An approximate resolution of a free boundary problem appearing in the equilibrium plasma by means of conformal mapping
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It is not easy, even numerically, to obtain shapes of one-component symmetric equilibrium plasma in the vessel which has a symmetric cross section. Because boundaries between plasma and vacuum are free boundaries, we are usually required to use a method with iteration technique to obtain them. However, the efficiency of these methods depends greatly on the choice of starting values. In the case of the polygonal fixed boundary, a method without iteration technique was developed by Demidov using a hodograph transformation. However, the polygonal fixed boundaries are too special as the fixed boundary prescribing the vessel. Therefore, in this paper, we propose an approximate resolution to obtain the shapes of plasma for the cases which have not the fixed boundary mentioned above. The characteristic of our method is to use the conformal mapping.
In order to check the validity of our method, we compare numerical results by Powell method using iteration technique with those by our method. As the starting value of the iteration in Powell method, our numerical results are used. We varied the value of flux constant κ to find that our new method is effective for a large flux constant κ. Even if κ is not so large, our results may still be useful as a starting value for numerical computations with iteration technique.
Key wordsequilibrium plasma free boundary conformal mapping harmonic function holomorphic function
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