Nicolas Salles et Benjamin Stamm
New analytical results on the convergence of a Convolution Quadrature method
Convolution Quadrature (CQ) methods are Laplace transform type methods for the solution of time-domain wave problems using Boundary Integration Equations. I will present new analytical results on the convergence of the numerical solution of acoustic problems computed using a CQ method with multistep schemes. We will discuss the connection between the radius of analyticity of the boundary condition, the radius of analyticity of the Laplace-domain solution (related to the scattering poles of the solution operator), the underlying time-stepping scheme and convergence of CQ methods for the acoustic equation. Numerical examples computed with BEM++ for the scattering by the unit sphere and by some trapping domains will be presented.
Mathematics meets chemistry: a new paradigm for implicit solvation models
Computational chemistry has attracted much attention since these years choice of the Nobel prize winners in chemistry for their contributions to multi-scale models such as the QM/MM method. The large majority of chemically interesting phenomena take place in condensed phase, where the environment (e.g., solvent) can play a crucial role in determining the structure, the properties and the dynamics of the system to be studied. In a practical context, accounting for all solvent molecules in an either Molecular Dynamics (MD) or even Quantum Mechanical (QM) computation is infeasible due to the complexity of the underlying equations. A particular choice of a multi-scale model is to model the solvent environment to be a conducting continuum medium (COSMO-model) and the resulting electrostatic energy contribution to the solvation energy can be computed by solving a Poisson equation in the Van der Waals cavity of the solute molecule.
The mathematical problem is therefore set (and well-posed) and we illustrate how to approximate its solution using Schwarz's domain decomposition method adapted to integral equations. In this manner, the problem can be solved iteratively and the coupling of the local problems is determined by the connectivity of the molecule (in contrast to fast multipole-based boundary element solvers). The resulting numerical scheme is extremely fast and a linear scaling of the computing resources with respect to the number of atoms of the solute molecule can be achieved. In numerical examples we show how this approach outperforms existing methods such as the fast multipole method.