An Efficient Numerical Method for Time Domain Electromagnetic Wave Propagation in Co-axial Cables

Abstract

In this work, we construct an efficient numerical method to solve 3D Maxwell’s equations in coaxial cables. Our strategy is based upon a hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objectives is to validate numerically generalized telegrapher’s models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article.

An implicit–explicit time discretization for elastic wave propagation problems in plates

“…The last time scheme we consider consists in treating implicitly the part of the stiffness bilinear form that depends solely on the derivative in the thickness direction while the remaining part is treated explicitly. This approach is linked to the one presented in References 21‐23 and is referred to as an implicit–explicit strategy. It reads $$$$ {m}_{\mathfrak{h}}\left(\frac{{\mathbf{u}}_{\mathfrak{h}}^{n+1}-2{\mathbf{u}}_{\mathfrak{h}}^n+{\mathbf{u}}_{\mathfrak{h}}^{n-1}}{\Delta {t}^2},{\mathbf{v}}_{\mathfrak{h}}\right)+\left({a}_{\mathfrak{h}}^{\delta }-{\delta}^{-2}{a}_{\nu \nu, \mathfrak{h}}\right)\left({\mathbf{u}}_{\mathfrak{h}}^n,{\mathbf{v}}_{\mathfrak{h}}\right)+{\delta}^{-2}{a}_{\nu \nu, \mathfrak{h}}\left({\left\{{\mathbf{u}}_{\mathfrak{h}}^n\right\}}_{\theta },{\mathbf{v}}_{\mathfrak{h}}\right)=\ell \left({t}^n;{\mathbf{v}}_{\mathfrak{h}}\right), $$$$ completed with null initialization (12).…”

“…Theorem 2. We denote by 𝜒 ⋆ the condition number of the linear system (23) preconditioned by the operator a 𝛿 𝔥,imex (⋅, ⋅), namely…”

“…The last time scheme we consider consists in treating implicitly the part of the stiffness bilinear form that depends solely on the derivative in the thickness direction while the remaining part is treated explicitly. This approach is linked to the one presented in [21,22,23] and is referred to as an implicit-explicit strategy. It reads…”

“…Applying the standard explicit scheme on this re‐scaled formulation shows that its related stability condition is proportionate to $$$ \delta $$$. After revisiting this well‐known constraint on the explicit scheme, and comparing it with the unconditionally stable implicit scheme, we apply an implicit–explicit scheme—see for example, the related prior works of References 21‐23. For this implicit–explicit scheme we are able to prove that there exists a sufficient stability condition that is independent of $$$ \delta $$$.…”

“…After revisiting this well-known constraint on the explicit scheme, and comparing it with the unconditionally stable implicit scheme, we apply an implicit-explicit scheme -see e.g. the related prior works of [21,22,23]. For this implicit-explicit scheme we are able to prove that there exists a sufficient stability condition that is independent of δ.…”