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## ‣ Simulation of axisymmetric stepped surfaces with a facet

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 266 p.; 13162424 bytes; 13180260 bytes; application/pdf; application/pdf

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A crystal lattice with a small miscut from the plane of symmetry has a surface which consists of a series of atomic height steps separated by terraces. If the surface of this crystal is not in equilibrium with the surrounding medium, then its evolution is strongly mediated by the presence of these steps, which act as sites for attachment and detachment of diffusing adsorbed atoms ('adatoms'). In the absence of material deposition and evaporation, steps move in response to two main physical effects: line tension, which is caused by curvature of the step edge, and step-step interactions which can arise because of thermal step fluctuations, or elastic effects. This thesis focuses on axisymmetric crystals, with the result that the position of a step is uniquely described by a single scalar variable, and the step positions obey a coupled system of "step-flow" Ordinary Differential Equations (step flow ODEs). Chapter 2 of this thesis concentrates on the derivation and numerical solution of these equations, and their properties in the limits of slow adatom terrace diffusion and slow adatom attachment-detachment. Chapter 3 focuses on the analysis carried out by Margetis, Aziz and Stone ('MAS') [78] on a Partial Differential Equation (PDE) description of surface evolution.; (cont.) Here...

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## ‣ Multi-Adaptive Galerkin Methods for ODEs II: Implementation and Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/05/2012
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Continuing the discussion of the multi-adaptive Galerkin methods mcG(q) and
mdG(q) presented in [A. Logg, SIAM J. Sci. Comput., 24 (2003), pp. 1879-1902],
we present adaptive algorithms for global error control, iterative solution
methods for the discrete equations, features of the implementation Tanganyika,
and computational results for a variety of ODEs. Examples include the Lorenz
system, the solar system, and a number of time-dependent PDEs.

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## ‣ Splitting and composition methods in the numerical integration of differential equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/12/2008
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We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.; Comment: Review paper; 56 pages...

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## ‣ Averaging along irregular curves and regularisation of ODEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+
dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent
vector field which possibly is only a distribution in the space variable. We
quantify the regularising properties of an arbitrary continuous path $w$ on the
existence and uniqueness of solutions to this equation. In this context we
introduce the notion of $\rho$-\tmtextit{irregularity} and show that it plays a
key role in some instances of the regularisation by noise phenomenon. In the
particular case of a function $w$ sampled according to the law of the
fractional Brownian motion of Hurst index $H \in (0,1)$, we prove that almost
surely the ODE admits a solution for all $b$ in the Besov-H\~A{\P}lder space
$B^{\alpha+1}_{\infty , \infty}$ with $\alpha >-1/2H$. If $\alpha >1-1/2H$ then
the solution is unique among a natural set of continuous solutions. If $H>1/3$
and $\alpha >3/2-1/2H$ or if $\alpha >2-1/2H$ then the equation admits a unique
Lipschitz flow. Note that when $\alpha <0$ the vector field $b$ is only a
distribution, nonetheless there exists a natural notion of solution for which
the above results apply.; Comment: 49 pages, small typos and minor corrections

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## ‣ Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Numerical Analysis#Mathematics - Dynamical Systems#Physics - Computational Physics#65P10, 70K70, 65C30, 65B99

We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible...

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## ‣ High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/07/2015
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In this paper we construct high order finite volume schemes on networks of
hyperbolic conservation laws with coupling conditions involving ODEs. We
consider two generalized Riemann solvers at the junction, one of Toro-Castro
type and a solver of Harten, Enquist, Osher, Chakravarthy type. The ODE is
treated with a Taylor method or an explicit Runge-Kutta scheme, respectively.
Both resulting high order methods conserve quantities exactly if the
conservation is part of the coupling conditions. Furthermore we present a
technique to incorporate lumped parameter models, which arise from simplifying
parts of a network. The high order convergence and the robust capturing of
shocks is investigated numerically in several test cases.

