Convergence of a Finite Difference Scheme for Two-Dimensional Incompressible Magnetohydrodynamics

We investigate a finite difference scheme for the two-dimensional, incompressible magnetohydrodynamics equations that was introduced in [J.-G. Liu and W.-C. Wang, J. Comput. Phys., 174 (2001), pp. 12--37]. It uses central difference and averaging operators on a staggered grid and was shown not only to keep the magnetic and the velocity field divergence free but, moreover, to preserve the energy exactly. We extend Liu and Wang's result by a discrete $H^1$ bound, or more precisely, we show that the discrete solution for the velocity and the magnetic field is bounded in $L^\infty(0,\infty;H^1(\Omega))$ and $L^2(0,T;H^2(\Omega))$. This bound allows us to prove the convergence of the scheme. The convergence is strong in $L^2$ and weak in $H^1$.

[1]  W. Habashi,et al.  A finite element method for magnetohydrodynamics , 2001 .

[2]  Jian-Guo Liu,et al.  Energy and helicity preserving schemes for hydro- and magnetohydro-dynamics flows with symmetry , 2004 .

[3]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[4]  Roger Temam,et al.  Some mathematical questions related to the MHD equations , 1983 .

[5]  Jian-Guo Liu,et al.  Convergence Analysis of the Energy and Helicity Preserving Scheme for Axisymmetric Flows , 2006, SIAM J. Numer. Anal..

[6]  Chi-Wang Shu,et al.  Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations , 2005, J. Sci. Comput..

[7]  Robert L. Pego,et al.  Stable discretization of magnetohydrodynamics in bounded domains , 2010 .

[8]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[9]  Jinchao Xu,et al.  Stable finite element methods preserving ∇·B=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \cdot \varvec{B}= , 2014, Numerische Mathematik.

[10]  G. Kanschat.  Divergence‐free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme , 2008 .

[11]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[12]  G. Gustafson,et al.  Boundary Value Problems of Mathematical Physics , 1998 .

[13]  Dominik Schötzau,et al.  Mixed finite element methods for stationary incompressible magneto–hydrodynamics , 2004, Numerische Mathematik.

[14]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[15]  J. L. Lions,et al.  Inéquations en thermoélasticité et magnétohydrodynamique , 1972 .

[16]  M. Gunzburger,et al.  On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics , 1991 .

[18]  Ming Wang,et al.  Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations , 2015, J. Sci. Comput..

[19]  X. Wu,et al.  Analysis and convergence of the MAC scheme. II. Navier-Stokes equations , 1996, Math. Comput..

[20]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[21]  D. Schötzau,et al.  A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics , 2010 .

[22]  Zhouping Xin,et al.  On the regularity of weak solutions to the magnetohydrodynamic equations , 2005 .

[23]  Magnus Svärd,et al.  Higher-order finite difference schemes for the magnetic induction equations with resistivity , 2011 .

[24]  Spencer Frei,et al.  Weak convergence methods for nonlinear partial differential equations , 2012 .

[25]  L. Driel-Gesztelyi.  An Introduction to Magnetohydrodynamics , 2004 .

[26]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[27]  Vivette Girault,et al.  Finite-element error estimates for the MAC scheme , 1996 .

[28]  Richard E. Mortensen,et al.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam) , 1991, SIAM Rev..

[29]  Houde Han,et al.  A New Mixed Finite Element Formulation and the MAC Method for the Stokes Equations , 1998 .

[30]  J. A. Shercliff,et al.  A Textbook of Magnetohydrodynamics , 1965 .

[31]  Jian-Guo Liu,et al.  An energy-preserving MAC-Yee scheme for the incompressible MHD equation , 2001 .

[32]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .