19,069 research outputs found
Post-Processing Techniques and Wavelet Applications for Hammerstein Integral Equations
This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation.
In the second half of this dissertation, the wavelet-collocation technique of solving nonlinear Hammerstein integral equation is discussed. The main objective is to establish a fast wavelet-collocation method for Hammerstein equation by using a \u27linearization\u27 technique. The sparsity in the Jacobian matrix takes place in the fast wavelet-collocation method for Hammerstein equation with smooth as well as weakly singular kernels. A fast algorithm is based upon the block truncation strategy which was recently proposed in [10]. A multilevel augmentation method for the linearized Hammerstein equation is subsequently proposed which further accelerates the solution process while maintaining the order of convergence. Numerical examples are given throughout this dissertation
Geostrophic balance preserving interpolation in mesh adaptive shallow-water ocean modelling
The accurate representation of geostrophic balance is an essential
requirement for numerical modelling of geophysical flows. Significant effort is
often put into the selection of accurate or optimal balance representation by
the discretisation of the fundamental equations. The issue of accurate balance
representation is particularly challenging when applying dynamic mesh
adaptivity, where there is potential for additional imbalance injection when
interpolating to new, optimised meshes.
In the context of shallow-water modelling, we present a new method for
preservation of geostrophic balance when applying dynamic mesh adaptivity. This
approach is based upon interpolation of the Helmholtz decomposition of the
Coriolis acceleration. We apply this in combination with a discretisation for
which states in geostrophic balance are exactly steady solutions of the
linearised equations on an f-plane; this method guarantees that a balanced and
steady flow on a donor mesh remains balanced and steady after interpolation
onto an arbitrary target mesh, to within machine precision. We further
demonstrate the utility of this interpolant for states close to geostrophic
balance, and show that it prevents pollution of the resulting solutions by
imbalanced perturbations introduced by the interpolation
Discontinuous collocation methods and gravitational self-force applications
Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure
Simulating water-entry/exit problems using Eulerian-Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library
In this paper we employ two implementations of the fictitious domain (FD)
method to simulate water-entry and water-exit problems and demonstrate their
ability to simulate practical marine engineering problems. In FD methods, the
fluid momentum equation is extended within the solid domain using an additional
body force that constrains the structure velocity to be that of a rigid body.
Using this formulation, a single set of equations is solved over the entire
computational domain. The constraint force is calculated in two distinct ways:
one using an Eulerian-Lagrangian framework of the immersed boundary (IB) method
and another using a fully-Eulerian approach of the Brinkman penalization (BP)
method. Both FSI strategies use the same multiphase flow algorithm that solves
the discrete incompressible Navier-Stokes system in conservative form. A
consistent transport scheme is employed to advect mass and momentum in the
domain, which ensures numerical stability of high density ratio multiphase
flows involved in practical marine engineering applications. Example cases of a
free falling wedge (straight and inclined) and cylinder are simulated, and the
numerical results are compared against benchmark cases in literature.Comment: The current paper builds on arXiv:1901.07892 and re-explains some
parts of it for the reader's convenienc
Destruction of Interstellar Dust in Evolving Supernova Remnant Shock Waves
Supernova generated shock waves are responsible for most of the destruction
of dust grains in the interstellar medium (ISM). Calculations of the dust
destruction timescale have so far been carried out using plane parallel steady
shocks, however that approximation breaks down when the destruction timescale
becomes longer than that for the evolution of the supernova remnant (SNR)
shock. In this paper we present new calculations of grain destruction in
evolving, radiative SNRs. To facilitate comparison with the previous study by
Jones et al. (1996), we adopt the same dust properties as in that paper. We
find that the efficiencies of grain destruction are most divergent from those
for a steady shock when the thermal history of a shocked gas parcel in the SNR
differs significantly from that behind a steady shock. This occurs in shocks
with velocities >~ 200 km/s for which the remnant is just beginning to go
radiative. Assuming SNRs evolve in a warm phase dominated ISM, we find dust
destruction timescales are increased by a factor of ~2 compared to those of
Jones et al. (1996), who assumed a hot gas dominated ISM. Recent estimates of
supernova rates and ISM mass lead to another factor of ~3 increase in the
destruction timescales, resulting in a silicate grain destruction timescale of
~2-3 Gyr. These increases, while not able resolve the problem of the discrepant
timescales for silicate grain destruction and creation, are an important step
towards understanding the origin, and evolution of dust in the ISM.Comment: 30 pages, 8 figures, accepted for publication in the Astrophysical
Journa
Distributed-memory parallelization of an explicit time-domain volume integral equation solver on Blue Gene/P
Two distributed-memory schemes for efficiently parallelizing the explicit marching-on in-time based solution of the time domain volume integral equation on the IBM Blue Gene/P platform are presented. In the first scheme, each processor stores the time history of all source fields and only the computationally dominant step of the tested field computations is distributed among processors. This scheme requires all-to-all global communications to update the time history of the source fields from the tested fields. In the second scheme, the source fields as well as all steps of the tested field computations are distributed among processors. This scheme requires sequential global communications to update the time history of the distributed source fields from the tested fields. Numerical results demonstrate that both schemes scale well on the IBM Blue Gene/P platform and the memory efficient second scheme allows for the characterization of transient wave interactions on composite structures discretized using three million spatial elements without an acceleration algorithm
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