209 research outputs found
Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators
We present a high-order accurate discontinuous Galerkin method for evolving
the spherically-reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system
expressed in terms of second-order spatial operators. Our multi-domain method
achieves global spectral accuracy and long-time stability on short
computational domains. We discuss in detail both our scheme for the BSSN system
and its implementation. After a theoretical and computational verification of
the proposed scheme, we conclude with a brief discussion of issues likely to
arise when one considers the full BSSN system.Comment: 35 pages, 6 figures, 1 table, uses revtex4. Revised in response to
referee's repor
Deep convolutional neural networks for estimating porous material parameters with ultrasound tomography
We study the feasibility of data based machine learning applied to ultrasound
tomography to estimate water-saturated porous material parameters. In this
work, the data to train the neural networks is simulated by solving wave
propagation in coupled poroviscoelastic-viscoelastic-acoustic media. As the
forward model, we consider a high-order discontinuous Galerkin method while
deep convolutional neural networks are used to solve the parameter estimation
problem. In the numerical experiment, we estimate the material porosity and
tortuosity while the remaining parameters which are of less interest are
successfully marginalized in the neural networks-based inversion. Computational
examples confirms the feasibility and accuracy of this approach
Persistent junk solutions in time-domain modeling of extreme mass ratio binaries
In the context of metric perturbation theory for non-spinning black holes,
extreme mass ratio binary (EMRB) systems are described by distributionally
forced master wave equations. Numerical solution of a master wave equation as
an initial boundary value problem requires initial data. However, because the
correct initial data for generic-orbit systems is unknown, specification of
trivial initial data is a common choice, despite being inconsistent and
resulting in a solution which is initially discontinuous in time. As is well
known, this choice leads to a "burst" of junk radiation which eventually
propagates off the computational domain. We observe another unintended
consequence of trivial initial data: development of a persistent spurious
solution, here referred to as the Jost junk solution, which contaminates the
physical solution for long times. This work studies the influence of both types
of junk on metric perturbations, waveforms, and self-force measurements, and it
demonstrates that smooth modified source terms mollify the Jost solution and
reduce junk radiation. Our concluding section discusses the applicability of
these observations to other numerical schemes and techniques used to solve
distributionally forced master wave equations.Comment: Uses revtex4, 16 pages, 9 figures, 3 tables. Document reformatted and
modified based on referee's report. Commentary added which addresses the
possible presence of persistent junk solutions in other approaches for
solving master wave equation
Numerical simulations with a first order BSSN formulation of Einstein's field equations
We present a new fully first order strongly hyperbolic representation of the
BSSN formulation of Einstein's equations with optional constraint damping
terms. We describe the characteristic fields of the system, discuss its
hyperbolicity properties, and present two numerical implementations and
simulations: one using finite differences, adaptive mesh refinement and in
particular binary black holes, and another one using the discontinuous Galerkin
method in spherical symmetry. The results of this paper constitute a first step
in an effort to combine the robustness of BSSN evolutions with very high
accuracy numerical techniques, such as spectral collocation multi-domain or
discontinuous Galerkin methods.Comment: To appear in Physical Review
Discontinuous Galerkin Discretizations of the Boltzmann Equations in 2D: semi-analytic time stepping and absorbing boundary layers
We present an efficient nodal discontinuous Galerkin method for approximating
nearly incompressible flows using the Boltzmann equations. The equations are
discretized with Hermite polynomials in velocity space yielding a first order
conservation law. A stabilized unsplit perfectly matching layer (PML)
formulation is introduced for the resulting nonlinear flow equations. The
proposed PML equations exponentially absorb the difference between the
nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic
time discretization methods to improve the time step restrictions in small
relaxation times. We also introduce a multirate semi-analytic Adams-Bashforth
method which preserves efficiency in stiff regimes. Accuracy and performance of
the method are tested using distinct cases including isothermal vortex, flow
around square cylinder, and wall mounted square cylinder test cases.Comment: 37 pages, 11 figure
Adaptive High-Order Finite-Difference Method for Nonlinear Wave Problems
