42,393 research outputs found
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
Inverse heat conduction problems by using particular solutions
Based on the method of fundamental solutions, we develop in this paper a new computational method to solve two-dimensional transient heat conduction inverse problems. The main idea is to use particular solutions as radial basis functions (PSRBF) for approximation of the solutions to the inverse heat conduction problems. The heat conduction equations are first analyzed in the Laplace transformed domain and the Durbin inversion method is then used to determine the solutions in the time domain. Least-square and singular value decomposition (SVD) techniques are adopted to solve the ill-conditioned linear system of algebraic equations obtained from the proposed PSRBF method. To demonstrate the effectiveness and simplicity of this approach, several numerical examples are given with satisfactory accuracy and stability.Peer reviewe
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally
challenging task, especially when a large set of eigenvalues is computed, or
just when particularly large eigenvalues are sought. This is a consequence of
the highly oscillatory behaviour of the solutions corresponding to high
eigenvalues, which forces a naive integrator to take increasingly smaller
steps. We will discuss some techniques that yield uniform approximation over
the whole eigenvalue spectrum and can take large steps even for high
eigenvalues. In particular, we will focus on methods based on coefficient
approximation which replace the coefficient functions of the Sturm-Liouville
problem by simpler approximations and then solve the approximating problem. The
use of (modified) Magnus or Neumann integrators allows to extend the
coefficient approximation idea to higher order methods
Spectral Methods for Time-Dependent Studies of Accretion Flows. II. Two-Dimensional Hydrodynamic Disks with Self-Gravity
Spectral methods are well suited for solving hydrodynamic problems in which
the self-gravity of the flow needs to be considered. Because Poisson's equation
is linear, the numerical solution for the gravitational potential for each
individual mode of the density can be pre-computed, thus reducing substantially
the computational cost of the method. In this second paper, we describe two
different approaches to computing the gravitational field of a two-dimensional
flow with pseudo-spectral methods. For situations in which the density profile
is independent of the third coordinate (i.e., an infinite cylinder), we use a
standard Poisson solver in spectral space. On the other hand, for situations in
which the density profile is a delta function along the third coordinate (i.e.,
an infinitesimally thin disk), or any other function known a priori, we perform
a direct integration of Poisson's equation using a Green's functions approach.
We devise a number of test problems to verify the implementations of these two
methods. Finally, we use our method to study the stability of polytropic,
self-gravitating disks. We find that, when the polytropic index Gamma is <=
4/3, Toomre's criterion correctly describes the stability of the disk. However,
when Gamma > 4/3 and for large values of the polytropic constant K, the
numerical solutions are always stable, even when the linear criterion predicts
the contrary. We show that, in the latter case, the minimum wavelength of the
unstable modes is larger than the extent of the unstable region and hence the
local linear analysis is inapplicable.Comment: 13 pages, 9 figures. To appear in the ApJ. High resolution plots and
animations of the simulations are available at
http://www.physics.arizona.edu/~chan/research/astro-ph/0512448/index.htm
Laplacian Growth and Whitham Equations of Soliton Theory
The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in
the case of zero surface tension is proven to be equivalent to an integrable
systems of Whitham equations known in soliton theory. The Whitham equations
describe slowly modulated periodic solutions of integrable hierarchies of
nonlinear differential equations. Through this connection the Laplacian growth
is understood as a flow in the moduli space of Riemann surfaces.Comment: 33 pages, 7 figures, typos corrected, new references adde
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