Decay of the Maxwell field on the Schwarzschild manifold

We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate $r$ ranges over $2M < r_1 < r < r_2$, we obtain a decay rate of $t^{-1}$ for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition. In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, $r_*>\epsilon t$, we obtain decay for the null components with rates of $|\phi_+| \sim |\alpha| < C r^{-5/2}$, $|\phi_0| \sim |\rho| + |\sigma| < C r^{-2} |t-r_*|^{-1/2}$, and $|\phi_{-1}| \sim |\underline{\alpha}| < C r^{-1} |t-r_*|^{-1}$. Along the event horizon and in ingoing regions, where $r_*<0$, and when $t+r_*1$, all components (normalized with respect to an ingoing null basis) decay at a rate of C \uout^{-1} with \uout=t+r_* in the exterior region.Comment: 37 pages, 5 figure

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Classical nonlinear response of a chaotic system: Langevin dynamics and spectral decomposition

We consider the classical response of a strongly chaotic Hamiltonian system. The spectrum of such a system consists of discrete complex Ruelle-Pollicott (RP) resonances which manifest themselves in the behavior of the correlation and response functions. We interpret the RP resonances as the eigenstates and eigenvalues of the Fokker-Planck operator obtained by adding an infinitesimal noise term to the first-order Liouville operator. We demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise. For the second-order response we establish an equivalence of the spectral decomposition with infinitesimal noise and the long-time asymptotic expansion for the deterministic case.Comment: 16 pages, 1 figur

Domain decomposition preconditioners for the spectral collocation method

Several block iteration preconditioners are proposed and analyzed for the solution of elliptic problems by spectral collocation methods in a region partitioned into several rectangles. It is shown that convergence is achieved with a rate which does not depend on the polynomial degree of the spectral solution. The iterative methods here presented can be effectively implemented on multiprocessor systems due to their high degree of parallelism

A quadtree-polygon-based scaled boundary finite element method for image-based mesoscale fracture modelling in concrete

A quadtree-polygon scaled boundary finite element-based approach for image-based modelling of concrete fracture at the mesoscale is developed. Digital images representing the two-phase mesostructure of concrete, which comprises of coarse aggregates and mortar are either generated using a take-and-place algorithm with a user-defined aggregate volume ratio or obtained from X-ray computed tomography as an input. The digital images are automatically discretised for analysis by applying a balanced quadtree decomposition in combination with a smoothing operation. The scaled boundary finite element method is applied to model the constituents in the concrete mesostructure. A quadtree formulation within the framework of the scaled boundary finite element method is advantageous in that the displacement compatibility between the cells are automatically preserved even in the presence of hanging nodes. Moreover, the geometric flexibility of the scaled boundary finite element method facilitates the use of arbitrary sided polygons, allowing better representation of the aggregate boundaries. The computational burden is significantly reduced as there are only finite number of cell types in a balanced quadtree mesh. The cells in the mesh are connected to each other using cohesive interface elements with appropriate softening laws to model the fracture of the mesostructure. Parametric studies are carried out on concrete specimens subjected to uniaxial tension to investigate the effects of various parameters e.g. aggregate size distribution, porosity and aggregate volume ratio on the fracture of concrete at the meso-scale. Mesoscale fracture of concrete specimens obtained from X-ray computed tomography scans are carried out to demonstrate its feasibility