A stable formulation of resonant Maxwell’s equations in cold plasma

Abstract

We consider a boundary value problem (BVP) for a reduced system of time harmonic Maxwell equations in magnetized plasma. The dielectric tensor is strongly anisotropic and the system admits resonant solutions in the context of the limit absorption principle. In particular, in the vanishing viscosity limit the normal component of the electric field becomes infinite and non integrable at the resonant point, and the system becomes ill-posed. In this article we recast the problem in the framework of mixed variational problems and we propose a well-posed formulation that characterizes the singular limit solutions. A key tool is the method of manufactured solutions [7] to construct an integral variational characterization of the jump conditions at the resonance. The well posedness is demonstrated and basic numerical results illustrate the robustness of our approach.

Degenerate elliptic equations for resonant wave problems

“…For the singular part, we will follow a limit absorption principle which relies on the regularized problem. The classical way is to introduce a complex shift α + iν [19,5], and then pass to the limit ν → 0 + . We will prove that for ν = 0 + the limit solution decomposes into a regular part in the weighted space Q plus a complementary singular part.…”

“…Resonant waves appear in various electromagnetic phenomenons, such as: fusion plasma heating where a wave is sent inside a plasma and transfers energy to the particles in a localized region [9,7,19]; cloaking devices where the transition between metamaterials of negative index and non-dissipative dielectrics is exploited [1,6,17]; or photoacoustic imaging of biological tissues where metallic nanoparticles' heating [22] is used. The model problem for resonant waves throughout this work is the degenerate equation, elliptic where α < 0, − div (α∇u) − u = 0 in Ω ⊂ R 2 , α∂ n u + iλu = f on Γ, (1.1) where λ > 0 is a positive scalar and f ∈ L 2 (Γ) is complex valued on the boundary Γ = ∂Ω.…”

“…For simplicity, we will consider from now on that δ is constant. In problem (1.2), the normal component of the electric field is expected [7,19] to have a singularity at Σ of the order of 1/α. This singularity does not belong to L 2 (Ω), nor to L 1 (Ω).…”

“…A classical mean to recover the correct solution of such time harmonic wave equations is to resort to a limiting absorption principle. In this work, as in previous 1D works [7,19], a small parameter ν > 0 is introduced, which regularizes the equation and corresponds to a negligible physical quantity. The regularized problem is then…”

Slab-geometry surface waves on steep gradients and the origin of related numerical issues in a variety of ICRF codes

“…In numerical ICRF calculations, non-physical surface waves often arise when an artificial discontinuity is introduced in the plasma density profile. A common reason to introduce this density jump is the desire to avoid numerical issues associated with the numerical resolution of the lower hybrid resonance, such as those described by Nicolopoulos, Campos-Pinto & Després (2019). In such calculations, much of edge plasma density gradient is replaced by a single density jump, on which numerical surface waves often occur.…”