September  2018, 23(7): 2803-2823. doi: 10.3934/dcdsb.2018106

Partitioned second order method for magnetohydrodynamics in Elsässer variables

Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260, USA

* Corresponding author

Received  March 2017 Revised  August 2017 Published  September 2018 Early access  April 2018

Fund Project: The first author was partially supported by the AFOSR under grant FA 9550-16-1-0355, and by the NSF grant DMS-1522574. The second author is partially supported by the AFOSR under grant FA 9550-12-1-0191, and by the NSF grant DMS-1522574.

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier-Stokes equations coupled with Maxwell equations via Lorentz force and Ohm's law. Monolithic methods, which solve fully coupled MHD systems, are computationally expensive. Partitioned methods, on the other hand, decouple the full system and solve subproblems in parallel, and thus reduce the computational cost.

This paper is devoted to the design and analysis of a partitioned method for the MHD system in the Elsässer variables. The stability analysis shows that for magnetic Prandtl number of order unity, the method is unconditionally stable. We prove the error estimates and present computational tests that support the theory.

Citation: Yong Li, Catalin Trenchea. Partitioned second order method for magnetohydrodynamics in Elsässer variables. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2803-2823. doi: 10.3934/dcdsb.2018106
References:
[1]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), p405.

[2]

L. BarleonV. Casal and L. Lenhart, MHD flow in liquid-metal-cooled blankets, Fusion Eng. Des., 14 (1991), 401-412. 

[3]

J. D. BarrowR. Maartens and C. G. Tsagas, Cosmology with inhomogeneous magnetic fields, Phys. Rep., 449 (2007), 131-171.  doi: 10.1016/j.physrep.2007.04.006.

[4]

D. Biskamp, Magnetohydrodynamic Turbulence, Cambridge University Press, 2003. doi: 10.1017/CBO9780511535222.

[5]

P. Bodenheimer, G. P. Laughlin, M. Różyczka and H. W. Yorke, Numerical Methods in Astrophysics, Series in Astronomy and Astrophysics, Taylor & Francis, New York, 2007.

[6]

J. ConnorsJ. Howell and W. Layton, Decoupled time stepping methods for fluid-fluid interaction, SIAM Journal on Numerical Analysis, 50 (2012), 1297-1319.  doi: 10.1137/090773362.

[7]

P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.

[8]

M. DobrowolnyA. Mangeney and P. Veltri, Fully developed anisotropic hydromagnetic turbulence in interplanetary space, Phys. Rev. Lett., 45 (1980), 144-147.  doi: 10.1103/PhysRevLett.45.144.

[9]

E. Dormy and M. Núñez, Introduction [Special issue: Magnetohydrodynamics in astrophysics and geophysics], Geophys. Astrophys. Fluid Dyn., 101 (2007), p169. doi: 10.1080/03091920701523287.

[10]

E. Dormy and Andrew M. Soward (eds.), Mathematical Aspects of Natural Dynamos, vol. 13 of Fluid Mechanics of Astrophysics and Geophysics, Grenoble Sciences. Universite Joseph Fourier, Grenoble, 2007. doi: 10.1201/9781420055269.

[11]

W. M. Elsässer, The hydromagnetic equations, Phys. Rev., 79 (1950), 183-183.  doi: 10.1103/PhysRev.79.183.

[12]

A. Fierros Palacios, The Hamilton-type Principle in Fluid Dynamics, Springer, Vienna, 2006, Fundamentals and applications to magnetohydrodynamics, thermodynamics, and astrophysics.

[13]

J. A. Font, General relativistic hydrodynamics and magnetohydrodynamics: Hyperbolic systems in relativistic astrophysics, in Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin, 2008, 3–17. doi: 10.1007/978-3-540-75712-2_1.

[14]

S. GaltierS. V. NazarenkoA. C. Newell and A. Pouquet, A weak turbulence theory for incompressible magnetohydrodynamics, Part of the Lecture Notes in Physics book series, 536 (2000), 291-330.  doi: 10.1007/3-540-47038-7_12.

