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

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

Thin Film Deposition Using Spray Pyrolysis

Spray pyrolysis has been applied to deposit a wide variety of thin films. These films were used in various devices such as solar cells, sensors, and solid oxide fuel cells. It is observed that often the properties of deposited thin films depend on the preparation conditions. An extensive review of the effects of spray parameters on film quality is given to demonstrate the importance of the process of optimization. The substrate surface temperature is the most critical parameter as it influences film roughness, cracking, crystallinity, etc. Processes involved in the spray pyrolysis technique are discussed in this review as wel

Metastable energy strata in numerical discretizations ofweakly 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