Coupling strategies of Compatible Discrete Operators schemes applied to Maxwell equations in time domain

Overview & Synopsis

In numerical simulations of Maxwell's equations, one of the main goals is to accurately represent the physical reality of electromagnetic fields while keeping a low computational cost. Numerous methods exist for solving the system in the time domain, each with its own strengths and weaknesses, depending on the situation. In this thesis, we focus on two coupling strategies of Compatible Discrete Operators (CDO) schemes applied to Maxwell's equations in time domain. The first consists in locally defining the metric of the scheme by considering the mesh geometry. In the second approach, the computational domain is partitioned in two subdomains and the coupling is achieved by defining operators on the interface. To this end, Maxwell's equations are studied in two parts: the topological relations and the constitutive relations. In the CDO framework, the topological relations are formulated using discrete differential operators corresponding to the discretization of the classical vector operators. In order to take into account non-homogeneous boundary conditions, these operators are extended using a dual boundary mesh. The constitutive relations are formulated using discrete Hodge operators. They define the metric of the scheme and depend on the material parameters. The discrete scheme in space and time is then analyzed in terms of stability and consistency. We then test it on different configurations using hybrid meshes.