166 research outputs found
Stochastic Variational Partitioned Runge-Kutta Integrators for Constrained Systems
Stochastic variational integrators for constrained, stochastic mechanical
systems are developed in this paper. The main results of the paper are twofold:
an equivalence is established between a stochastic Hamilton-Pontryagin (HP)
principle in generalized coordinates and constrained coordinates via Lagrange
multipliers, and variational partitioned Runge-Kutta (VPRK) integrators are
extended to this class of systems. Among these integrators are first and
second-order strongly convergent RATTLE-type integrators. We prove order of
accuracy of the methods provided. The paper also reviews the deterministic
treatment of VPRK integrators from the HP viewpoint.Comment: 26 pages, 2 figure
Discrete mechanics and variational integrators
This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented
Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems
In this work, the stochastic version of the variational principle is established, important for stochastic symplectic integration, and for structure-preserving algorithms of stochastic dynamical systems. Based on it, the stochastic variational integrators in formulation of stochastic Lagrangian functions are proposed, and some applications to symplectic integrations are given. Three types of generating functions in the cases of one and two noises are discussed for constructing new schemes
R-adaptive multisymplectic and variational integrators
Moving mesh methods (also called r-adaptive methods) are space-adaptive
strategies used for the numerical simulation of time-dependent partial
differential equations. These methods keep the total number of mesh points
fixed during the simulation, but redistribute them over time to follow the
areas where a higher mesh point density is required. There are a very limited
number of moving mesh methods designed for solving field-theoretic partial
differential equations, and the numerical analysis of the resulting schemes is
challenging. In this paper we present two ways to construct r-adaptive
variational and multisymplectic integrators for (1+1)-dimensional Lagrangian
field theories. The first method uses a variational discretization of the
physical equations and the mesh equations are then coupled in a way typical of
the existing r-adaptive schemes. The second method treats the mesh points as
pseudo-particles and incorporates their dynamics directly into the variational
principle. A user-specified adaptation strategy is then enforced through
Lagrange multipliers as a constraint on the dynamics of both the physical field
and the mesh points. We discuss the advantages and limitations of our methods.
Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure
Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems
The purpose of this work is twofold. First, we demonstrate analytically
that the classical Newmark family as well as related integration
algorithms are variational in the sense of the Veselov formulation of
discrete mechanics. Such variational algorithms are well known to be
symplectic and momentum preserving and to often have excellent global
energy behavior. This analytical result is veried through numerical examples
and is believed to be one of the primary reasons that this class
of algorithms performs so well.
Second, we develop algorithms for mechanical systems with forcing,
and in particular, for dissipative systems. In this case, we develop integrators
that are based on a discretization of the Lagrange d'Alembert
principle as well as on a variational formulation of dissipation. It is
demonstrated that these types of structured integrators have good numerical
behavior in terms of obtaining the correct amounts by which
the energy changes over the integration run
Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise
One- and multi-dimensional stochastic Maxwell equations with additive noise
are considered in this paper. It is known that such system can be written in
the multi-symplectic structure, and the stochastic energy increases linearly in
time. High order discontinuous Galerkin methods are designed for the stochastic
Maxwell equations with additive noise, and we show that the proposed methods
satisfy the discrete form of the stochastic energy linear growth property and
preserve the multi-symplectic structure on the discrete level. Optimal error
estimate of the semi-discrete DG method is also analyzed. The fully discrete
methods are obtained by coupling with symplectic temporal discretizations. One-
and two-dimensional numerical results are provided to demonstrate the
performance of the proposed methods, and optimal error estimates and linear
growth of the discrete energy can be observed for all cases
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