Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Second-Order Symplectic Integrator for Guiding-Center Equations
127
0
0.0
(
0
)
Authors
John R. Cary
Ask ChatGPT about the research
No Arabic abstract
This paper had no abstract originally. A second-order symplectic integration algorithm for guiding center motion is presented. The algorithm is based on the Poincare (mid-point) generating function.
rate research
Read More
The difference between the guiding center phase-space Lagrangians derived in [J.W. Burby, J. Squire, and H. Qin, Phys. Plasmas {bf 20}, 072105 (2013)] and [F.I. Parra, and I. Calvo, Plasma Phys. Control. Fusion {bf 53}, 045001 (2011)] is due to a different definition of the guiding center coordinates. In this brief communication the difference between the guiding center coordinates is calculated explicitly.
We present a reversible and symplectic algorithm called ROLL, for integrating the equations of motion in molecular dynamics simulations of simple fluids on a hypersphere $mathcal{S}^d$ of arbitrary dimension $d$. It is derived in the framework of geometric algebra and shown to be mathematically equivalent to algorithm RATTLE. An application to molecular dynamics simulation of the one component plasma is briefly discussed.
In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schrodinger equation on the $d$ dimensional torus $mathbb T^d$. The scheme has recently also been derived in a wider context of decorated trees in [Y. Bruned and K. Schratz, arXiv:2005.01649]. It is explicit and efficient to implement. Here, we present an alternative derivation, and we give a rigorous error analysis. In particular, we prove second-order convergence in $H^gamma(mathbb T^d)$ for initial data in $H^{gamma+2}(mathbb T^d)$ for any $gamma > d/2$. This improves the previous work in [Knoller, A. Ostermann, and K. Schratz, SIAM J. Numer. Anal. 57 (2019), 1967-1986]. The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.
First-order accurate degenerate variational integration (DVI) was introduced in C. L. Ellison et. al, Phys. Plasmas 25, 052502 (2018) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introducing second order accurate DVI schemes, both with and without non-uniform time stepping. We show that it is not in general possible to construct a second order scheme with a preserved two-form by composing a first order scheme with its adjoint, and discuss the conditions under which such a composition is possible. We build two classes of second order accurate DVI schemes. We test these second order schemes numerically on two systems having noncanonical variables, namely the magnetic field line and guiding center systems. Variational integration for Hamiltonian systems with nonuniform time steps, in terms of an extended phase space Hamiltonian, is generalized to noncanonical variables. It is shown that preservation of proper degeneracy leads to single-step methods without parasitic modes, i.e. to non-uniform time step DVIs. This extension applies to second order accurate as well as first order schemes, and can be applied to adapt the time stepping to an error estimate.
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems. In this way we develop an approach to a class of fourth order elliptic partial differential equations using the idea of splitting into coupled second order equations. Our main motivation is to treat certain fourth order surface equations arising in the modelling of biomembranes but the approach may be applied more generally. In particular, we are interested in equations with non-smooth right hand sides and operators which have non-trivial kernels.The theory for well posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
