1,418 research outputs found
Some recent developments in spectral methods
This paper is solely devoted to spectral iterative methods including spectral multigrid methods. These techniques are explained with reference to simple model problems. Some Navier-Stokes algorithms based on these techniques are mentioned. Results on transition simulation using these algorithms are presented
Black hole evolution by spectral methods
Current methods of evolving a spacetime containing one or more black holes
are plagued by instabilities that prohibit long-term evolution. Some of these
instabilities may be due to the numerical method used, traditionally finite
differencing. In this paper, we explore the use of a pseudospectral collocation
(PSC) method for the evolution of a spherically symmetric black hole spacetime
in one dimension using a hyperbolic formulation of Einstein's equations. We
demonstrate that our PSC method is able to evolve a spherically symmetric black
hole spacetime forever without enforcing constraints, even if we add dynamics
via a Klein-Gordon scalar field. We find that, in contrast to
finite-differencing methods, black hole excision is a trivial operation using
PSC applied to a hyperbolic formulation of Einstein's equations. We discuss the
extension of this method to three spatial dimensions.Comment: 20 pages, 17 figures, submitted to PR
On the Existence of Energy-Preserving Symplectic Integrators Based upon Gauss Collocation Formulae
We introduce a new family of symplectic integrators depending on a real
parameter. When the paramer is zero, the corresponding method in the family
becomes the classical Gauss collocation formula of order 2s, where s denotes
the number of the internal stages. For any given non-null value of the
parameter, the corresponding method remains symplectic and has order 2s-2:
hence it may be interpreted as an order 2s-2 (symplectic) perturbation of the
Gauss method. Under suitable assumptions, we show that the free parameter may
be properly tuned, at each step of the integration procedure, so as to
guarantee energy conservation in the numerical solution. The resulting
symplectic, energy conserving method shares the same order 2s as the generating
Gauss formula.Comment: 19 pages, 7 figures; Sections 1, 2, and 6 sliglthly modifie
A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations
In this paper we introduce a numerical method for solving nonlinear Volterra
integro-differential equations. In the first step, we apply implicit trapezium
rule to discretize the integral in given equation. Further, the Daftardar-Gejji
and Jafari technique (DJM) is used to find the unknown term on the right side.
We derive existence-uniqueness theorem for such equations by using Lipschitz
condition. We further present the error, convergence, stability and bifurcation
analysis of the proposed method. We solve various types of equations using this
method and compare the error with other numerical methods. It is observed that
our method is more efficient than other numerical methods
Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods
In this paper we are concerned with the analysis of a class of geometric
integrators, at first devised in [14, 18], which can be regarded as an
energy-conserving variant of Gauss collocation methods. With these latter they
share the property of conserving quadratic first integrals but, in addition,
they also conserve the Hamiltonian function itself. We here reformulate the
methods in a more convenient way, and propose a more refined analysis than that
given in [18] also providing, as a by-product, a practical procedure for their
implementation. A thorough comparison with the original Gauss methods is
carried out by means of a few numerical tests solving Hamiltonian and Poisson
problems.Comment: 28 pages, 2 figures, 4 table
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