[15]

P. Goldreich and S. Sridhar, Toward a theory of interstellar turbulence. Ⅱ: Strong Alfvénic turbulence, ApJ, 438 (1995), 763-775. 

[16]

H. Hashizume, Numerical and experimental research to solve MHD problem in liquid blanket system, Fusion Eng. Des., 81 (2006), 1431-1438.  doi: 10.1016/j.fusengdes.2005.08.086.

[17]

N. Haugen, A. Brandenburg and W. Dobler, Simulations of nonhelical hydromagnetic turbulence, Phys. Rev. E, 70 (2004), 016308. doi: 10.1103/PhysRevE.70.016308.

[18]

T. HeisterM. Mohebujjaman and L. G. Rebholz, Decoupled, unconditionally stable, higher order discretizations for mhd flow simulation, Journal of Scientific Computing, 71 (2017), 21-43.  doi: 10.1007/s10915-016-0288-4.

[19]

W. Hillebrandt and F. Kupka (eds.), Interdisciplinary Aspects of Turbulence, vol. 756 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009.

[20]

P. S. Iroshnikov, Turbulence of a conducting fluid in a strong magnetic field, Soviet Astronom. AJ, 7 (1964), 566-571. 

[21]

R. H. Kraichnan, Inertial-range spectrum of hydromagnetic turbulence, Phys. Fluids, 8 (1965), 1385-1387.  doi: 10.1063/1.1761412.

[22]

W. LaytonH. Tran and C. Trenchea, Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows, Numer. Methods Partial Differential Equations, 30 (2014), 1083-1102.  doi: 10.1002/num.21857.

[23]

T. Lin, J. Gilbert, R. Kossowsky and P. S. U. S. COLLEGE., Sea-Water Magnetohydrodynamic Propulsion for Next-Generation Undersea Vehicles, Defense Technical Information Center, 1990, URL http://books.google.com/books?id=GvhwNwAACAAJ.

[24]

E. Marsch, Turbulence in the solar wind, in Reviews in Modern Astronomy (ed. G. Klare), vol. 4 of Reviews in Modern Astronomy, Springer Berlin Heidelberg, 1991,145–156. doi: 10.1007/978-3-642-76750-0_10.

[25]

M. MeneguzziU. Frisch and A. Pouquet, Helical and nonhelical turbulent dynamos, Phys. Rev. Lett., 47 (1981), 1060-1064.  doi: 10.1103/PhysRevLett.47.1060.

[26]

D. Mitchell and D. Gubser, Magnetohydrodynamic ship propulsion with superconducting magnets, J. Supercond., 1 (1988), 349-364.  doi: 10.1007/BF00618593.

[27]

B. Punsly, Black Hole Gravitohydromagnetics, vol. 355 of Astrophysics and Space Science Library, 2nd edition, Springer-Verlag, Berlin, 2008.

[28]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[29]

J. V. ShebalinW. H. Matthaeus and D. Montgomery, Anisotropy in MHD turbulence due to a mean magnetic field, J. Plasma Phys., 29 (1983), 525-547.  doi: 10.1017/S0022377800000933.

[30]

S. SmolentsevR. MoreauL. Bühler and C. Mistrangelo, MHD thermofluid issues of liquid-metal blankets: Phenomena and advances, Fusion Eng. Des., 85 (2010), 1196-1205.  doi: 10.1016/j.fusengdes.2010.02.038.

[31]

D. Sondak and A. A. Oberai, Large eddy simulation models for incompressible magnetohydrodynamics derived from the variational multiscale formulation, Phys. Plasmas, 19 (2012), 102308. doi: 10.1063/1.4759157.

[32]

C. Trenchea, Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows, Appl. Math. Lett., 27 (2014), 97-100.  doi: 10.1016/j.aml.2013.06.017.

[33]

M. K. Verma, Statistical theory of magnetohydrodynamic turbulence: Recent results, Phys. Rep., 401 (2004), 229-380.  doi: 10.1016/j.physrep.2004.07.007.

