L²-stability of the upwind first order finite volume scheme for the Maxwell equations in two and three dimensions on arbitrary unstructured meshes

Abstract

Abstract.We investigate sufficient and possibly necessary conditions for the L 2 stability of the upwind first order finite volume scheme for Maxwell equations, with metallic and absorbing boundary conditions. We yield a very general sufficient condition, valid for any finite volume partition in two and three space dimensions. We show this condition is necessary for a class of regular meshes in two space dimensions. However, numerical tests show it is not necessary in three space dimensions even on regular meshes. Stability limits for time and space schemes with higher orders of accuracy are numerically investigated. Résumé. Nous cherchonsàétablir des conditions suffisantes etéventuellement nécessaires de stabilitéL 2 pour le schéma en volumes finis décentré du premier ordre, appliqué auxéquations de Maxwell, avec des conditions aux limites absorbantes et métalliques. En deux et trois dimensions d'espace, nous proposons une condition suffisante de stabilité d'une grande généralité, puisqu'elle est valable pour toute forme de volumes finis. En deux dimensions, nous montrons que cette condition estégalement nécessaire pour une classe de maillages réguliers. En trois dimensions, des tests numériques montrent que cette condition n'est pas nécessaire, même pour des maillages réguliers. Nous recherchons les limites de stabilité observées numériquement pour des schémas plus précis en temps et en espace.

Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes

“…We notice here that the situation is quite different from the proof of the L 2 -stability of the first-order upwind finite-volume scheme of [19], where the energy was obviously a positive definite quadratic form of all unknowns. At the same time, the energy proposed here depends explicitly on the numerical scheme, since it can be only written as a quadratic form of unknowns (…”

“…We use the same kind of energy approach as in [19], where a quadratic form plays the role of a Lyapunov function of the whole set of numerical unknowns.…”

“…piecewise constant, discontinuous, Galerkin-type finite element approximation) have been developed on body-fitted coordinates [23], on unstructured finite element triangulations [4] or on totally destructured meshes [1]. First-order conservative upwind schemes, for which stability [19], convergence [8] and L 1 error estimates of h 1/2 [25] were proven, are too dissipative to be used for the Keywords and phrases. Electromagnetics, finite volume methods, discontinuous Galerkin methods, centered fluxes, leap-frog time scheme, L 2 stability, unstructured meshes, absorbing boundary condition, convergence, divergence preservation.…”

Discontinuous Galerkin frequency domain forward modelling for the inversion of electric permittivity in the 2D case

“… a centred flux (see Fezoui et al 2005) for the time‐domain equivalent): an upwind flux (see Piperno 2000; Ern and Guermond 2006): …”

The finite-volume time-domain algorithm using least square method in solving Maxwell’s equations

“…The general stability condition of explicit Runge-Kutta scheme for two-dimensional and three-dimensional FVTD method on arbitrary unstructured mesh can be found in reference [31].…”