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## ‣ Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/04/2011
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#Mathematics - Numerical Analysis#Mathematics - Analysis of PDEs#Mathematics - Dynamical Systems#Physics - Computational Physics

We present a new class of integrators for stiff PDEs. These integrators are
generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs
introduced in [Tao, Owhadi and Marsden 2010] with the following properties: (i)
Multiscale: they are based on flow averaging and have a computational cost
determined by mesoscopic steps in space and time instead of microscopic steps
in space and time; (ii) Versatile: the method is based on averaging the flows
of the given PDEs (which may have hidden slow and fast processes). This
bypasses the need for identifying explicitly (or numerically) the slow
variables or reduced effective PDEs; (iii) Nonintrusive: A pre-existing
numerical scheme resolving the microscopic time scale can be used as a black
box and easily turned into one of the integrators in this paper by turning the
large coefficients on over a microscopic timescale and off during a mesoscopic
timescale; (iv) Convergent over two scales: strongly over slow processes and in
the sense of measures over fast ones; (v) Structure-preserving: for stiff
Hamiltonian PDEs (possibly on manifolds), they can be made to be
multi-symplectic, symmetry-preserving (symmetries are group actions that leave
the system invariant) in all variables and variational.

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## ‣ What is wrong with the Lax-Richtmyer fundamental theorem of linear numerical analysis ?

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We show that the celebrated 1956 Lax-Richtmyer linear theorem in Numerical
Analysis - often called the Fundamental Theorem of Numerical Analysis - is in
fact wrong. Here "wrong" does not mean that its statement is false
mathematically, but that it has a limited practical relevance as it
misrepresents what actually goes on in the numerical analysis of partial
differential equations. Namely, the assumptions used in that theorem are
excessive to the extent of being unrealistic from practical point of view. The
two facts which the mentioned theorem gets wrong from practical point of view
are : - the relationship between the convergence and stability of numerical
methods for linear PDEs, - the effect of the propagation of round-off errors in
such numerical methods. The mentioned theorem leads to a result for PDEs which
is unrealistically better than the well known best possible similar result in
the numerical analysis of ODEs. Strangely enough, this fact seems not to be
known well enough in the literature. Once one becomes aware of the above, new
avenues of both practical and theoretical interest can open up in the numerical
analysis of PDEs.

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## ‣ The infinite Arnoldi exponential integrator for linear inhomogeneous ODEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Exponential integrators that use Krylov approximations of matrix functions
have turned out to be efficient for the time-integration of certain ordinary
differential equations (ODEs). This holds in particular for linear homogeneous
ODEs, where the exponential integrator is equivalent to approximating the
product of the matrix exponential and a vector. In this paper, we consider
linear inhomogeneous ODEs, $y'(t)=Ay(t)+g(t)$, where the function $g(t)$ is
assumed to satisfy certain regularity conditions. We derive an algorithm for
this problem which is equivalent to approximating the product of the matrix
exponential and a vector using Arnoldi's method. The construction is based on
expressing the function $g(t)$ as a linear combination of given basis functions
$[\phi_i]_{i=0}^\infty$ with particular properties. The properties are such
that the inhomogeneous ODE can be restated as an infinite-dimensional linear
homogeneous ODE. Moreover, the linear homogeneous infinite-dimensional ODE has
properties that directly allow us to extend a Krylov method for
finite-dimensional linear ODEs. Although the construction is based on an
infinite-dimensional operator, the algorithm can be carried out with operations
involving matrices and vectors of finite size. This type of construction
resembles in many ways the infinite Arnoldi method for nonlinear eigenvalue
problems. We prove convergence of the algorithm under certain natural
conditions...

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## ‣ An efficient algorithm for locating and continuing connecting orbits

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/12/1998
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A successive continuation method for locating connecting orbits in
parametrized systems of autonomous ODEs was considered in [9]. In this paper we
present an improved algorithm for locating and continuing connecting orbits,
which includes a new algorithm for the continuation of invariant subspaces. The
latter algorithm is of independent interest, and can be used in different
contexts than the present one.; Comment: 16 pages, 2 figures, 14 refs, E-mails dieci@math.gatech.edu,
friedman@math.uah.edu

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## ‣ Efficient Gluing of Numerical Continuation and a Multiple Solution Method for Elliptic PDEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Dynamical Systems#Computer Science - Mathematical Software#Mathematics - Numerical Analysis#Nonlinear Sciences - Pattern Formation and Solitons#Physics - Computational Physics