[34]

I. Veselovsky, Turbulence and waves in the solar wind formation region and the heliosphere, Astrophys. Space Sci., 277 (2001), 219-224.  doi: 10.1007/978-94-010-0904-1_28.

[35]

N. WilsonA. Labovsky and C. Trenchea, High accuracy method for magnetohydrodynamics system in Elsässer variables, Comput. Methods Appl. Math., 15 (2015), 97-110.  doi: 10.1515/cmam-2014-0023.

[36]

G. Yuksel and R. Ingram, Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers, Int. J. Numer. Anal. Model., 10 (2013), 74-98. 

show all references

References:
[1]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), p405.

[2]

L. BarleonV. Casal and L. Lenhart, MHD flow in liquid-metal-cooled blankets, Fusion Eng. Des., 14 (1991), 401-412. 

[3]

J. D. BarrowR. Maartens and C. G. Tsagas, Cosmology with inhomogeneous magnetic fields, Phys. Rep., 449 (2007), 131-171.  doi: 10.1016/j.physrep.2007.04.006.

[4]

D. Biskamp, Magnetohydrodynamic Turbulence, Cambridge University Press, 2003. doi: 10.1017/CBO9780511535222.

[5]

P. Bodenheimer, G. P. Laughlin, M. Różyczka and H. W. Yorke, Numerical Methods in Astrophysics, Series in Astronomy and Astrophysics, Taylor & Francis, New York, 2007.

[6]

J. ConnorsJ. Howell and W. Layton, Decoupled time stepping methods for fluid-fluid interaction, SIAM Journal on Numerical Analysis, 50 (2012), 1297-1319.  doi: 10.1137/090773362.

[7]

P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.

[8]

M. DobrowolnyA. Mangeney and P. Veltri, Fully developed anisotropic hydromagnetic turbulence in interplanetary space, Phys. Rev. Lett., 45 (1980), 144-147.  doi: 10.1103/PhysRevLett.45.144.

[9]

E. Dormy and M. Núñez, Introduction [Special issue: Magnetohydrodynamics in astrophysics and geophysics], Geophys. Astrophys. Fluid Dyn., 101 (2007), p169. doi: 10.1080/03091920701523287.

[10]

E. Dormy and Andrew M. Soward (eds.), Mathematical Aspects of Natural Dynamos, vol. 13 of Fluid Mechanics of Astrophysics and Geophysics, Grenoble Sciences. Universite Joseph Fourier, Grenoble, 2007. doi: 10.1201/9781420055269.

[11]

W. M. Elsässer, The hydromagnetic equations, Phys. Rev., 79 (1950), 183-183.  doi: 10.1103/PhysRev.79.183.

[12]

A. Fierros Palacios, The Hamilton-type Principle in Fluid Dynamics, Springer, Vienna, 2006, Fundamentals and applications to magnetohydrodynamics, thermodynamics, and astrophysics.

[13]

J. A. Font, General relativistic hydrodynamics and magnetohydrodynamics: Hyperbolic systems in relativistic astrophysics, in Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin, 2008, 3–17. doi: 10.1007/978-3-540-75712-2_1.

[14]

S. GaltierS. V. NazarenkoA. C. Newell and A. Pouquet, A weak turbulence theory for incompressible magnetohydrodynamics, Part of the Lecture Notes in Physics book series, 536 (2000), 291-330.  doi: 10.1007/3-540-47038-7_12.

[15]

P. Goldreich and S. Sridhar, Toward a theory of interstellar turbulence. Ⅱ: Strong Alfvénic turbulence, ApJ, 438 (1995), 763-775. 

[16]

H. Hashizume, Numerical and experimental research to solve MHD problem in liquid blanket system, Fusion Eng. Des., 81 (2006), 1431-1438.  doi: 10.1016/j.fusengdes.2005.08.086.

[17]

N. Haugen, A. Brandenburg and W. Dobler, Simulations of nonhelical hydromagnetic turbulence, Phys. Rev. E, 70 (2004), 016308. doi: 10.1103/PhysRevE.70.016308.

