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Numerical methods for thermally driven incompressible Magnetohydrodynamic problems

Abstract :  We study numerical methods for  the time-dependent magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation, in which the Joule effect and Viscous heating are taken into account.  To overcome the difficulties of  very low regularity of  the heat source terms, a regularized weak system is proposed to deal with Joule and Viscous heating terms.  We  consider an Euler semi-implicit  semi-discrete scheme  for the regularized system.  As both discrete parameter and regularization parameter tend to zero, we prove that the discrete solution converges to a weak solution of the original problem. Next, we consider the fully discrete Euler semi-implicit scheme based on  the mixed finite method to approximate the  fluid  equation and N$/mathrm{/acute{e}}$d$/mathrm{/acute{e}}$lec edge element to  the magnetic induction. The fully discrete scheme requires only solving a linear system per time step.   The error estimates for  the velocity, magnetic induction and temperature are derived under  a proper regularity  assumption for the exact solution. Finally, several numerical examples are performed to demonstrate  both accuracy and efficiency of our proposed scheme.

个人简介:毛士鹏,中国科学院数学与系统科学研究院研究员。2008年获中国科学院数学与系统科学研究院理学博士学位, 然后留所工作至今,期间于2008-2012年分别在法国的INRIA以及在瑞士苏黎世联邦理工学院(ETH Zurich) 做博士后。主要研究工作有限元方法及其应用, 磁流体力学计算等方面, 至今已在在 Numer. Math.、Math. Comp., SIAM. J. Numer. Math., SIAM J.Sci.Computing,M3AS等杂志上发表论文60余篇。 曾入选中科院青年创新促进会会员和获得中科院朱李月华优秀教师奖。