We discuss a scheme for the numerical solution of one-dimensional initial value problems exhibiting strongly localized solutions or finite-time singularities. To accurately and efficiently model such phenomena we present a full space-time adaptive scheme, based on a variable order spatial finite-difference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance with the local behavior of the solution. Through several examples, taken from gasdynamics and nonlinear optics, we illustrate the performance of the scheme, the use of which results in several orders of magnitude reduction in the required degrees of freedom to solve a problem to a particular fidelity
Discontinuous Galerkin methods for the time-domain Maxwell's equations
We discuss the basic elements of the discontinuous Galerkin methods for the time-domain Maxwell's equations. A one-dimensional example is developed in detail from which the extension to two- and three-dimensional algorithms are minimal. A few examples are offered as well as guidelines for extensions, generalizations, and helpful software resources
A geometrically motivated coordinate system for exploring spacetime dynamics in numerical-relativity simulations using a quasi-Kinnersley tetrad
We investigate the suitability and properties of a quasi-Kinnersley tetrad
and a geometrically motivated coordinate system as tools for quantifying both
strong-field and wave-zone effects in numerical relativity (NR) simulations. We
fix the radial and latitudinal coordinate degrees of freedom of the metric,
using the Coulomb potential associated with the quasi-Kinnersley transverse
frame. These coordinates are invariants of the spacetime and can be used to
unambiguously fix the outstanding spin-boost freedom associated with the
quasi-Kinnersley frame (resulting in a preferred quasi-Kinnersley tetrad
(QKT)). In the limit of small perturbations about a Kerr spacetime, these
coordinates and QKT reduce to Boyer-Lindquist coordinates and the Kinnersley
tetrad, irrespective of the simulation gauge choice. We explore the properties
of this construction both analytically and numerically, and we gain insights
regarding the propagation of radiation described by a super-Poynting vector. We
also quantify in detail the peeling properties of the chosen tetrad and gauge.
We argue that these choices are particularly well suited for a rapidly
converging wave-extraction algorithm as the extraction location approaches
infinity, and we explore numerically the extent to which this property remains
applicable on the interior of a computational domain. Using a number of
additional tests, we verify that the prescription behaves as required in the
appropriate limits regardless of simulation gauge. We explore the behavior of
the geometrically motivated coordinate system in dynamical binary-black-hole NR
mergers, and find them useful for visualizing features in NR simulations such
as the spurious "junk" radiation. Finally, we carefully scrutinize the head-on
collision of two black holes and, for example, the way in which the extracted
waveform changes as it moves through the computational domain.Comment: 30 pages, 17 figures, 2 table
A natural-norm Successive Constraint Method for inf-sup lower bounds
We present a new approach for the construction of lower bounds for the inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. We combine the “linearized” inf-sup statement of the natural-norm approach with the approximation procedure of the Successive Constraint Method (SCM): the former (natural-norm) provides an economical parameter expansion and local concavity in parameter—a small(er) optimization problem which enjoys intrinsic lower bound properties; the latter (SCM) provides a systematic optimization framework—a Linear Program (LP) relaxation which readily incorporates continuity and stability constraints. The natural-norm SCM requires a parameter domain decomposition: we propose a greedy algorithm for selection of the SCM control points as well as adaptive construction of the optimal subdomains. The efficacy of the natural-norm SCM is illustrated through numerical results for two types of non-coercive problems: the Helmholtz equation (for acoustics, elasticity, and electromagnetics), and the convection–diffusion equation.United States. Air Force Office of Scientific Research (Grant No. FA 9550-07-1-0425
Efficient preconditioning of hphp-FEM matrix sequences with slowly-varying coefficients:An application to topology optimization
We previously introduced a preconditioner that has proven effective for hp-FEM dis- cretizations of various challenging elliptic and hyperbolic problems. The construc- tion is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi- Separable matrices. The preconditioner is built as an approximate LDMt factorization through a divide-and-conquer approach. This implies an enhanced flexibility which allows to handle unstructured geometric meshes, anisotropies, and discontinuities. We build on our previous numerical experiments and develop a preconditioner- update strategy that allows us handle time-varying problems. We investigate the performance of the precondition along with the update strategy in context of topology optimization of an acoustic cavity
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