98 research outputs found
The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity
When numerically integrating canonical Hamiltonian systems, the long-term
conservation of some of its invariants, among which the Hamiltonian function
itself, assumes a central role. The classical approach to this problem has led
to the definition of symplectic methods, among which we mention Gauss-Legendre
collocation formulae. Indeed, in the continuous setting, energy conservation is
derived from symplecticity via an infinite number of infinitesimal contact
transformations. However, this infinite process cannot be directly transferred
to the discrete setting. By following a different approach, in this paper we
describe a sequence of methods, sharing the same essential spectrum (and, then,
the same essential properties), which are energy preserving starting from a
certain element of the sequence on, i.e., after a finite number of steps.Comment: 15 page
\^A-and \^I-stability of collocation Runge-Kutta methods
This paper deals with stability of classical Runge-Kutta collocation methods.
When such methods are embedded in linearly implicit methods as developed in
[12] and used in [13] for the time integration of nonlinear evolution PDEs, the
stability of these methods has to be adapted to this context. For this reason,
we develop in this paper several notions of stability, that we analyze. We
provide sufficient conditions that can be checked algorithmically to ensure
that these stability notions are fulfilled by a given Runge-Kutta collocation
method. We also introduce examples and counterexamples used in [13] to
highlight the necessity of these stability conditions in this context
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
Numerical comparisons among some methods for Hamiltonian problems
We report a few sumerical tests comparing some newly defined
energy-preserving integrators and symplectic methods, using either constant and
variable stepsize.Comment: 5 pages, 8 figure
Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations
We are concerned with the efficient implementation of symplectic implicit
Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian)
ordinary differential equations by means of Newton-like iterations. We pay
particular attention to symmetric symplectic IRK schemes (such as collocation
methods with Gaussian nodes). For a -stage IRK scheme used to integrate a
-dimensional system of ordinary differential equations, the application of
simplified versions of Newton iterations requires solving at each step several
linear systems (one per iteration) with the same real
coefficient matrix. We propose rewriting such -dimensional linear systems
as an equivalent -dimensional systems that can be solved by performing
the LU decompositions of real matrices of size . We
present a C implementation (based on Newton-like iterations) of Runge-Kutta
collocation methods with Gaussian nodes that make use of such a rewriting of
the linear system and that takes special care in reducing the effect of
round-off errors. We report some numerical experiments that demonstrate the
reduced round-off error propagation of our implementation
Shooting methods for a PT-symmetric periodic eigenvalue problem
We present a rigorous analysis of the performance of some one-step discretization schemes for a class of PT-symmetric singular boundary eigenvalue problem which encompasses a number of different problems whose investigation has been inspired by the 2003 article of Benilov et al. (J Fluid Mech 497:201-224, 2003). These discretization schemes are analyzed as initial value problems rather than as discrete boundary problems, since this is the setting which ties in most naturally with the formulation of the problem which one is forced to adopt due to the presence of an interior singularity. We also devise and analyze a variable step scheme for dealing with the singular points. Numerical results show better agreement between our results and those obtained from small-ε asymptotics than has been shown in results presented hitherto
Conservation laws of semidiscrete canonical Hamiltonian equations
There are many evolution partial differential equations which can be cast
into Hamiltonian form. Conservation laws of these equations are related to
one-parameter Hamiltonian symmetries admitted by the PDEs. The same result
holds for semidiscrete Hamiltonian equations. In this paper we consider
semidiscrete canonical Hamiltonian equations. Using symmetries, we find
conservation laws for the semidiscretized nonlinear wave equation and
Schrodinger equation.Comment: 19 pages, 2 table
Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations
We are concerned with the efficient implementation of symplectic
implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily
Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular attention to symmetric symplectic IRK schemes
(such as collocation methods with Gaussian nodes). For a s-stage IRK scheme
used to integrate a d-dimensional system of ordinary differential equations,
the application of simplified versions of Newton iterations requires solving at
each step several linear systems (one per iteration) with the same sd × sd real
coefficient matrix. We propose rewriting such sd-dimensional linear systems as
an equivalent (s + 1)d-dimensional systems that can be solved by performing
the LU decompositions of [s/2] + 1 real matrices of size d × d. We present a
C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the
linear system and that takes special care in reducing the effect of round-off
errors. We report some numerical experiments that demonstrate the reduced
round-off error propagation of our implementation.Project of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU).
Project MTM2013-46553-C3-2-P from Spanish Ministry of Economy and Trade.
Consolidated Research Group IT649-13 from the Basque Government
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