SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Preserving Taylor's constraint in magnetohydrodynamics

J. Niesen and P. Livermore

Abstract
The Earth's magnetic field is generated in the outer core, whose behaviour is described by a combination of the Navier-Stokes equation and the Maxwell equation. A problem when solving these equation is that a certain number, the Ekman number, is very small (around $10^{-15}$). Usually, people use big computers to solve the equations, but even then the Ekman number in the simulations can only be taken to be $10^{-6}$. The alternative is to take the Ekman number equal to zero. This corresponds to a singular limit, in which the equations degenerate into a partial differential algebraic system. Taylor's constraint is an infinite number of quadratic conserved quantities in this limit and it is thought that this constraint causes instabilities in earlier attempt to solve the equations. I will describe our on-going efforts to design and implement a numerical method for solving the equations which preserves Taylor's constraint, making it hopefully stable.

Organized by         Universidad de Valladolid     IMUVA