Numerical Analysis of Backward-Euler Method for Simplified MagnetoHydroDynamics (SMHD) with Linear Time Relaxation

Year 2021, Volume: 13 Issue: 1, 145 - 161, 30.06.2021

Gamze Yüksel Mustafa Hicret Yaman

Abstract

In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with linear time relaxation term. Therefore, the differential filter $\kappa(u-\bar{u})$ term is added to SMHD equations for numerical regularization and it is introduced SMHD Linear Time Relaxation Model (SMHDLTRM). The SMHDLTRM model is discretized by Backward-Euler (BE) method to obtain finite element solutions. The stability and convergency of the method is also conducted for SMHDLTRM. The present method is unconditionally stable and convergent with small time step condition. Additionally, the effectiveness of the method has presented with some numerical examples. The BE solutions of the SMHDLTRM are compared with the BE and the Crank-Nicolson (CN) solutions of the SMHD equations. This method can be applied to some problems when necessary more time steps to get accuracy or numerical solutions blow up for classical methods (BE or CN). All computations are conducted by using FreeFem++.

Keywords

MagnetoHydroDynamics, backward-Euler method, linear time relaxation, finite element method

Project Number

Project Grant Number: 17/225 and

References

• [1] Adams, N.A., Stolz, S., Deconvolution Methods for Subgrid-Scale Approximation in LES, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, 2001.
• [2] Akbas, M., Mohebujjaman, M., Rebholz, L.G., Xiao, M., High order algebraic splitting for magnetohydrodynamics simulation, Journal of Computational and Applied Mathematics, 321(2017).
• [3] Belenli, M.A., Kaya, S., Rebholz, L.G.,Wilson, N.E., A subgrid stabilization finite element method for incompressible magnetohydrodynamics, Int. J. Comp. Math., 90(7)(2013), 1506–1523.
• [4] Breckling, S., Neda, M., Hill, T., A review of time relaxation methods, Fluids. 2 (2017), 40.
• [5] 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).
• [6] Davidson, P.A., An Introduction to Magnetohydrodynamics, Cambridge University Press, United Kingdom, 2001.
• [7] Erkmen, D., Kaya, S., Cibik, A., A second order decoupled penalty projection method based on deferred correction for MHD in Elsasser variable, Journal of Computational and Applied Mathematics, 371(2020).
• [8] Gunzburger, M.D., Meir, A.J., Peterson, J.S., On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comput., 56(194)(1991), 523–563.
• [9] Isik, O.R., Spin up problem and accelerating convergence to steady state, Applied Mathematical Modelling, 37(2013), 3242–3253.
• [10] Isik, O.R., Yuksel, G., Demir, B., Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation, Num. Meth. Part. Di. Equa., 34(6)(2018), 2060–2078.
• [11] Layton, W., Neda, M., Truncation of scales by time relaxation, Journal of Mathematical Analysis and Applications, 325(2007), 788–807.
• [12] Layton, W., Introduction to the Numerical Analysis of Incompressible Viscous Flows, SIAM, Philadelphia, 2008.
• [13] Layton, W., David Pruett, C., Rebholz, Leo G., Temporally regularized direct numerical simulation, Appl. Math. Comput., 216(2010), 3728– 3738.
• [14] Layton, W., Rebholz, L., Approximate Deconvolution Models of Turbulence, Analysis, Phenomenology and Numerical Analysis, Springer, 2012.
• [15] Layton, W., Tran, H., Trenchea, C., Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows, Numer. Meth. Part. D. E., 30(4)(2014), 1083–1102.
• [16] Özbunar, M., Finite Element Solution of Navier-Stokes Time RelaxationModel Euler Time Discretization, M.Sc. Thesis, Grad. Sch. Nat. Appl. Sci., 2017.
• [17] Pakzad, A., On the long time behavior of time relaxation model of fluids, arXiv:1903.12339.
• [18] Peterson, J.S., On the finite element approximation of incompressible flows of an electrically conducting fluid, Numer. Meth. Part. D. E., 4(1)(1988), 57–68.
• [19] Rosenau, P., Extending hydrodynamics via the regularization of the Chapman–Enskog expansion, Phys. Rev. A., 40(1989), 7193–7196.
• [20] Schochet, S., Tadmor, E., The regularized Chapman–Enskog expansion for scalar conservation laws, Arch. Ration. Mech. Anal., 119(1992), 95–107.
• [21] Stolz, S., Adams, N.A., Kleiser, L., The approximate deconvolution model for LES of compressible flows and its application to shock-turbulentboundary layer interaction, Phys. Fluids, 13(2001), 2985.
• [22] Yuksel, G., Ingram, R., Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flow at small magnetic Reynolds number, Int. J. Num. Anal. Model., 10(1)(2013), 74–98.
• [23] Yuksel, G., Isik, O.R., Numerical analysis of Backward-Euler discretization for simplified magnetohydrodynamic flows, Appl. Math. Modell., 39(7)(2015), 1889–1898.