Error analysis of trigonometric integrators for semilinear wave equations

An error analysis of trigonometric integrators (or exponential integrators) applied to spatial semi-discretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given. In particular, optimal second-order convergence is shown requiring only that the exact solution is of finite energy. The analysis is uniform in the spatial discretization parameter. It covers the impulse method which coincides with the method of Deuflhard and the mollified impulse method of Garc\'ia-Archilla, Sanz-Serna & Skeel as well as the trigonometric methods proposed by Hairer & Lubich and by Grimm & Hochbruck. The analysis can also be used to explain the convergence behaviour of the St\"ormer-Verlet/leapfrog discretization in time.Comment: 25 page

Metastable energy strata in numerical discretizations of weakly nonlinear wave equations

The quadratic nonlinear wave equation on a one-dimensional torus with small initial values located in a single Fourier mode is considered. In this situation, the formation of metastable energy strata has recently been described and their long-time stability has been shown. The topic of the present paper is the correct reproduction of these metastable energy strata by a numerical method. For symplectic trigonometric integrators applied to the equation, it is shown that these energy strata are reproduced even on long time intervals in a qualitatively correct way.Comment: 28 pages, 9 figure

Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions

For trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with one or several constant high frequencies, near-conservation of the total and oscillatory energies are shown over time scales that cover arbitrary negative powers of the step size. This requires non-resonance conditions between the step size and the frequencies, but in contrast to previous results the results do not require any non-resonance conditions among the frequencies. The proof uses modulated Fourier expansions with appropriately modified frequencies.Comment: 26 page

Stochastic Generation of Particle Structures with Controlled Degree of Heterogeneity

The recently developed void expansion method (VEM) allows for an efficient generation of porous packings of spherical particles over a wide range of volume fractions. The method is based on a random placement of the structural particles under addition of much smaller "void-particles" whose radii are repeatedly increased during the void expansion. Thereby, they rearrange the structural particles until formation of a dense particle packing and introduce local heterogeneities in the structure. In this paper, microstructures with volume fractions between 0.4 and 0.6 produced by VEM are analyzed with respect to their degree of heterogeneity (DOH). In particular, the influence of the void- to structural particle number ratio, which constitutes a principal VEM-parameter, on the DOH is studied. The DOH is quantified using the pore size distribution, the Voronoi volume distribution and the density-fluctuation method in conjunction with fit functions or integral measures. This analysis has revealed that for volume fractions between 0.4 and 0.55 the void-particle number allows for a quasi-continuous adjustment of the DOH. Additionally, the DOH-range of VEM-generated microstructures with a volume fraction of 0.4 is compared to the range covered by microstructures generated using previous Brownian dynamics simulations, which represent the structure of coagulated colloidal suspensions. Both sets of microstructures cover similarly broad and overlapping DOH-ranges, which allows concluding that VEM is an efficient method to stochastically reproduce colloidal microstructures with varying DOH.Comment: 10 pages, 7 figure

Microstructures and Mechanical Properties of Dense Particle Gels: Microstructural Characterization

The macroscopic mechanical properties of densely packed coagulated colloidal particle gels strongly depend on the local arrangement of the powder particles on length scales of a few particle diameters. Heterogeneous microstructures exhibit up to one order of magnitude higher elastic properties and yield strengths than their homogeneous counterparts. The microstructures of these gels are analyzed by the straight path method quantifying quasi-linear particle arrangements of particles. They show similar characteristics than force chains bearing the mechanical load in granular material. Applying this concept to gels revealed that heterogeneous colloidal microstructures show a significantly higher straight paths density and exhibit longer straight paths than their homogeneous counterparts.Comment: 7 pages, 9 figure

Sintering Behavior of Cobalt Oxide Doped Ceria Powders of Different Particle Sizes

The effect of cobalt oxide as sintering aid for CeO2 and Ce0.9Gd0.1O1.95 powders was studied as a function of initial powder particle size. The sintering effect of cobalt oxide, measured as the difference in temperature of maximum shrinkage rate between undoped and 1 cat% cobalt oxide doped powders, decreased with initial particle size. Almost no effect was found for the smallest particle sizes investigated (d 100 nm. For the smaller particles, only doping of much higher concentration produces significant decreases in the temperature of maximum shrinkage rate. This observation suggests that the effectiveness of sintering depends on the specific surface area of the starting powder and points at the decisive role of the doping method used. Dense microstructures with average grain sizes smaller than 100 nm are obtained by doping very fine powders with cobalt oxid