Partitioned second order method for magnetohydrodynamics in Elsässer variables

Yong Li; Catalin Trenchea

doi:10.3934/dcdsb.2018106

Article Contents

2018, Volume 23, Issue 7: 2803-2823. Doi: 10.3934/dcdsb.2018106

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Partitioned second order method for magnetohydrodynamics in Elsässer variables

Yong Li and
Catalin Trenchea^,

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

* Corresponding author
* Corresponding author

Received: February 28, 2017

Revised: July 31, 2017

Early access: April 2018

Published: July 2018

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.

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Abstract

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.

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Mathematics Subject Classification: 76W05, 65M12, 65M60, 76D05, 35Q30, 35Q61.

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Figure 1. Log-log plot of the error in Elsässer variables as a function of time step $\Delta t$.

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Figure 2. Energy of the numerical solution.

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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

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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

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