391 research outputs found
High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method
We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to
solve numerically the second-order one-dimensional hyperbolic telegraph
equation provided with some initial and Dirichlet boundary conditions. The
framework of the numerical scheme involves the recast of the problem into its
integral formulation followed by its discretization into a system of
well-conditioned linear algebraic equations. The integral operators are
numerically approximated using some novel shifted Gegenbauer operational
matrices of integration. We derive the error formula of the associated
numerical quadratures. We also present a method to optimize the constructed
operational matrix of integration by minimizing the associated quadrature error
in some optimality sense. We study the error bounds and convergence of the
optimal shifted Gegenbauer operational matrix of integration. Moreover, we
construct the relation between the operational matrices of integration of the
shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive
the global collocation matrix of the SGPM, and construct an efficient
computational algorithm for the solution of the collocation equations. We
present a study on the computational cost of the developed computational
algorithm, and a rigorous convergence and error analysis of the introduced
method. Four numerical test examples have been carried out in order to verify
the effectiveness, the accuracy, and the exponential convergence of the method.
The SGPM is a robust technique, which can be extended to solve a wide range of
problems arising in numerous applications.Comment: 36 pages, articl
Fast Mesh Refinement in Pseudospectral Optimal Control
Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy
--- simply increase the order of the Lagrange interpolating polynomial and
the mathematics of convergence automates the distribution of the grid points.
Unfortunately, as increases, the condition number of the resulting linear
algebra increases as ; hence, spectral efficiency and accuracy are lost in
practice. In this paper, we advance Birkhoff interpolation concepts over an
arbitrary grid to generate well-conditioned PS optimal control discretizations.
We show that the condition number increases only as in general, but
is independent of for the special case of one of the boundary points being
fixed. Hence, spectral accuracy and efficiency are maintained as increases.
The effectiveness of the resulting fast mesh refinement strategy is
demonstrated by using \underline{polynomials of over a thousandth order} to
solve a low-thrust, long-duration orbit transfer problem.Comment: 27 pages, 12 figures, JGCD April 201
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
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
Optimal Control of a Parabolic Distributed Parameter System Using a Barycentric Shifted Gegenbauer Pseudospectral Method
In this paper, we introduce a novel pseudospectral method for the numerical
solution of optimal control problems governed by a parabolic distributed
parameter system. The infinite-dimensional optimal control problem is reduced
into a finite-dimensional nonlinear programming problem through shifted
Gegenbauer quadratures constructed using a stable barycentric representation of
Lagrange interpolating polynomials and explicit barycentric weights for the
shifted Gegenbauer-Gauss (SGG) points. A rigorous error analysis of the method
is presented, and a numerical test example is given to show the accuracy and
efficiency of the proposed pseudospectral method.Comment: 15 pages, 3 figure
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