10,254 research outputs found
Numerical Approximations Using Chebyshev Polynomial Expansions
We present numerical solutions for differential equations by expanding the
unknown function in terms of Chebyshev polynomials and solving a system of
linear equations directly for the values of the function at the extrema (or
zeros) of the Chebyshev polynomial of order N (El-gendi's method). The
solutions are exact at these points, apart from round-off computer errors and
the convergence of other numerical methods used in connection to solving the
linear system of equations. Applications to initial value problems in
time-dependent quantum field theory, and second order boundary value problems
in fluid dynamics are presented.Comment: minor wording changes, some typos have been eliminate
Spectral Methods for Numerical Relativity. The Initial Data Problem
Numerical relativity has traditionally been pursued via finite differencing.
Here we explore pseudospectral collocation (PSC) as an alternative to finite
differencing, focusing particularly on the solution of the Hamiltonian
constraint (an elliptic partial differential equation) for a black hole
spacetime with angular momentum and for a black hole spacetime superposed with
gravitational radiation. In PSC, an approximate solution, generally expressed
as a sum over a set of orthogonal basis functions (e.g., Chebyshev
polynomials), is substituted into the exact system of equations and the
residual minimized. For systems with analytic solutions the approximate
solutions converge upon the exact solution exponentially as the number of basis
functions is increased. Consequently, PSC has a high computational efficiency:
for solutions of even modest accuracy we find that PSC is substantially more
efficient, as measured by either execution time or memory required, than finite
differencing; furthermore, these savings increase rapidly with increasing
accuracy. The solution provided by PSC is an analytic function given
everywhere; consequently, no interpolation operators need to be defined to
determine the function values at intermediate points and no special
arrangements need to be made to evaluate the solution or its derivatives on the
boundaries. Since the practice of numerical relativity by finite differencing
has been, and continues to be, hampered by both high computational resource
demands and the difficulty of formulating acceptable finite difference
alternatives to the analytic boundary conditions, PSC should be further pursued
as an alternative way of formulating the computational problem of finding
numerical solutions to the field equations of general relativity.Comment: 15 pages, 5 figures, revtex, submitted to PR
Spectral methods for partial differential equations
Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized
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
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
In this paper, the fractional order of rational Bessel functions collocation
method (FRBC) to solve Thomas-Fermi equation which is defined in the
semi-infinite domain and has singularity at and its boundary condition
occurs at infinity, have been introduced. We solve the problem on semi-infinite
domain without any domain truncation or transformation of the domain of the
problem to a finite domain. This approach at first, obtains a sequence of
linear differential equations by using the quasilinearization method (QLM),
then at each iteration solves it by FRBC method. To illustrate the reliability
of this work, we compare the numerical results of the present method with some
well-known results in other to show that the new method is accurate, efficient
and applicable
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
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
A rational spectral collocation method with adaptively transformed Chebyshev grid points
A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are approximated by the locations of poles of Chebyshev-Padé approximants. Numerical experiments on two time-dependent problems, one with finite time blow-up and one with a moving front, indicate that the method far outperforms the standard Chebyshev spectral collocation method for problems whose solutions have singularities in the complex plan close to [-1,1]
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