S. Rjasanow | Searcharxiv

arXiv2024

Generalised Adaptive Cross Approximation for Convolution Quadrature based Boundary Element Formulation

The acoustic wave equation is solved in time domain with a boundary element formulation. The time discretisation is performed with the generalised convolution quadrature method and for the spatial approximation standard lowest order elements are used. Collocation and Galerkin methods are applied. In the interest of increasing the efficiency of the boundary element method, a low-rank approximation such as the adaptive cross approximation (ACA) is carried out. We discuss about a generalisation of the ACA to approximate a three-dimensional array of data, i.e., usual boundary element matrices at several complex frequencies. This method is used within the generalised convolution quadrature (gCQ) method to obtain a real time domain formulation. The behaviour of the proposed method is studied with three examples, a unit cube, a unit cube with a reentrant corner, and a unit ball. The properties of the method are preserved in the data sparse representation and a significant reduction in storage is obtained.

math.NA↗

arXiv2004

A novel formulation of nonlocal electrostatics

The accurate modeling of the dielectric properties of water is crucial for many applications in physics, computational chemistry and molecular biology. This becomes possible in the framework of nonlocal electrostatics, for which we propose a novel formulation allowing for numerical solutions for the nontrivial molecular geometries arising in the applications mentioned before. Our approach is based on the introduction of a secondary field, $ψ$, which acts as the potential for the rotation free part of the dielectric displacement field ${\bf D}$. For many relevant models, the dielectric function of the medium can be expressed as the Green's function of a local differential operator. In this case, the resulting coupled Poisson (-Boltzmann) equations for $ψ$ and the electrostatic potential $ϕ$ reduce to a system of coupled PDEs. The approach is illustrated by its application to simple geometries.

physics.class-ph↗