58 research outputs found
Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs
Several recently developed multisymplectic schemes for Hamiltonian PDEs have
been shown to preserve associated local conservation laws and constraints very
well in long time numerical simulations. Backward error analysis for PDEs, or
the method of modified equations, is a useful technique for studying the
qualitative behavior of a discretization and provides insight into the
preservation properties of the scheme. In this paper we initiate a backward
error analysis for PDE discretizations, in particular of multisymplectic box
schemes for the nonlinear Schrodinger equation. We show that the associated
modified differential equations are also multisymplectic and derive the
modified conservation laws which are satisfied to higher order by the numerical
solution. Higher order preservation of the modified local conservation laws is
verified numerically.Comment: 12 pages, 6 figures, accepted Math. and Comp. Simul., May 200
Geometric numerical schemes for the KdV equation
Geometric discretizations that preserve certain Hamiltonian structures at the
discrete level has been proven to enhance the accuracy of numerical schemes. In
particular, numerous symplectic and multi-symplectic schemes have been proposed
to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this
work, we show that geometrical schemes are as much robust and accurate as
Fourier-type pseudo-spectral methods for computing the long-time KdV dynamics,
and thus more suitable to model complex nonlinear wave phenomena.Comment: 22 pages, 14 figures, 74 references. Other author's papers can be
downloaded at http://www.lama.univ-savoie.fr/~dutykh
Backward error analysis for variational discretisations of partial differential equations
In backward error analysis, an approximate solution to an equation is
compared to the exact solution to a nearby "modified" equation. In numerical
ordinary differential equations, the two agree up to any power of the step
size. If the differential equation has a geometric property then the modified
equation may share it. In this way, known properties of differential equations
can be applied to the approximation. But for partial differential equations,
the known modified equations are of higher order, limiting applicability of the
theory. Therefore, we study symmetric solutions of discretized partial
differential equations that arise from a discrete variational principle. These
symmetric solutions obey infinite-dimensional functional equations. We show
that these equations admit second-order modified equations which are
Hamiltonian and also possess first-order Lagrangians in modified coordinates.
The modified equation and its associated structures are computed explicitly for
the case of rotating travelling waves in the nonlinear wave equation
Asynchronous Variational Integrators
We describe a new class of asynchronous variational integrators (AVI) for nonlinear
elastodynamics. The AVIs are distinguished by the following attributes: (i)
The algorithms permit the selection of independent time steps in each element, and
the local time steps need not bear an integral relation to each other; (ii) the algorithms
derive from a spacetime form of a discrete version of Hamilton’s variational
principle. As a consequence of this variational structure, the algorithms conserve
local momenta and a local discrete multisymplectic structure exactly.
To guide the development of the discretizations, a spacetime multisymplectic
formulation of elastodynamics is presented. The variational principle used incorporates
both configuration and spacetime reference variations. This allows a unified
treatment of all the conservation properties of the system.A discrete version of reference
configuration is also considered, providing a natural definition of a discrete
energy. The possibilities for discrete energy conservation are evaluated.
Numerical tests reveal that, even when local energy balance is not enforced
exactly, the global and local energy behavior of the AVIs is quite remarkable, a
property which can probably be traced to the symplectic nature of the algorith
Multisymplectic geometry, variational integrators, and nonlinear PDEs
This paper presents a geometric-variational approach to continuous and
discrete mechanics and field theories. Using multisymplectic geometry, we show
that the existence of the fundamental geometric structures as well as their
preservation along solutions can be obtained directly from the variational
principle. In particular, we prove that a unique multisymplectic structure is
obtained by taking the derivative of an action function, and use this structure
to prove covariant generalizations of conservation of symplecticity and
Noether's theorem. Natural discretization schemes for PDEs, which have these
important preservation properties, then follow by choosing a discrete action
functional. In the case of mechanics, we recover the variational symplectic
integrators of Veselov type, while for PDEs we obtain covariant spacetime
integrators which conserve the corresponding discrete multisymplectic form as
well as the discrete momentum mappings corresponding to symmetries. We show
that the usual notion of symplecticity along an infinite-dimensional space of
fields can be naturally obtained by making a spacetime split. All of the
aspects of our method are demonstrated with a nonlinear sine-Gordon equation,
including computational results and a comparison with other discretization
schemes.Comment: LaTeX2E, 52 pages, 11 figures, to appear in Comm. Math. Phy
Energy conservation issues in the numerical solution of the semilinear wave equation
In this paper we discuss energy conservation issues related to the numerical
solution of the nonlinear wave equation. As is well known, this problem can be
cast as a Hamiltonian system that may be autonomous or not, depending on the
specific boundary conditions at hand. We relate the conservation properties of
the original problem to those of its semi-discrete version obtained by the
method of lines. Subsequently, we show that the very same properties can be
transferred to the solutions of the fully discretized problem, obtained by
using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value
Methods) class. Similar arguments hold true for different types of Hamiltonian
Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation.Comment: 41 pages, 11 figur
Variational Structures in Cochain Projection Based Variational Discretizations of Lagrangian PDEs
Compatible discretizations, such as finite element exterior calculus, provide
a discretization framework that respect the cohomological structure of the de
Rham complex, which can be used to systematically construct stable mixed finite
element methods. Multisymplectic variational integrators are a class of
geometric numerical integrators for Lagrangian and Hamiltonian field theories,
and they yield methods that preserve the multisymplectic structure and
momentum-conservation properties of the continuous system. In this paper, we
investigate the synthesis of these two approaches, by constructing
discretization of the variational principle for Lagrangian field theories
utilizing structure-preserving finite element projections. In our
investigation, compatible discretization by cochain projections plays a pivotal
role in the preservation of the variational structure at the discrete level,
allowing the discrete variational structure to essentially be the restriction
of the continuum variational structure to a finite-dimensional subspace. The
preservation of the variational structure at the discrete level will allow us
to construct a discrete Cartan form, which encodes the variational structure of
the discrete theory, and subsequently, we utilize the discrete Cartan form to
naturally state discrete analogues of Noether's theorem and multisymplecticity,
which generalize those introduced in the discrete Lagrangian variational
framework by Marsden et al. [29]. We will study both covariant spacetime
discretization and canonical spatial semi-discretization, and subsequently
relate the two in the case of spacetime tensor product finite element spaces.Comment: 44 pages, 1 figur
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