Cite
Cited By
Mugla Journal of Science and Technology (MJST) is licensed under the Creative Commons Attribution-Noncommercial-Pseudonymity License 4.0 international license.