2,783 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
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
Energy conserving schemes for the simulation of musical instrument contact dynamics
Collisions are an innate part of the function of many musical instruments.
Due to the nonlinear nature of contact forces, special care has to be taken in
the construction of numerical schemes for simulation and sound synthesis.
Finite difference schemes and other time-stepping algorithms used for musical
instrument modelling purposes are normally arrived at by discretising a
Newtonian description of the system. However because impact forces are
non-analytic functions of the phase space variables, algorithm stability can
rarely be established this way. This paper presents a systematic approach to
deriving energy conserving schemes for frictionless impact modelling. The
proposed numerical formulations follow from discretising Hamilton's equations
of motion, generally leading to an implicit system of nonlinear equations that
can be solved with Newton's method. The approach is first outlined for point
mass collisions and then extended to distributed settings, such as vibrating
strings and beams colliding with rigid obstacles. Stability and other relevant
properties of the proposed approach are discussed and further demonstrated with
simulation examples. The methodology is exemplified through a case study on
tanpura string vibration, with the results confirming the main findings of
previous studies on the role of the bridge in sound generation with this type
of string instrument
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