We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem (GEP). The layered piezo device is studied for resonance frequencies using different meshes, sparse matrix representations, and numerical methods in COMSOL Multiphysics and ACELAN-COMPOS packages. Specifically, the matrix-vector and matrix-matrix product implementation for large sparse matrices is discussed. The shift-and-invert Lanczos method is used to solve the partial symmetric GEP numerically. Different solvers are compared in terms of the efficiency. The results of numerical experiments are presented.

Energy-conserving Hermite methods for solving Maxwell’s equations in dielectric and dispersive media are described and analyzed. In three space dimensions, methods of order 2m to

require

{(m+1)}^3

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(m+1)^3$$\end{document}

degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of m. We prove the stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special semi-norm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of the electromagnetic wave propagation over thousands of wavelengths.

We introduce in this paper two time discretization schemes tailored for a range of Wasserstein gradient flows. These schemes are designed to preserve mass and positivity and to be uniquely solvable. In addition, they also ensure energy dissipation in many typical scenarios. Through extensive numerical experiments, we demonstrate the schemes’ robustness, accuracy, and efficiency.