Numerical solution of a nonlocal problem arising in plasma physics

Bermúdez, Alfredo; Seoane, M. L.

doi:10.1016/s0895-7177(98)00017-x

Cited by 4 publications

(6 citation statements)

References 4 publications

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“…From the above a priori estimate, we deduce that solutions to (8) are defined in the whole interval [τ, T ], and the sequence {u n } is bounded in…”

Section: Existence Of a Weak Solutionmentioning

confidence: 99%

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Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

2015

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In this paper, we consider a non-autonomous nonlocal reactiondiffusion equation with a small perturbation in the nonlocal diffusion term and the non-autonomous force. Under the assumptions imposed on the viscosity function, the uniqueness of weak solutions cannot be guaranteed. In this multi-valued framework, the existence of weak solutions and minimal pullback attractors in the L 2 -norm are analysed. In addition, some relationships between the attractors of the universe of fixed bounded sets and those associated to a universe given by a tempered condition are established. Finally, the upper semicontinuity property of pullback attractors w.r.t. the parameter is proved. Indeed, under suitable assumptions, we prove that the family of pullback attractors converges to the corresponding global compact attractor associated to the autonomous nonlocal limit problem when the parameter goes to zero.Keywords nonlocal diffusion · reaction-diffusion equations without uniqueness · pullback attractors · upper semicontinuity of attractors · multivalued dynamical systems.

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“…From the above a priori estimate, we deduce that solutions to (8) are defined in the whole interval [τ, T ], and the sequence {u n } is bounded in…”

Section: Existence Of a Weak Solutionmentioning

confidence: 99%

“…t ∈ (τ, T n ), where (τ, T n ) is an interval of existence of solutions to (8) by the Carathéodory Theorem. From (4),…”

Section: Existence Of a Weak Solutionmentioning

confidence: 99%

Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

2015

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“…Analogous discretization and approximation techniques can be employed for 2 − D and 3 − D nonsymmetric problems (see [5] for (PS)).…”

Section: The Nonsymmetric Casementioning

confidence: 99%

Numerical Approximations of the Relative Rearrangement: The piecewise linear case. Application to some Nonlocal Problems

Rakotoson

Seoane

2000

ESAIM: M2AN

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Abstract.We first prove an abstract result for a class of nonlocal problems using fixed point method.We apply this result to equations revelant from plasma physic problems. These equations contain terms like monotone or relative rearrangement of functions. So, we start the approximation study by using finite element to discretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.

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“…As usual we have replace the physical notation of the unknown ψ by the more common u in mathematics. The equivalent problem to (14), (19), (18) but in Cartesian coordinate, is problem (P I ), what stated at the beginning of this section.…”

Section: Introductionmentioning

confidence: 99%

“…To study problem (P I ), following [8], we will reformulate it in terms of a new problem (P * ), of nonlocal nature, where we replace the unknown function F , by a term involving the function u and some functions associated to u: its decreasing rearrangement u * , and the relative rearrangement of b with respect of u, b * u . In order to justify it, we observe that by the Cavalieri Theorem, condition (18) can be written in terms of…”

Section: Introductionmentioning

confidence: 99%

On a mathematical model arising in MHD perturbed equilibrium for Stellarator devices. A numerical approach

Díaz

Sastre²,

Padial³

2012

2012 International Conference on High Performance Computing &Amp; Simulation (HPCS)

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Abstract-We consider a mathematical model related to the stationary regime of a plasma of fusion nuclear, magnetically confined in a Stellarator device. Using the geometric properties of the fusion device, a suitable system of coordinates and averaging methods, the mathematical problem may be reduced to a two dimensional free boundary problem of nonlocal type, where the corresponding differential equation is of the Grad-Shafranov type. The current balance within each flux magnetic gives us the possibility to define the third covariant magnetic field component with respect to the averaged poloidal flux function. We present here some numerical experiences and we give some numerical approach for the averaged poloidal flux and for the third covariant magnetic field component.

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Numerical solution of a nonlocal problem arising in plasma physics

Cited by 4 publications

References 4 publications

Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

Numerical Approximations of the Relative Rearrangement: The piecewise linear case. Application to some Nonlocal Problems

On a mathematical model arising in MHD perturbed equilibrium for Stellarator devices. A numerical approach

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