2,362 research outputs found
Poisson integrators
An overview of Hamiltonian systems with noncanonical Poisson structures is
given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are
presented. Numerical integrators using generating functions, Hamiltonian
splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems
and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems
and the discrete gradient methods are also presented.Comment: 30 page
High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels
The solution of the Volterra integral equation, where , and are smooth functions, can be represented as ,, where , are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate via , in a neighborhood of the origin and use (*) on the rest of the interval . In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson's rule are treated in detail. The schemes are shown to be convergent of order . Asymptotic error estimates are derived in order to examine the numerical stability of the 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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