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By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, the Godunov scheme applied to the resolution of the compressible Euler or Navier-Stokes system is accurate at low Mach number when the mesh is triangular or tetrahedral and is not accurate when the mesh is a 2D (or 3D) cartesian mesh. This approach confirms also the fact that a Godunov scheme remains accurate when it is modified by simply centering the discretization of the pressure gradient.

A new finite volume method is presented for discretizing the two-dimensional Maxwell equations. This method may be seen as an extension of the covolume type methods to arbitrary, possibly non-conforming or even non-convex, n-sided polygonal meshes, thanks to an appropriate choice of degrees of freedom. An equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles. The main properties of the scheme are its energy conservation, its stability under a CFL-like condition, and the fact that it preserves Gauss' law and divergence free magnetic fields. Second-order convergence is demonstrated numerically on non-conforming and distorted meshes.

A self-consistent nonlinear model of an isotope separation process based on selective ion cyclotron resonance heating in a magnetized plasma is presented, and its numerical resolution is described. The response of the electrons to the electromagnetic field is modeled by a cold and linear conductivity tensor, while a particle method is used to solve nonlinear Vlasov equations for the ions. The resolution of the time-harmonic Maxwell equations is achieved by a finite-element method. Both steps are coupled by an iterative procedure, which shows fast convergence. Results are presented for the case of a solenoidal launching antenna.

Usually, non-stationary numerical calculations in electromagnetics are based on the hyperbolic evolution equations for the electric and magnetic fields and leave Gauss' law out of consideration because the latter is a consequence of the former and of the charge conservation equation in the continuous case. However, in the simulation of the self-consistent movement of charged particles in electromagnetic fields, it is a well-known fact that the approximation of the particle motion introduces numerical errors and that, consequently, the charge conservation equation is not satisfied on the dicrete level. Then, in order to avoid the increase of errors in Gauss' law, a divergence cleaning step which solves a Poisson equation for a correction potential is often added. In the present paper, a new method for incorporating Gauss' law into non-stationary electromagnetic simulation codes is developed, starting from a constrained formulation of the Maxwell equations. The resulting system is hyperbolic, and the divergence errors propagate with the speed of light to the boundary of the computational domain. Furthermore, the basic ideas of the numerical approximation are introduced and the extended hyperbolic system is treated numerically within the framework of high-resolution finite-volume schemes. Simulation results obtained with this new technique for pure electromagnetic wave propagation and for an electromagnetic particle-in-cell computation are presented and compared with other methods.