1,700 research outputs found
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
Chebyshev model arithmetic for factorable functions
This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden
A New Supersymmetric Index
We show that is an index for =2
supersymmetric theories in two dimensions, in the sense that it is independent
of almost all deformations of the theory. This index is related to the geometry
of the vacua (Berry's curvature) and satisfies an exact differential equation
as a function of . For integrable theories we can also compute the index
thermodynamically, using the exact -matrix. The equivalence of these two
results implies a highly non-trivial equivalence of a set of coupled integral
equations with these differential equations, among them Painleve III and the
affine Toda equations.Comment: 60 page
High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures
The work reported in this article presents a high-order, stable, and
efficient Gegenbauer pseudospectral method to solve numerically a wide variety
of mathematical models. The proposed numerical scheme exploits the stability
and the well-conditioning of the numerical integration operators to produce
well-conditioned systems of algebraic equations, which can be solved easily
using standard algebraic system solvers. The core of the work lies in the
derivation of novel and stable Gegenbauer quadratures based on the stable
barycentric representation of Lagrange interpolating polynomials and the
explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous
error and convergence analysis of the proposed quadratures is presented along
with a detailed set of pseudocodes for the established computational
algorithms. The proposed numerical scheme leads to a reduction in the
computational cost and time complexity required for computing the numerical
quadrature while sharing the same exponential order of accuracy achieved by
Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical
test examples to assess the efficiency and accuracy of the numerical scheme.
The present method provides a strong addition to the arsenal of numerical
pseudospectral methods, and can be extended to solve a wide range of problems
arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl
Efficient solution of parabolic equations by Krylov approximation methods
Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms
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