arXiv subjects

M. Brio

Publications and source records attributed to M. Brio.

3 recordsLinked to original sources

arXiv2022

Localized eigenvectors on metric graphs

Using our previously published algorithm, we analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. As expected, localization of an eigenvector is rare and the network should be tuned to observe exactly localized eigenvectors. We derive the resonance conditions to obtain localized eigenvectors for various geometric configurations and their combinations to form more complicated resonant structures. These localized eigenvectors suggest a new localization indicator based on the $L_2$ norm. They also can be excited, even with leaky boundary conditions, as shown by the numerical solution of the time-dependent wave equation on the metric graph. Finally, the study suggests practical ways to make resonating systems based on metric graphs.

math-ph↗

arXiv2021

Spectral solutions of PDEs on networks

To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields eigenvalues and eigenvectors of arbitary order with machine precision and converges exponentially. These eigenvectors provide a Fourier-like basis on which to expand the solution; however, more complex coupling conditions require additional research. The discontinuous Galerkin method provides approximations of arbitrary polynomial order; however computing high-order eigenvalues accurately requires the respective eigenvector to be well-resolved. The method allows arbitrary non-Kirchhoff flux conditions and requires special penalty terms at the vertices to enforce continuity of the solutions. For the finite difference method, the standard one-sided second-order finite difference stencil reduces the accuracy of the vertex solution to $ O(h^{3/2})$. To preserve overall second-order accuracy, we used ghost cells for each edge. For all three methods we provide the implementation details, their validation, and examples illustrating their performance for the eigenproblem, Poisson equation, and the wave equation.

math.NA↗

arXiv2018

Stable 3D FDTD Method for arbitrary Fully Electric and Magnetic Anisotropic Maxwell Equations

We have developed a new fully anisotropic 3D FDTD Maxwell solver for arbitrary electrically and magnetically anisotropic media for piecewise constant electric and magnetic materials that are co-located over the primary computational cells. Two numerical methods were developed that are called non-averaged and averaged methods, respectively. The non-averaged method is first order accurate, while the averaged method is second order accurate for smoothly-varying materials and reduces to first order for discontinuous material distributions. For the standard FDTD field locations with the co-location of the electric and magnetic materials at the primary computational cells, the averaged method require development of the different inversion algorithms of the constitutive relations for the electric and magnetic fields. We provide a mathematically rigorous stability proof followed by extensive numerical testing that includes long-time integration, eigenvalue analysis, tests with extreme, randomly placed material parameters, and various boundary conditions. For accuracy evaluation we have constructed a test case with an explicit analytic solution. Using transformation optics, we have constructed complex, spatially inhomogeneous geometrical object with fully anisotropic materials and a large dynamic range of $\underline{\epsilon}$ and $\underline{\mu}$, such that a plane wave incident on the object is perfectly reconstructed downstream. In our implementation, the considerable increase in accuracy of the averaged method only increases the computational run time by 20%.

math.NA↗