633 research outputs found
The Hamiltonian BVMs (HBVMs) Homepage
Hamiltonian Boundary Value Methods (in short, HBVMs) is a new class of
numerical methods for the efficient numerical solution of canonical Hamiltonian
systems. In particular, their main feature is that of exactly preserving, for
the numerical solution, the value of the Hamiltonian function, when the latter
is a polynomial of arbitrarily high degree. Clearly, this fact implies a
practical conservation of any analytical Hamiltonian function. In this notes,
we collect the introductory material on HBVMs contained in the HBVMs Homepage,
available at http://web.math.unifi.it/users/brugnano/HBVM/index.htmlComment: 49 pages, 16 figures; Chapter 4 modified; minor corrections to
Chapter 5; References update
High-order convergent deferred correction schemes based on parameterized Runge-Kutta-Nyström methods for second-order boundary value problems
AbstractIterated deferred correction is a widely used approach to the numerical solution of first-order systems of nonlinear two-point boundary value problems. Normally, the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and, as a direct result, each time deferred correction is used the order of the overall scheme is increased by a maximum of 2. In [16], however, it has been shown that there exist schemes based on parameterized Runge–Kutta methods, which allow a higher increase of the overall order. A first example of such a high-order convergent scheme which allows an increase of 4 orders per deferred correction was based on two mono-implicit Runge–Kutta methods. In the present paper, we will investigate the possibility for high-order convergence of schemes for the numerical solution of second-order nonlinear two-point boundary value problems not containing the first derivative. Two examples of such high-order convergent schemes, based on parameterized Runge–Kutta-Nyström methods of orders 4 and 8, are analysed and discussed
Legendre-Gauss-Lobatto grids and associated nested dyadic grids
Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral
methods for the numerical solution of partial differential equations. They not
only provide efficient high-order quadrature rules, but give also rise to norm
equivalences that could eventually lead to efficient preconditioning techniques
in high-order methods. Unfortunately, a serious obstruction to fully exploiting
the potential of such concepts is the fact that LGL grids of different degree
are not nested. This affects, on the one hand, the choice and analysis of
suitable auxiliary spaces, when applying the auxiliary space method as a
principal preconditioning paradigm, and, on the other hand, the efficient
solution of the auxiliary problems. As a central remedy, we consider certain
nested hierarchies of dyadic grids of locally comparable mesh size, that are in
a certain sense properly associated with the LGL grids. Their actual
suitability requires a subtle analysis of such grids which, in turn, relies on
a number of refined properties of LGL grids. The central objective of this
paper is to derive just these properties. This requires first revisiting
properties of close relatives to LGL grids which are subsequently used to
develop a refined analysis of LGL grids. These results allow us then to derive
the relevant properties of the associated dyadic grids.Comment: 35 pages, 7 figures, 2 tables, 2 algorithms; Keywords:
Legendre-Gauss-Lobatto grid, dyadic grid, graded grid, nested grid
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