Numerical continuation calculations for ordinary differential equations
(ODEs) are, by now, an established tool for bifurcation analysis in dynamical
systems theory as well as across almost all natural and engineering sciences.
Although several excellent standard software packages are available for ODEs,
there are - for good reasons - no standard numerical continuation toolboxes
available for partial differential equations (PDEs), which cover a broad range
of different classes of PDEs automatically. A natural approach to this problem
is to look for efficient gluing computation approaches, with independent
components developed by researchers in numerical analysis, dynamical systems,
scientific computing and mathematical modelling. In this paper, we shall study
several elliptic PDEs (Lane-Emden-Fowler, Lane-Emden-Fowler with microscopic
force, Caginalp) via the numerical continuation software pde2path and develop a
gluing component to determine a set of starting solutions for the continuation
by exploting the variational structures of the PDEs. In particular, we solve
the initialization problem of numerical continuation for PDEs via a minimax
algorithm to find multiple unstable solution. Furthermore, for the Caginalp
system, we illustrate the efficient gluing link of pde2path to the underlying
mesh generation and the FEM MatLab pdetoolbox. Even though the approach works
efficiently due to the high-level programming language and without developing
any new algorithms...

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## ‣ A quantitative investigation into the accumulation of rounding errors in the numerical solution of ODEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/2005
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We examine numerical rounding errors of some deterministic solvers for
systems of ordinary differential equations (ODEs). We show that the
accumulation of rounding errors results in a solution that is inherently random
and we obtain the theoretical distribution of the trajectory as a function of
time, the step size and the numerical precision of the computer. We consider,
in particular, systems which amplify the effect of the rounding errors so that
over long time periods the solutions exhibit divergent behaviour. By performing
multiple repetitions with different values of the time step size, we observe
numerically the random distributions predicted theoretically. We mainly focus
on the explicit Euler and RK4 methods but also briefly consider more complex
algorithms such as the implicit solvers VODE and RADAU5.; Comment: 17 pages, 7 figures

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## ‣ Foundational aspects of singular integrals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We investigate integration of classes of real-valued continuous functions on
(0,1]. Of course difficulties arise if there is a non-$L^1$ element in the
class, and the Hadamard finite part integral ({\em p.f.}) does not apply. Such
singular integrals arise naturally in many contexts including PDEs and singular
ODEs.
The Lebesgue integral as well as $p.f.$, starting at zero, obey two
fundamental conditions: (i) they act as antiderivatives and, (ii) if $f =g$ on
$(0,a)$, then their integrals from $0$ to $x$ coincide for any $x\in (0,a)$.
We find that integrals from zero with the essential properties of $p.f.$,
plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions
on $(0,1]$ which are $L^1((\epsilon,1])$ for all $\epsilon>0$. However, this
existence proof does not provide a satisfactory construction. Without some
regularity at $0$, the existence of general antiderivatives which satisfy only
(i) and (ii) above on classes with a non-$L^1$ element is independent of ZF
(the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with
the Axiom of Dependent Choice). Moreover we show that there is no mathematical
description that can be proved (within ZFC or even extensions of ZFC with large
cardinal hypotheses) to uniquely define such an antiderivative operator.
Such results are precisely formulated for a variety of sets of functions...

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## ‣ Analytical and numerical aspects on motion of polygonal curves with constant area speed

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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General area-preserving motion of polygonal curves is formulated as a system
of ODEs. Solution polygonal curves belong to a prescribed polygonal class,
which is similar to the admissible class used in the crystalline curvature
flow. The ODEs are discretized implicitly in time keeping a given constant area
speed while solution polygonal curves keep belonging to the polygonal class.; Comment: Proceedings of Slovak-Austrian Mathematical Congress, within MAGIA
2007, Dept. of Mathematics and Descriptive Geometry, Faculty of Civil
Engineering, Slovak University of Technology, ISBN 978-80-227-2796-9 (2007)
127--141

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## ‣ Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/04/2015
Português

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#Mathematics - Spectral Theory#Mathematics - Dynamical Systems#Mathematics - Functional Analysis#34B09, 34C27, 34L05

The present paper is devoted to studying the Hill-type formula and Krein-type
trace formula for ODE, which is a continuous work of our previous work for
Hamiltonian systems \cite{HOW}. Hill-type formula and Krein-type trace formula
are given by Hill at 1877 and Krein in 1950's separately. Recently, we find
that there is a closed relationship between them \cite{HOW}. In this paper, we
will obtain the Hill-type formula for the $S$-periodic orbits of the first
order ODEs. Such a kind of orbits is considered naturally to study the
symmetric periodic and quasi-periodic solutions. By some similar idea in
\cite{HOW}, based on the Hill-type formula, we will build up the Krein-type
trace formula for the first order ODEs, which can be seen as a non-self-adjoint
version of the case of Hamiltonian system.; Comment: 23 pages

