240 research outputs found
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
Efficient PML for the wave equation
In the last decade, the perfectly matched layer (PML) approach has proved a
flexible and accurate method for the simulation of waves in unbounded media.
Most PML formulations, however, usually require wave equations stated in their
standard second-order form to be reformulated as first-order systems, thereby
introducing many additional unknowns. To circumvent this cumbersome and
somewhat expensive step, we instead propose a simple PML formulation directly
for the wave equation in its second-order form. Inside the absorbing layer, our
formulation requires only two auxiliary variables in two space dimensions and
four auxiliary variables in three space dimensions; hence it is cheap to
implement. Since our formulation requires no higher derivatives, it is also
easily coupled with standard finite difference or finite element methods.
Strong stability is proved while numerical examples in two and three space
dimensions illustrate the accuracy and long time stability of our PML
formulation.Comment: 16 pages, 6 figure
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
On local nonreflecting boundary conditions for time dependent wave propagation
The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded exterior and limit the computation to a finite region. At the artificial boundary a boundary condition is then needed, which allows the propagating waves to exit the computational domain without spurious reflection. In 1977, Engquist and Majda proposed the first hierarchy of absorbing boundary conditions, which allows a systematic reduction of spurious reflection without moving the artificial boundary farther away from the scatterer. Their pioneering work, which initiated an entire research area, is reviewed here from a modern perspective. Recent developments such as high-order local conditions and their extension to multiple scattering are also presented. Finally, the accuracy of high-order local conditions is demonstrated through numerical experiment
Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation
In Grote et al. (SIAM J.Numer.Anal., 44:2408-2431, 2006) a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frog” scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+Δt 2), where p denotes the polynomial degree, h the mesh size, and Δt the time ste
Discrete nonlinear Schrödinger equations for periodic optical systems : pattern formation in \chi(3) coupled waveguide arrays
Discrete nonlinear Schrödinger equations have
been used for many years to model the propagation of light in optical architectures whose refractive index profile is modulated periodically
in the transverse direction. Typically, one considers a modal decomposition of the electric field
where the complex amplitudes satisfy a coupled
system that accommodates nearest neighbour
linear interactions and a local intensity dependent term whose origin lies in the χ
(3) contribution to the medium's dielectric response.
In this presentation, two classic continuum
configurations are discretized in ways that have
received little attention in the literature: the
ring cavity and counterpropagating waves. Both
of these systems are defined by distinct types of
boundary condition. Moreover, they are susceptible to spatial instabilities that are ultimately
responsible for generating spontaneous patterns
from arbitrarily small background disturbances.
Good agreement between analytical predictions
and simulations will be demonstrated
Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates
AbstractWe develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the energy norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space
Stabilized leapfrog based local time-stepping method for the wave equation
Local time-stepping methods permit to overcome the severe stability
constraint on explicit methods caused by local mesh refinement without
sacrificing explicitness. In \cite{DiazGrote09}, a leapfrog based explicit
local time-stepping (LF-LTS) method was proposed for the time integration of
second-order wave equations. Recently, optimal convergence rates were proved
for a conforming FEM discretization, albeit under a CFL stability condition
where the global time-step, , depends on the smallest elements in the
mesh \cite{grote_sauter_1}. In general one cannot improve upon that stability
constraint, as the LF-LTS method may become unstable at certain discrete values
of . To remove those critical values of , we apply a slight
modification (as in recent work on LF-Chebyshev methods \cite{CarHocStu19}) to
the original LF-LTS method which nonetheless preserves its desirable
properties: it is fully explicit, second-order accurate, satisfies a three-term
(leapfrog like) recurrence relation, and conserves the energy. The new
stabilized LF-LTS method also yields optimal convergence rates for a standard
conforming FE discretization, yet under a CFL condition where no
longer depends on the mesh size inside the locally refined region
Uncertainty quantification by multilevel Monte Carlo and local time-stepping
Because of their robustness, efficiency, and non intrusiveness, Monte Carlo methods are probablythe most popular approach in uncertainty quantification for computing expected values of quantitiesof interest. Multilevel Monte Carlo (MLMC) methods significantly reduce the computational costby distributing the sampling across a hierarchy of discretizations and allocating most samples tothe coarser grids. For time dependent problems, spatial coarsening typically entails an increasedtime step. Geometric constraints, however, may impede uniform coarsening thereby forcing someelements to remain small across all levels. If explicit time-stepping is used, the time step will thenbe dictated by the smallest element on each level for numerical stability. Hence, the increasinglystringent CFL condition on the time step on coarser levels significantly reduces the advantages of themultilevel approach. To overcome that bottleneck we propose to combine the multilevel approach ofMLMC with local time-stepping. By adapting the time step to the locally refined elements on eachlevel, the efficiency of MLMC methods is restored even in the presence of complex geometry withoutsacrificing the explicitness and inherent parallelism. In a careful cost comparison, we quantify thereduction in computational cost for local refinement either inside a small fixed region or towards areentrant corner
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