[18]

T. HeisterM. Mohebujjaman and L. G. Rebholz, Decoupled, unconditionally stable, higher order discretizations for mhd flow simulation, Journal of Scientific Computing, 71 (2017), 21-43.  doi: 10.1007/s10915-016-0288-4.

[19]

W. Hillebrandt and F. Kupka (eds.), Interdisciplinary Aspects of Turbulence, vol. 756 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009.

[20]

P. S. Iroshnikov, Turbulence of a conducting fluid in a strong magnetic field, Soviet Astronom. AJ, 7 (1964), 566-571. 

[21]

R. H. Kraichnan, Inertial-range spectrum of hydromagnetic turbulence, Phys. Fluids, 8 (1965), 1385-1387.  doi: 10.1063/1.1761412.

[22]

W. LaytonH. Tran and C. Trenchea, Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows, Numer. Methods Partial Differential Equations, 30 (2014), 1083-1102.  doi: 10.1002/num.21857.

[23]

T. Lin, J. Gilbert, R. Kossowsky and P. S. U. S. COLLEGE., Sea-Water Magnetohydrodynamic Propulsion for Next-Generation Undersea Vehicles, Defense Technical Information Center, 1990, URL http://books.google.com/books?id=GvhwNwAACAAJ.

[24]

E. Marsch, Turbulence in the solar wind, in Reviews in Modern Astronomy (ed. G. Klare), vol. 4 of Reviews in Modern Astronomy, Springer Berlin Heidelberg, 1991,145–156. doi: 10.1007/978-3-642-76750-0_10.

[25]

M. MeneguzziU. Frisch and A. Pouquet, Helical and nonhelical turbulent dynamos, Phys. Rev. Lett., 47 (1981), 1060-1064.  doi: 10.1103/PhysRevLett.47.1060.

[26]

D. Mitchell and D. Gubser, Magnetohydrodynamic ship propulsion with superconducting magnets, J. Supercond., 1 (1988), 349-364.  doi: 10.1007/BF00618593.

[27]

B. Punsly, Black Hole Gravitohydromagnetics, vol. 355 of Astrophysics and Space Science Library, 2nd edition, Springer-Verlag, Berlin, 2008.

[28]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[29]

J. V. ShebalinW. H. Matthaeus and D. Montgomery, Anisotropy in MHD turbulence due to a mean magnetic field, J. Plasma Phys., 29 (1983), 525-547.  doi: 10.1017/S0022377800000933.

[30]

S. SmolentsevR. MoreauL. Bühler and C. Mistrangelo, MHD thermofluid issues of liquid-metal blankets: Phenomena and advances, Fusion Eng. Des., 85 (2010), 1196-1205.  doi: 10.1016/j.fusengdes.2010.02.038.

[31]

D. Sondak and A. A. Oberai, Large eddy simulation models for incompressible magnetohydrodynamics derived from the variational multiscale formulation, Phys. Plasmas, 19 (2012), 102308. doi: 10.1063/1.4759157.

[32]

C. Trenchea, Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows, Appl. Math. Lett., 27 (2014), 97-100.  doi: 10.1016/j.aml.2013.06.017.

[33]

M. K. Verma, Statistical theory of magnetohydrodynamic turbulence: Recent results, Phys. Rep., 401 (2004), 229-380.  doi: 10.1016/j.physrep.2004.07.007.

[34]

I. Veselovsky, Turbulence and waves in the solar wind formation region and the heliosphere, Astrophys. Space Sci., 277 (2001), 219-224.  doi: 10.1007/978-94-010-0904-1_28.

[35]

N. WilsonA. Labovsky and C. Trenchea, High accuracy method for magnetohydrodynamics system in Elsässer variables, Comput. Methods Appl. Math., 15 (2015), 97-110.  doi: 10.1515/cmam-2014-0023.

[36]

G. Yuksel and R. Ingram, Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers, Int. J. Numer. Anal. Model., 10 (2013), 74-98. 

