# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - B, 2018, 23 (7) : 2803-2823. doi: 10.3934/dcdsb.2018106
##### References:

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##### References:
Log-log plot of the error in Elsässer variables as a function of time step $\Delta t$.
Energy of the numerical solution.
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
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
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