On one-dimensional stretching functions for finite-difference calculations

The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the one-dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent

On One-Dimensional Stretching Functions for Finite-Difference Calculations

The class of one dimensional stretching function used in finite difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two sided stretching function, for which the arbitrary slopes at the two ends of the one dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. The general two sided function has many applications in the construction of finite difference grids

Driven Dynamics of Periodic Elastic Media in Disorder

We analyze the large-scale dynamics of vortex lattices and charge density waves driven in a disordered potential. Using a perturbative coarse-graining procedure we present an explicit derivation of non-equilibrium terms in the renormalized equation of motion, in particular Kardar-Parisi-Zhang non-linearities and dynamic strain terms. We demonstrate the absence of glassy features like diverging linear friction coefficients and transverse critical currents in the drifting state. We discuss the structure of the dynamical phase diagram containing different elastic phases very small and very large drive and plastic phases at intermediate velocity.Comment: 21 pages Latex with 4 figure

Slow Crack Propagation in Heterogeneous Materials

Statistics and thermally activated dynamics of crack nucleation and propagation in a two-dimensional heterogeneous material containing quenched randomly distributed defects are studied theoretically. Using the generalized Griffith criterion we derive the equation of motion for the crack tip position accounting for dissipation, thermal noise and the random forces arising from the defects. We find that aggregations of defects generating long-range interaction forces (e.g., clouds of dislocations) lead to anomalously slow creep of the crack tip or even to its complete arrest. We demonstrate that heterogeneous materials with frozen defects contain a large number of arrested microcracks and that their fracture toughness is enhanced to the experimentally accessible time scales.Comment: 5 pages, 1 figur

Fluctuation-induced noise in out-of-equilibrium disordered superconducting films

We study out-of-equilibrium transport in disordered superconductors close to the superconducting transition. We consider a thin film connected by resistive tunnel interfaces to thermal reservoirs having different chemical potentials and temperatures. The nonequilibrium longitudinal current-current correlation function is calculated within the nonlinear sigma model description and nonlinear dependence on temperatures and chemical potentials is obtained. Different contributions are calculated, originating from the fluctuation-induced suppression of the quasiparticle density of states, Maki- Thompson and Aslamazov-Larkin processes. As a special case of our results, close-to-equilibrium we obtain the longitudinal ac conductivity using the fluctuation-dissipation theorem