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## ‣ $\mathcal C^1$-HO - an implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We propose a new algorithm for computing validated bounds for the solutions
to the first order variational equations associated to ODEs. The method uses a
high order Taylor method as a predictor step and an implicit method based on
the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm
is an improvement of the $\mathcal C^1$-Lohner algorithm by Zgliczy\'nski and
by its very construction it cannot produce worse bounds.
As an application of the algorithm we give a computer assisted proof of the
existence of an attractor in the Rossler system and we show that the attractor
contains an invariant and uniformly hyperbolic subset on which the dynamics is
chaotic, i.e. conjugated to the Bernoulli shift on two symbols.; Comment: 30 pages, 9 figures

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## ‣ Multi-Adaptive Galerkin Methods for ODEs I

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/05/2012
Português

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We present multi-adaptive versions of the standard continuous and
discontinuous Galerkin methods for ODEs. Taking adaptivity one step further, we
allow for individual time-steps, order and quadrature, so that in particular
each individual component has its own time-step sequence. This paper contains a
description of the methods, an analysis of their basic properties, and a
posteriori error analysis. In the accompanying paper [A. Logg, SIAM J. Sci.
Comput., 27 (2003), pp. 741-758], we present adaptive algorithms for
time-stepping and global error control based on the results of the current
paper.

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## ‣ Geometric integration on spheres and some interesting applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/12/2011
Português

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Geometric integration theory can be employed when numerically solving ODEs or
PDEs with constraints. In this paper, we present several one-step algorithms of
various orders for ODEs on a collection of spheres. To demonstrate the
versatility of these algorithms, we present representative calculations for
reduced free rigid body motion (a conservative ODE) and a discretization of
micromagnetics (a dissipative PDE). We emphasize the role of isotropy in
geometric integration and link numerical integration schemes to modern
differential geometry through the use of partial connection forms; this
theoretical framework generalizes moving frames and connections on principal
bundles to manifolds with nonfree actions.; Comment: This paper appeared in print

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## ‣ Alternating minimal energy approach to ODEs and conservation laws in tensor product formats

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Numerical Analysis#Condensed Matter - Strongly Correlated Electrons#Physics - Computational Physics#15A69, 33F05, 65F10, 65L05, 65M70, 34C14

We propose an algorithm for solution of high-dimensional evolutionary
equations (ODEs and discretized time-dependent PDEs) in tensor product formats.
The solution must admit an approximation in a low-rank separation of variables
framework, and the right-hand side of the ODE (for example, a matrix) must be
computable in the same low-rank format at a given time point. The time
derivative is discretized via the Chebyshev spectral scheme, and the solution
is sought simultaneously for all time points from the global space-time linear
system. To compute the solution adaptively in the tensor format, we employ the
Alternating Minimal Energy algorithm, the DMRG-flavored alternating iterative
technique.
Besides, we address the problem of maintaining system invariants inside the
approximate tensor product scheme. We show how the conservation of a linear
function, defined by a vector given in the low-rank format, or the second norm
of the solution may be accurately and elegantly incorporated into the tensor
product method.
We present a couple of numerical experiments with the transport problem and
the chemical master equation, and confirm the main beneficial properties of the
new approach: conservation of invariants up to the machine precision...

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## ‣ Projection methods and discrete gradient methods for preserving first integrals of ODEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/02/2013
Português

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In this paper we study linear projection methods for approximating the
solution and simultaneously preserving first integrals of autonomous ordinary
differential equations. We show that (linear) projection methods are a subset
of discrete gradient methods. In particular, each projection method is
equivalent to a class of discrete gradient methods (where the choice of
discrete gradient is arbitrary) and earlier results for discrete gradient
methods also apply to projection methods. Thus we prove that for the case of
preserving one first integral, under certain mild conditions, the numerical
solution for a projection method exists and is locally unique, and preserves
the order of accuracy of the underlying method.
In the case of preserving multiple first integrals the relationship between
projection methods and discrete gradient methods persists. Moreover, numerical
examples show that similar existence and order results should also hold for the
multiple integral case.
For completeness we show how existing projection methods from the literature
fit into our general framework.; Comment: 31 pages, 4 figures

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