This thesis is concerned with numerical approximation of linear stochastic partial differential equations driven by additive noise. In the first part, we develop a framework for the analysis of weak convergence and within this framework we analyze the stochastic heat equation, the stochastic wave equation, and the linearized stochastic Cahn-Hilliard, or the linearized Cahn-Hilliard-Cook equatio...

The accuracy of atomistic-to-continuum hybrid methods can be guaranteed only for deformations where the lattice configuration is stable for both the atomistic energy and the hybrid energy. For this reason, a sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation of lattice defects. We formulate a simple one...

A sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation and motion of lattice defects. We formulate a simple one-dimensional model problem and give a detailed analysis of the stability of the force-based quasicontinuum (QCF) method. The focus of the analysis is the question whether the QCF method is able t...

<p style='text-indent:20px;'>In the following work, we consider Boltzmann equation that models a diatomic gas by representing microscopic internal energy continuous variable I. Under some convenient assumptions on collision cross-section <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, prove linearized ope...

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker-Planck-Landau operator 8], we establish in this paper analogous results for two more realistic collision operators: The rst one is the Fokker-Planck-Landau collision operator obtained by relativistic calculations of binary interactions , and the second is a collision operator (of Fokker-Planck...

We consider a class of physiologically structured population models, a first order nonlinear partial differential equation equipped with a nonlocal boundary condition, with a constant external inflow of individuals. We prove that the linearized system is governed by a quasicontraction semigroup. We also establish that linear stability of equilibrium solutions is governed by a generalized net re...

In this paper we show how the simplest wave structure that balances compression and rarefaction in the nonlinear compressible Euler equations can be represented in a solution of the linearized compressible Euler equations. Such waves are exact solutions of the equations obtained by linearizing the compressible Euler equations about the periodic extension of two constant states separated by entr...

In [13] Prüß investigated a very general nonlinear age-structured model consisting of n species. He proved the principle of linearized stability, which means that the asymptotic behavior of stationary solutions of the nonlinear system is determined by the spectrum of the linearized operator. In other words the stability is determined by the roots of a complex valued characteristic function as i...