Jorge I. Rubiano-Murcia

arXiv2023

A Comparison of Two Lattice Boltzmann Models for Electrodynamics

In recent years, various Lattice Boltzmann models for electrodynamics have been developed as alternatives to classical methods such as Finite Difference Time Domain (FDTD) and Finite Element Methods (FEM). However, there has been a lack of systematic comparisons between these models. This paper addresses this gap by comparing two specific Lattice Boltzmann models, published by Mendoza and Muñoz (MM), and Hauser and Verhey (HV), respectively. To compare the models, we utilize time and memory as indicators, considering the same achieved error, in four standard tests: a dielectric pulse traveling through two interfaces, the skin effect, the Hertz dipole, and a dielectric pulse traveling through several interfaces. The results indicate that both methods accurately simulate the tests and exhibit convergence as the mesh is refined. However, the MM method outperforms the HV method regarding time, while its memory efficiency was lower. The modified Hauser-Verhey model demonstrates itself to be a promising alternative to the Mendoza-Muñoz model. These findings contribute to the ongoing development and optimization of numerical methods for electromagnetics simulations.

physics.comp-ph↗

arXiv2023

Higher order derivatives of the adjugate matrix and the Jordan form

In this short note, we show that the higher-order derivatives of the adjugate matrix $\mbox{Adj}(z-A)$, are related to the nilpotent matrices and projections in the Jordan decomposition of the matrix $A$. These relations appear as a factorization of the derivative of the adjugate matrix as a product of factors related to the eigenvalues, nilpotent matrices and projectors. The novel relations are obtained using the Riesz projector and functional calculus. The results presented here can be considered to be a generalization of Thompson and McEnteggert's theorem relating the adjugate matrix to the orthogonal projection on the eigenspace of simple eigenvalues for symmetric matrices. They can also be seen as a complement to some earlier results by B. Parisse, M. Vaughan that relate derivatives of the adjugate matrix to the invariant subspaces associated with an eigenvalue. Our results can also be interpreted as a general eigenvector-eigenvalue identity. Many previous works have dealt with relations between the projectors on the eigenspaces and the derivatives of the adjugate matrix with the characteristic spaces but it seems that there is no explicit mention in the literature of the factorization of the higher-order derivatives of the adjugate matrix as a matrix multiplication involving nilpotent and projector matrices, which appear in the Jordan decomposition theorem.

math.FA↗