144 research outputs found
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
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
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
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 MLMC and Local Time-stepping For Wave Propagation
Because of their robustness, efficiency and non-intrusiveness, Monte Carlo
methods are probably the most popular approach in uncertainty quantification to
computing expected values of quantities of interest (QoIs). Multilevel Monte
Carlo (MLMC) methods significantly reduce the computational cost by
distributing the sampling across a hierarchy of discretizations and allocating
most samples to the coarser grids. For time dependent problems, spatial
coarsening typically entails an increased time-step. Geometric constraints,
however, may impede uniform coarsening thereby forcing some elements to remain
small across all levels. If explicit time-stepping is used, the time-step will
then be dictated by the smallest element on each level for numerical stability.
Hence, the increasingly stringent CFL condition on the time-step on coarser
levels significantly reduces the advantages of the multilevel approach. By
adapting the time-step to the locally refined elements on each level, local
time-stepping (LTS) methods permit to restore the efficiency of MLMC methods
even in the presence of complex geometry without sacrificing the explicitness
and inherent parallelism
Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates
We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375-386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L2-norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetic
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