volume | PIER Journals

Abstract

We discuss the ab initio rendering of four-dimensional (4-d) spacetime of Maxwell's equations on random (irregular) lattices. This rendering is based on casting Maxwell's equations in the framework of the exterior calculus of differential forms, and a translation thereof to a simplicial complex whereby fields and causative sources are represented as differential p-forms and paired with the oriented p-dimensional geometrical objects that comprise the set of spacetime lattice cells (simplices). We pay particular attention to the case of simplicial spacetime lattices because these can serve as building blocks of lattices made of more generic cells (polygons). The generalized Stokes' theorem is used to construct discrete calculus operations on the lattice based upon combinatorial relations depending solely on the connectivity and relative orientation among simplices. This rendering provides a natural factorization of (lattice) 4-d spacetime Maxwell's equations into a metric-free part and a metric-dependent part. The latter is encoded by discrete Hodge star operators which are built using Whitney forms, i.e., canonical interpolants for discrete differential forms. The derivation of Whitney forms is illustrated here using a geometrical construction based on the concept of barycentric coordinates to represent a point on a simplex, and the generalization thereof to represent higher-dimensional objects (lines, areas, volumes, and hypervolumes) in 4-d. We stress the role of the primal lattice, the barycentric dual lattice, and the barycentric decomposition lattice in achieving a complete description of the lattice theory. Lattice Maxwell's equations based on the exterior calculus of differential forms and on the use of Whitney forms as field interpolants inherits the symplectic structure and discrete analogues of conservation laws present in the continuum theory, such as energy and charge conservation. This framework also provides precise localization rules for the degrees of freedom associated with the different fields and sources on the lattice, and design principles for constructing consistent numerical solution methods that are free from spurious modes, spectral pollution, and (unconditional) numerical instabilities. We also brie y consider the relationship between lattice 4-d Maxwell's equations and some incarnations of discretization schemes for Maxwell's equations in (3+1)-d, such as finite-differences and finite-elements.

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Prof. Fernando Lisboa Teixeira