2,978 research outputs found
Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation
Local adaptivity and mesh refinement are key to the efficient simulation of
wave phenomena in heterogeneous media or complex geometry. Locally refined
meshes, however, dictate a small time-step everywhere with a crippling effect
on any explicit time-marching method. In [18] a leap-frog (LF) based explicit
local time-stepping (LTS) method was proposed, which overcomes the severe
bottleneck due to a few small elements by taking small time-steps in the
locally refined region and larger steps elsewhere. Here a rigorous convergence
proof is presented for the fully-discrete LTS-LF method when combined with a
standard conforming finite element method (FEM) in space. Numerical results
further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of
corner singularities
Time-Dependent Density Functional Theory with Ultrasoft Pseudopotential: Real-Time Electron Propagation across Molecular Junction
A practical computational scheme based on time-dependent density functional
theory (TDDFT) and ultrasoft pseudopotential (USPP) is developed to study
electron dynamics in real time. A modified Crank-Nicolson time-stepping
algorithm is adopted, under planewave basis. The scheme is validated by
calculating the optical absorption spectra for sodium dimer and benzene
molecule. As an application of this USPP-TDDFT formalism, we compute the time
evolution of a test electron packet at the Fermi energy of the left metallic
lead crossing a benzene-(1,4)-dithiolate junction. A transmission probability
of 5-7%, corresponding to a conductance of 4.0-5.6muS, is obtained. These
results are consistent with complex band structure estimates, and Green's
function calculation results at small bias voltages
An Energy- and Charge-conserving, Implicit, Electrostatic Particle-in-Cell Algorithm
This paper discusses a novel fully implicit formulation for a 1D
electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier
implicit electrostatic PIC approaches (which are based on a linearized
Vlasov-Poisson formulation), ours is based on a nonlinearly converged
Vlasov-Amp\`ere (VA) model. By iterating particles and fields to a tight
nonlinear convergence tolerance, the approach features superior stability and
accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit
PIC implementations. In particular, the formulation is stable against temporal
(CFL) and spatial (aliasing) instabilities. It is charge- and energy-conserving
to numerical roundoff for arbitrary implicit time steps. While momentum is not
exactly conserved, errors are kept small by an adaptive particle sub-stepping
orbit integrator, which is instrumental to prevent particle tunneling. The VA
model is orbit-averaged along particle orbits to enforce an energy conservation
theorem with particle sub-stepping. As a result, very large time steps,
constrained only by the dynamical time scale of interest, are possible without
accuracy loss. Algorithmically, the approach features a Jacobian-free
Newton-Krylov solver. A main development in this study is the nonlinear
elimination of the new-time particle variables (positions and velocities). Such
nonlinear elimination, which we term particle enslavement, results in a
nonlinear formulation with memory requirements comparable to those of a fluid
computation, and affords us substantial freedom in regards to the particle
orbit integrator. Numerical examples are presented that demonstrate the
advertised properties of the scheme. In particular, long-time ion acoustic wave
simulations show that numerical accuracy does not degrade even with very large
implicit time steps, and that significant CPU gains are possible.Comment: 29 pages, 8 figures, submitted to Journal of Computational Physic
Explicit local time-stepping methods for time-dependent wave propagation
Semi-discrete Galerkin formulations of transient wave equations, either with
conforming or discontinuous Galerkin finite element discretizations, typically
lead to large systems of ordinary differential equations. When explicit time
integration is used, the time-step is constrained by the smallest elements in
the mesh for numerical stability, possibly a high price to pay. To overcome
that overly restrictive stability constraint on the time-step, yet without
resorting to implicit methods, explicit local time-stepping schemes (LTS) are
presented here for transient wave equations either with or without damping. In
the undamped case, leap-frog based LTS methods lead to high-order explicit LTS
schemes, which conserve the energy. In the damped case, when energy is no
longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS
schemes of arbitrarily high accuracy. When combined with a finite element
discretization in space with an essentially diagonal mass matrix, the resulting
time-marching schemes are fully explicit and thus inherently parallel.
Numerical experiments with continuous and discontinuous Galerkin finite element
discretizations validate the theory and illustrate the usefulness of these
local time-stepping methods.Comment: overview paper, typos added, references updated. arXiv admin note:
substantial text overlap with arXiv:1109.448
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