Figure 1.  Log-log plot of the error in Elsässer variables as a function of time step $\Delta t$.
Figure 2.  Energy of the numerical solution.
Table 1.  Convergence rate for algorithm (3.1).
$ \Delta t=h$ $\|z^{+}-z^{+}_{h}\|_{\infty}$ rate $\|\nabla z^{+}-\nabla z^{+}_{h}\|_{2}$ rate $\|z^{-}-z^{-}_{h}\|_{\infty}$ rate $\|\nabla z^{-}-\nabla z^{-}_{h}\|_{2}$ rate
1/16 4.047e-2 - 2.978e+0 - 3.653e-2 - 2.028e+0 -
1/32 6.701e-3 2.59 8.755e-1 1.77 8.536e-3 2.10 7.035e-1 1.53
1/64 1.360e-3 2.30 1.676e-1 2.38 2.101e-3 2.02 1.812e-1 1.96
1/128 3.359e-4 2.02 2.930e-2 2.51 5.217e-4 2.01 4.497e-2 2.01
$ \Delta t=h$ $\|z^{+}-z^{+}_{h}\|_{\infty}$ rate $\|\nabla z^{+}-\nabla z^{+}_{h}\|_{2}$ rate $\|z^{-}-z^{-}_{h}\|_{\infty}$ rate $\|\nabla z^{-}-\nabla z^{-}_{h}\|_{2}$ rate
1/16 4.047e-2 - 2.978e+0 - 3.653e-2 - 2.028e+0 -
1/32 6.701e-3 2.59 8.755e-1 1.77 8.536e-3 2.10 7.035e-1 1.53
1/64 1.360e-3 2.30 1.676e-1 2.38 2.101e-3 2.02 1.812e-1 1.96
1/128 3.359e-4 2.02 2.930e-2 2.51 5.217e-4 2.01 4.497e-2 2.01
Table 2.  Convergence rate for algorithm (3.1).
$ \Delta t=h$ $\|z^{+}_{T}-z^{+}_{T,h}\|_{2}$ rate $\|z^{-}_{T}-z^{-}_{T,h}\|_{2}$ rate
1/10 8.4849e-3 - 8.4844e-3 -
1/20 1.0152e-3 3.0651 1.0143e-3 3.0510
1/30 3.0062e-4 3.0174 2.9832e-4 3.0180
1/40 1.3455e-4 2.7345 1.2995e-4 2.7996
$ \Delta t=h$ $\|z^{+}_{T}-z^{+}_{T,h}\|_{2}$ rate $\|z^{-}_{T}-z^{-}_{T,h}\|_{2}$ rate
1/10 8.4849e-3 - 8.4844e-3 -
1/20 1.0152e-3 3.0651 1.0143e-3 3.0510
1/30 3.0062e-4 3.0174 2.9832e-4 3.0180
1/40 1.3455e-4 2.7345 1.2995e-4 2.7996
[1]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905

[2]

Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465

[3]

Fasma Diele, Angela Martiradonna, Catalin Trenchea. Stability and errors estimates of a second-order IMSP scheme. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022076

[4]

Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic and Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047

[5]

O. Guès, G. Métivier, M. Williams, K. Zumbrun. Boundary layer and long time stability for multi-D viscous shocks. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 131-160. doi: 10.3934/dcds.2004.11.131

[6]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551

[7]

Amjad Khan, Dmitry E. Pelinovsky. Long-time stability of small FPU solitary waves. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2065-2075. doi: 10.3934/dcds.2017088

[8]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401

[9]

Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

[10]

Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems and Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052

[11]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683

[12]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[13]

Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems and Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427

[14]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203

[15]

Margaret Beck. Stability of nonlinear waves: Pointwise estimates. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 191-211. doi: 10.3934/dcdss.2017010

[16]

Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems and Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039

[17]

Z. B. Ibrahim, N. A. A. Mohd Nasir, K. I. Othman, N. Zainuddin. Adaptive order of block backward differentiation formulas for stiff ODEs. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 95-106. doi: 10.3934/naco.2017006

[18]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481

[19]

Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062

[20]

Wojciech M. Zajączkowski. Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1447-1482. doi: 10.3934/cpaa.2019070

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (246)
  • HTML views (441)
  • Cited by (0)

Other articles
by authors

[Back to Top]