30 research outputs found
An Arbitrary Curvilinear Coordinate Method for Particle-In-Cell Modeling
A new approach to the kinetic simulation of plasmas in complex geometries,
based on the Particle-in- Cell (PIC) simulation method, is explored. In the two
dimensional (2d) electrostatic version of our method, called the Arbitrary
Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are
carried out in 2d on a uniform grid on the unit square logical domain, and
mapped to a nonuniform boundary-fitted grid on the physical domain. As the
resulting logical grid equations of motion are not separable, we have developed
an extension of the semi-implicit Modified Leapfrog (ML) integration technique
to preserve the symplectic nature of the logical grid particle mover. A
generalized, curvilinear coordinate formulation of Poisson's equations to solve
for the electrostatic fields on the uniform logical grid is also developed. By
our formulation, we compute the plasma charge density on the logical grid based
on the particles' positions on the logical domain. That is, the plasma
particles are weighted to the uniform logical grid and the self-consistent mean
electrostatic fields obtained from the solution of the logical grid Poisson
equation are interpolated to the particle positions on the logical grid. This
process eliminates the complexity associated with the weighting and
interpolation processes on the nonuniform physical grid and allows us to run
the PIC method on arbitrary boundary-fitted meshes.Comment: Submitted to Computational Science & Discovery December 201
Symplectic integrators with adaptive time steps
In recent decades, there have been many attempts to construct symplectic
integrators with variable time steps, with rather disappointing results. In
this paper we identify the causes for this lack of performance, and find that
they fall into two categories. In the first, the time step is considered a
function of time alone, \Delta=\Delta(t). In this case, backwards error
analysis shows that while the algorithms remain symplectic, parametric
instabilities arise because of resonance between oscillations of \Delta(t) and
the orbital motion. In the second category the time step is a function of phase
space variables \Delta=\Delta(q,p). In this case, the system of equations to be
solved is analyzed by introducing a new time variable \tau with dt=\Delta(q,p)
d\tau. The transformed equations are no longer in Hamiltonian form, and thus
are not guaranteed to be stable even when integrated using a method which is
symplectic for constant \Delta. We analyze two methods for integrating the
transformed equations which do, however, preserve the structure of the original
equations. The first is an extended phase space method, which has been
successfully used in previous studies of adaptive time step symplectic
integrators. The second, novel, method is based on a non-canonical
mixed-variable generating function. Numerical trials for both of these methods
show good results, without parametric instabilities or spurious growth or
damping. It is then shown how to adapt the time step to an error estimate found
by backward error analysis, in order to optimize the time-stepping scheme.
Numerical results are obtained using this formulation and compared with other
time-stepping schemes for the extended phase space symplectic method.Comment: 23 pages, 9 figures, submitted to Plasma Phys. Control. Fusio
A spectral method for elliptic equations: the Dirichlet problem
An elliptic partial differential equation Lu=f with a zero Dirichlet boundary
condition is converted to an equivalent elliptic equation on the unit ball. A
spectral Galerkin method is applied to the reformulated problem, using
multivariate polynomials as the approximants. For a smooth boundary and smooth
problem parameter functions, the method is proven to converge faster than any
power of 1/n with n the degree of the approximate Galerkin solution. Examples
in two and three variables are given as numerical illustrations. Empirically,
the condition number of the associated linear system increases like O(N), with
N the order of the linear system.Comment: This is latex with the standard article style, produced using
Scientific Workplace in a portable format. The paper is 22 pages in length
with 8 figure