High-Order Time-Domain Methods for Maxwells Equations

Abstract

Abstract : A concentrated effort has recently been initiated to develop methods for solving the time domain Maxwells equations in complex geometries and media using an fully unstructured grid. The USEMe code (Unstructured Spectral Element Method) is the resulting implementation of a numerical scheme which employs nodal high-order/spectral multi-domain methods. Each component of the code has been carefully designed with efficiency, accuracy, flexibility, and robustness in mind. The standard geometric building block by which the computational domain is gridded in USEME is the triangular element in 2D, and the tetrahedral element in 3D. This allows for the use of standard finite element or finite volume type meshes using these elements to describe arbitrarily complex domains, hence overcoming one of the main concerns associated with the structured grid methods. The use of triangles and tetrahedrons remains of the most important design principles of the method as it enables a tight integration with industry standard mesh generators. USEMe is based on penalty/discontinuous Galerkin (DGM) type scheme, which is a particular form of a penalty scheme that guarantees elemental conservation and global stability. Neighboring elements are patched together through the use of penalty terms to account for jumps across their shared boundary. This formulation has the advantage that it allows material coefficients to vary discontinuously at the interface between two elements, while still preserving high order accuracy. Additionally this allows for efficient communication in parallel computations.