In this paper, we design a decoupled, linear, stable scheme with variable time step for solving a ferrohydrodynamics system. Based on the backward Euler scheme with variable time step for time discretization, this scheme deals with nonlinear terms by explicit treatment. Meanwhile, we show the stability of the proposed scheme. Finally, some numerical experiments are provided to illustrate the accuracy and stability of the proposed scheme.
Aytura KERAM
,
Pengzhan HUANG
,
Yinnian HE
. ON THE STABILITY OF A VARIABLE TIME STEP SCHEME FOR THE FERROHYDRODYNAMICS FLOW[J]. Acta mathematica scientia, Series B, 2026
, 46(2)
: 971
-992
.
DOI: 10.1007/s10473-026-0223-1
[1] Amirat Y, Hamdache K. Global weak solutions to a ferrofluid flow model. Mathematical Methods in the Applied Sciences, 2008, 31: 123-151
[2] Amirat Y, Hamdache K. Strong solutions to the equations of a ferrofluid flow model. Journal of Mathematical Analysis and Applications, 2009, 353: 271-294
[3] Bai F, Han D Z, He X M, Yang X F. Deformation and coalescence of ferrodroplets in Rosensweig model using the phase field and modified level set approaches under uniform magnetic fields. Communications in Nonlinear Science and Numerical Simulation, 2020, 85: 105213
[4] Besier M, Rannacher R. Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow. International Journal for Numerical Methods in Fluids, 2012, 70: 1139-1166
[5] DeCaria V, Layton W, Zhao H Y. A time-accurate, adaptive discretization for fluid flow problems. International Journal of Numerical Analysis and Modeling, 2020, 17: 254-280
[6] DeCaria V, Schneier M. An embedded variable step IMEX scheme for the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 2021, 376: 113661
[7] Ferdows M, Murtaza M G, Tzirtzilakis E E, Alzahrani F. Numerical study of blood flow and heat transfer through stretching cylinder in the presence of a magnetic dipole. Zeitschrift für Angewandte Mathematik und Mechanik,2020, 100: e201900278
[8] John V, Rang J. Adaptive time step control for the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 2010, 199: 514-524
[9] Kay D A, Gresho P M, Griffiths D F, Silvester D J. Adaptive time-stepping for incompressible flow Part II: Navier-Stokes equations. SIAM Journal on Scientific Computing, 2010, 32: 111-128
[10] Keram A, Huang P. Numerical simulation of the ferrohydrodynamics flow using an unconditionally stable second-order scheme. Zeitschrift für Angewandte Mathematik und Mechanik,2024, 104: e202400025
[11] Keram A, Huang P, He Y N. An unconditionally stable variable time step scheme for two-phaseferrofluid flows. Computer Methods in Applied Mechanics and Engineering, 2025, 442: 118018
[12] Li Y, Hou Y R, Layton W, Zhao H. Adaptive partitioned methods for the time accurate approximation of the evolutionary Stokes-Darcy system. Computer Methods in Applied Mechanics and Engineering, 2020, 364: 112923
[13] Latorre M, Rinaldi C. Applications of magnetic nanoparticles in medicine: Magnetic fluid hyperthermia. Puerto Rico Health Sciences Journal, 2009, 28: 227-238
[14] Liu H D, Xu W, Wang S G, Ke Z J. Hydrodynamic modeling of ferrofluid flow in magnetic targeting drug delivery. Applied Mathematics and Mechanics, 2008, 29: 1341-1349
[15] Mefford O T, Woodward R C, Goff J D, et al. Field-induced motion of ferrofluids through immiscible viscous media: Testbed for restorative treatment of retinal detachment. Journal of Magnetism and Magnetic Materials, 2007, 311: 347-353
[16] Neuringer J L, Rosensweig R E.Ferrohydrodynamics. Physics of Fluids, 1964, 7: 1927-1937
[17] Nochetto R H, Salgado A J, Tomas I. A diffuse interface model for two-phase ferrofluid flows. Computer Methods in Applied Mechanics and Engineering, 2016, 309: 497-531
[18] Oliveira J C. Strong solutions for ferrofluid equations in exterior domains. Acta Applicandae Mathematicae, 2018, 156: 1-14
[19] Rosensweig R E.Ferrohydrodynamics. Cambridge: Cambridge University Press, 1985
[20] Scrobogna S. Zero limit of entropic relaxation time for the Shliomis model of ferrofluids. Journal of Mathematical Analysis and Applications, 2021, 501: 125-213
[21] Shliomis M I. Effective viscosity of magnetic suspensions (Translated by J. G. Adashko). Soviet Physics JETP, 1972, 34: 1291-1294
[22] Shliomis M I.Ferrofluids: Magnetically Controllable Fluids and Their Applications. Berlin: Springer Verlag, 2002
[23] Tan Z, Wang Y J. Global analysis for strong solutions to the equations of a ferrofluid flow model. Journal of Mathematical Analysis and Applications, 2010, 364: 424-436
[24] Wang S, Liu H, Xu W. Hydrodynamic modelling and CFD simulation of ferrofluids flow in magnetic targeting drug delivery. International Journal of Computational Fluid Dynamics, 2008, 22: 659-667
[25] Wang Y J, Tan Z. Global existence and asymptotic analysis of weak solutions to the equations of ferrohydrodynamics. Nonlinear Analysis: Real World Applications, 2010, 11: 4254-4268
[26] Wu Y K, Xie X P. Mixed finite element methods for the ferrofluid model with magnetization paralleled to the magnetic field. Numerical Mathematics: Theory, Methods and Applications, 2023, 16: 489-510
[27] Wu Y K, Xie X P. Energy-stable mixed finite element methods for a ferrofluid flow model. Communications in Nonlinear Science and Numerical Simulation, 2023, 125: 107330
[28] Xie C G. Global strong solutions to the Shliomis system for ferrofluids in a bounded domain. Mathematical Methods in the Applied Sciences, 2019, 42: 6021-6028
[29] Yang X. A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model. Journal of Computational Physics, 2021, 432: 110015
[30] Yang X. On a novel fully decoupled, second-order accurate energy stable numerical scheme for a binary fluid-surfactant phase-field model. SIAM Journal on Scientific Computing, 2021, 43: B479-B507
[31] Yang W M. A finite volume method for ferrohydrodynamic problems coupled with microscopic magnetization dynamics. Applied Mathematics and Computation, 2023, 441: 127-704
[32] Zhang G, He X, Yang X. Decoupled, linear,unconditionally energy stable fully discrete finite element numerical scheme for a two-phase ferrohydrodynamics model. SIAM Journal on Scientific Computing, 2021, 43: B167-B193
[33] Zhang G, He X, Yang X. Reformulated weak formulation and efficient fully discrete finite element method for a two-phase ferrohydrodynamics shliomis model. SIAM Journal on Scientific Computing, 2023, 45: B253-B282
[34] Zhang G, He X, Yang X. A unified framework of the SAV-ZEC method for a mass-conserved Allen-Cahn type two-phase ferrofluid flow model. SIAM Journal on Scientific Computing, 2024, 46: B77-B106
