80 research outputs found
Exponential energy-preserving methods for charged-particle dynamics in a strong and constant magnetic field
In this paper, exponential energy-preserving methods are formulated and
analysed for solving charged-particle dynamics in a strong and constant
magnetic field. The resulting method can exactly preserve the energy of the
dynamics. Moreover, it is shown that the magnetic moment of the considered
system is nearly conserved over a long time along this exponential
energy-preserving method, which is proved by using modulated Fourier
expansions. Other properties of the method including symmetry and convergence
are also studied. An illustrated numerical experiment is carried out to
demonstrate the long-time behaviour of the method
Energy preserving methods for nonlinear Schr\"odinger equations
This paper is concerned with the numerical integration in time of nonlinear
Schr\"odinger equations using different methods preserving the energy or a
discrete analog of it. The Crank-Nicolson method is a well known method of
order 2 but is fully implicit and one may prefer a linearly implicit method
like the relaxation method introduced in [10] for the cubic nonlinear
Schr{\"o}dinger equation. This method is also an energy preserving method and
numerical simulations have shown that its order is 2. In this paper we give a
rigorous proof of the order of this relaxation method and propose a generalized
version that allows to deal with general power law nonlinearites. Numerical
simulations for different physical models show the efficiency of these methods
Discrete line integral method for the Lorentz force system
In this paper, we apply the Boole discrete line integral to solve the Lorentz
force system which is written as a non-canonical Hamiltonian system. The method
is exactly energy-conserving for polynomial Hamiltonians of degree . In any other case, the energy can also be conserved approximatively. With
comparison to well-used Boris method, numerical experiments are presented to
demonstrate the energy-preserving property of the method
Energy-Preserving Integrators Applied to Nonholonomic Systems
We introduce energy-preserving integrators for nonholonomic mechanical
systems. We will see that the nonholonomic dynamics is completely determined by
a triple , where is the
dual of the vector bundle determined by the nonholonomic constraints, is
an almost-Poisson bracket (the nonholonomic bracket) and is a Hamiltonian function. For this triple, we
can apply energy-preserving integrators, in particular, we show that discrete
gradients can be used in the numerical integration of nonholonomic dynamics. By
construction, we achieve preservation of the constraints and of the energy of
the nonholonomic system. Moreover, to facilitate their applicability to complex
systems which cannot be easily transformed into the aforementioned
almost-Poisson form, we rewrite our integrators using just the initial
information of the nonholonomic system. The derived procedures are tested on
several examples: A chaotic quartic nonholonomic mechanical system, the
Chaplygin sleigh system, the Suslov problem and a continuous gearbox driven by
an asymmetric pendulum. Their performace is compared with other standard
methods in nonholonomic dynamics, and their merits verified in practice
Geometric Exponential Integrators
In this paper, we consider exponential integrators for semilinear Poisson
systems. Two types of exponential integrators are constructed, one preserves
the Poisson structure, and the other preserves energy. Numerical experiments
for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs
are presented. These geometric exponential integrators exhibit better long time
stability properties as compared to non-geometric integrators, and are
computationally more efficient than traditional symplectic integrators and
energy-preserving methods based on the discrete gradient method.Comment: 18 pages, 11 figure
A characterization of energy-preserving methods and the construction of parallel integrators for Hamiltonian systems
High order energy-preserving methods for Hamiltonian systems are presented.
For this aim, an energy-preserving condition of continuous stage Runge--Kutta
methods is proved. Order conditions are simplified and parallelizable
conditions are also given. The computational cost of our high order methods is
comparable to that of the average vector field method of order two
Exponential collocation methods for conservative or dissipative systems
In this paper, we propose and analyse a novel class of exponential
collocation methods for solving conservative or dissipative systems based on
exponential integrators and collocation methods. It is shown that these novel
methods can be of arbitrarily high order and exactly or nearly preserve first
integrals or Lyapunov functions. We also consider order estimates of the new
methods. Furthermore, we explore and discuss the application of our methods in
important stiff gradient systems, and it turns out that our methods are
unconditionally energy-diminishing and strongly damped even for very stiff
gradient systems. Practical examples of the new methods are derived and the
efficiency and superiority are confirmed and demonstrated by three numerical
experiments including a nonlinear Schr\"{o}dinger equation. As a byproduct of
this paper, arbitrary-order trigonometric/RKN collocation methods are also
presented and analysed for second-order highly oscillatory/general systems. The
paper is accompanied by numerical results that demonstrate the great potential
of this work
Conservative methods for stochastic differential equations with a conserved quantity
This paper proposes a novel conservative method for numerical computation of
general stochastic differential equations in the Stratonovich sense with a
conserved quantity. We show that the mean-square order of the method is if
noises are commutative and that the weak order is also . Since the proposed
method may need the computation of a deterministic integral, we analyse the
effect of the use of quadrature formulas on the convergence orders.
Furthermore, based on the splitting technique of stochastic vector fields, we
construct conservative composition methods with similar orders as the above
method. Finally, numerical experiments are presented to support our theoretical
results
A linearized and conservative Fourier pseudo-spectral method for the damped nonlinear Schr\"{o}dinger equation in three dimensions
In this paper, we propose a linearized Fourier pseudo-spectral method, which
preserves the total mass and energy conservation laws, for the damped nonlinear
Schr\"{o}dinger equation in three dimensions. With the aid of the semi-norm
equivalence between the Fourier pseudo-spectral method and the finite
difference method, an optimal -error estimate for the proposed method
without any restriction on the grid ratio is established by analyzing the real
and imaginary parts of the error function. Numerical results are addressed to
confirm our theoretical analysis.Comment: 29 pages, 2 figure
Energy preserving moving mesh methods applied to the BBM equation
Energy preserving numerical methods for a certain class of PDEs are derived,
applying the partition of unity method. The methods are extended to also be
applicable in combination with moving mesh methods by the rezoning approach.
These energy preserving moving mesh methods are then applied to the
Benjamin--Bona--Mahony equation, resulting in schemes that exactly preserve an
approximation to one of the Hamiltonians of the system. Numerical experiments
that demonstrate the advantages of the methods are presented.Comment: 13 pages; 7 figures, 13 subfigure
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