840 research outputs found
Phase-fitted Discrete Lagrangian Integrators
Phase fitting has been extensively used during the last years to improve the
behaviour of numerical integrators on oscillatory problems. In this work, the
benefits of the phase fitting technique are embedded in discrete Lagrangian
integrators. The results show improved accuracy and total energy behaviour in
Hamiltonian systems. Numerical tests on the long term integration (100000
periods) of the 2-body problem with eccentricity even up to 0.95 show the
efficiency of the proposed approach. Finally, based on a geometrical evaluation
of the frequency of the problem, a new technique for adaptive error control is
presented
GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
We present a novel framework for Finite Element Particle-in-Cell methods
based on the discretization of the underlying Hamiltonian structure of the
Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains
the defining properties of a bracket, anti-symmetry and the Jacobi identity, as
well as conservation of its Casimir invariants, implying that the semi-discrete
system is still a Hamiltonian system. In order to obtain a fully discrete
Poisson integrator, the semi-discrete bracket is used in conjunction with
Hamiltonian splitting methods for integration in time. Techniques from Finite
Element Exterior Calculus ensure conservation of the divergence of the magnetic
field and Gauss' law as well as stability of the field solver. The resulting
methods are gauge invariant, feature exact charge conservation and show
excellent long-time energy and momentum behaviour. Due to the generality of our
framework, these conservation properties are guaranteed independently of a
particular choice of the Finite Element basis, as long as the corresponding
Finite Element spaces satisfy certain compatibility conditions.Comment: 57 Page
Multi-symplectic discretisation of wave map equations
We present a new multi-symplectic formulation of constrained Hamiltonian
partial differential equations, and we study the associated local conservation
laws. A multi-symplectic discretisation based on this new formulation is
exemplified by means of the Euler box scheme. When applied to the wave map
equation, this numerical scheme is explicit, preserves the constraint and can
be seen as a generalisation of the Shake algorithm for constrained mechanical
systems. Furthermore, numerical experiments show excellent conservation
properties of the numerical solutions
Geometric Numerical Integration (hybrid meeting)
The topics of the workshop
included interactions between geometric numerical integration and numerical partial differential equations;
geometric aspects of stochastic differential equations;
interaction with optimisation and machine learning;
new applications of geometric integration in physics;
problems of discrete geometry, integrability, and algebraic aspects
Energy Minimization Scheme for Split Potential Systems Using Exponential Variational Integrators
From MDPI via Jisc Publications RouterHistory: accepted 2021-06-18, pub-electronic 2021-06-24Publication status: PublishedIn previous works we developed a methodology of deriving variational integrators to provide numerical solutions of systems having oscillatory behavior. These schemes use exponential functions to approximate the intermediate configurations and velocities, which are then placed into the discrete Lagrangian function characterizing the physical system. We afterwards proved that, higher order schemes can be obtained through the corresponding discrete Euler–Lagrange equations and the definition of a weighted sum of “continuous intermediate Lagrangians” each of them evaluated at an intermediate time node. In the present article, we extend these methods so as to include Lagrangians of split potential systems, namely, to address cases when the potential function can be decomposed into several components. Rather than using many intermediate points for the complete Lagrangian, in this work we introduce different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials. Finally, we assess the accuracy, convergence and computational time of the proposed technique by testing and comparing them with well known standards
High order and energy preserving discontinuous Galerkin methods for the Vlasov-Poisson system
We present a computational study for a family of discontinuous Galerkin
methods for the one dimensional Vlasov-Poisson system that has been recently
introduced. We introduce a slight modification of the methods to allow for
feasible computations while preserving the properties of the original methods.
We study numerically the verification of the theoretical and convergence
analysis, discussing also the conservation properties of the schemes. The
methods are validated through their application to some of the benchmarks in
the simulation of plasma physics.Comment: 44 pages, 28 figure
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