LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Abstract

Abstract-We discuss the ab initio rendering 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 pdimensional 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. It 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 briefly 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.

Finite element time-domain body-of-revolution Maxwell solver based on discrete exterior calculus

“…In this paper, we present a mixed FE BOR solver for time-domain Maxwell's curl equations based on transformation optics (TO) [31,32,33,34,35,36] and discretization principles based on the discrete exterior calculus (DEC) of differential forms [23,27,37,38,39,40,41,42,43,44]. We explore TO principles to map the original threedimensional (3-D) BOR problem to an equivalent problem on the 2-D meridian plane where the resulting metric is not the cylindrical one but instead the Cartesian one (i.e., with no radial factors present).…”

Diagnosing numerical Cherenkov instabilities in relativistic plasma simulations based on general meshes

“…The present FETD field solver is based on the exterior calculus representation of Maxwell's equations [43,44,45,46], and on expressing the field unknowns in terms of discrete differential forms, also known as Whitney forms [47,48]. On a general mesh, the electric field intensity (1-form) and the magnetic flux density (2-form) can be expanded using a linear combination of Whitney 1-forms w (1) j associated with edges (1-cells) j = 1, .…”

“…k , can be found in [45]. By substituting (A.1) and (A.2) into Faraday's law, applying the Generalized Stokes' Theorem [45,48,49], and using a leap-frog time integration scheme, a discrete representation of Faraday's law on a general mesh is obtained as…”

“…where the superscript n denotes the time step index, e = [e 1 , e 2 , ..., e N1 ] T , b = [b 1 , b 2 , ..., b N2 ] T , and C is an incidence matrix encoding the discrete exterior derivative on the mesh [45,48,50,51]. The elements of C are in the set {−1, 0, 1}.…”

“…where [ ] and µ −1 are discrete Hodge operators (square positive-definite matrices) encoding both the constitutive properties of the background medium and metric information of the mesh [45,48,47,30,52,53,31]. The elements of [ ] and µ −1 can be computed by the integrals [48,30,52,31].…”