12 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
hp-version time domain boundary elements for the wave equation on quasi-uniform meshes
Solutions to the wave equation in the exterior of a polyhedral domain or a
screen in exhibit singular behavior from the edges and corners.
We present quasi-optimal -explicit estimates for the approximation of the
Dirichlet and Neumann traces of these solutions for uniform time steps and
(globally) quasi-uniform meshes on the boundary. The results are applied to an
-version of the time domain boundary element method. Numerical examples
confirm the theoretical results for the Dirichlet problem both for screens and
polyhedral domains.Comment: 41 pages, 11 figure
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
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
Error Estimates for FE-BE Coupling of Scattering of Waves in the Time Domain
This article considers a coupled finite and boundary element method for an interface problem for the acoustic wave equation. Well-posedness, a priori and a posteriori error estimates are discussed for a symmetric space-time Galerkin discretization related to the energy. Numerical experiments in three dimensions illustrate the performance of the method in model problems. © 2022 Walter de Gruyter GmbH, Berlin/Boston 2022
Multi-level local time-stepping methods of Runge-Kutta type forwave equations
Local mesh refinement significantly in uences the performance of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods improve the efficiency by using smaller time-steps precisely where the smallest mesh elements are located, thus permitting a larger time-step in the coarser regions of the mesh without violating the stability condition. However, when the mesh contains nested patches of refinement, any local time-step will be unnecessarily small in some regions. To allow for an appropriate time-step at each level of mesh refinement, multi-level local time-stepping (MLTS) methods have been proposed. Starting from the Runge{Kutta-based LTS methods derived by Grote et al. [17], we propose explicit MLTS methods of arbitrarily high accuracy. Numerical experiments with finite difference and continuous finite element spatial discretizations illustrate the usefulness of the novel MLTS methods and show that they retain the high accuracy and stability of the underlying Runge{Kutta methods
A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
Trefftz methods are high-order Galerkin schemes in which all discrete
functions are elementwise solution of the PDE to be approximated. They are
viable only when the PDE is linear and its coefficients are piecewise constant.
We introduce a 'quasi-Trefftz' discontinuous Galerkin method for the
discretisation of the acoustic wave equation with piecewise-smooth wavespeed:
the discrete functions are elementwise approximate PDE solutions. We show that
the new discretisation enjoys the same excellent approximation properties as
the classical Trefftz one, and prove stability and high-order convergence of
the DG scheme. We introduce polynomial basis functions for the new discrete
spaces and describe a simple algorithm to compute them. The technique we
propose is inspired by the generalised plane waves previously developed for
time-harmonic problems with variable coefficients; it turns out that in the
case of the time-domain wave equation under consideration the quasi-Trefftz
approach allows for polynomial basis functions.Comment: 25 pages, 9 figure
Boundary elements with mesh refinements for the wave equation
The solution of the wave equation in a polyhedral domain in
admits an asymptotic singular expansion in a neighborhood of the corners and
edges. In this article we formulate boundary and screen problems for the wave
equation as equivalent boundary integral equations in time domain, study the
regularity properties of their solutions and the numerical approximation.
Guided by the theory for elliptic equations, graded meshes are shown to recover
the optimal approximation rates known for smooth solutions. Numerical
experiments illustrate the theory for screen problems. In particular, we
discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann
operator and applications to the sound emission of tires.Comment: 45 pages, to appear in Numerische Mathemati
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains ,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.Comment: 38 pages, 8 figure
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains ,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.Comment: 38 pages, 8 figure