The inverse problem for the Grad – Shafranov equation with application to magnetic field computation in tokamakдоклад на конференции

• Авторы:
• Международная Конференция : Mathematical Modeling and Computational Physics (MMCP'2013)
• Даты проведения конференции: 8-12 июля 2013
• Тип доклада: не указан
• Докладчик: не указан
• Место проведения: Дубна, Russia
• Аннотация доклада:

The well--known model [1] of magnetic field in tokamak reduces to solving the Grad - Shafranov equation $\Delta u (x) = a u (x) + b$ in the cross--section $G$ of plasma coil with homogeneous Dirichlet condition on its boundary $\Gamma$, which is supposed to be piece-wise $C^{3, \alpha}$--smooth; constants $a$, $b$ are unknown. In [2] this statement was supplemented with the non--local condition $\int_\Gamma \partial_\nu u (x)\, ds = 1$, which physically means prescribing the value of full current; here $ds$ is length element of arc $\Gamma$, \partial_\nu$is normal derivative to$\Gamma$. The non--local condition states an explicit relation$b = b (a)$between parameters$a$and$b$of the equation, and so makes the problem stated above depending only on parameter$a$. The inverse problem for the Grad - Shafranov equation with non-local condition consists in finding parameter$a$via the value of normal derivative$\partial_\nu u (x)$in any point$x$belonging to special subset$\widetilde{\Gamma}$of boundary$\Gamma$. In the present work the necessary and sufficient conditions of unque solvability of the inverse problem are stated. An effective analytic--numerical method is elaborated for finding parameter$a$, including an algorithm of constructing subset$\widetilde{\Gamma}$. Those results were obtained by the use of the multipole method [3] that ensures high precision computation of normal derivative$\partial_\nu u (x)$and by the use of asymptotics [4] as$a \to \infty$for$\partial_\nu u (x)$and$\frac{d}{da}\, \partial_\nu u (x)$,$x \in \Gamma\$. [1] A.Demidov, 2nd Conference on Inverse Proplems, Control and Shape Optimization. Carthage, Tunisie, April 10-12, 2002, Abstracts, 93--94. [2] V.I.Vlasov, Boundary value problems in domains with curvilinear boundary, Computing Centre AS USSR, Moscow, (1987) [in Russian]. [3] A.S.Demidov, M.Moussaoui, Inverse Problems, 20 (2004), 137-154.

• Добавил в систему: Безродных Сергей Игоревич