7,742 research outputs found
Implementation of standard testbeds for numerical relativity
We discuss results that have been obtained from the implementation of the
initial round of testbeds for numerical relativity which was proposed in the
first paper of the Apples with Apples Alliance. We present benchmark results
for various codes which provide templates for analyzing the testbeds and to
draw conclusions about various features of the codes. This allows us to sharpen
the initial test specifications, design a new test and add theoretical insight.Comment: Corrected versio
A spectral solver for evolution problems with spatial S3-topology
We introduce a single patch collocation method in order to compute solutions
of initial value problems of partial differential equations whose spatial
domains are 3-spheres. Besides the main ideas, we discuss issues related to our
implementation and analyze numerical test applications. Our main interest lies
in cosmological solutions of Einstein's field equations. Motivated by this, we
also elaborate on problems of our approach for general tensorial evolution
equations when certain symmetries are assumed. We restrict to U(1)- and Gowdy
symmetry here.Comment: 29 pages, 11 figures, uses psfrag and hyperref, large parts rewritten
in order to match to the requirements of the journal, conclusions unchanged;
J. Comput. Phys. (2009
Designing Illumination Lenses and Mirrors by the Numerical Solution of Monge-Amp\`ere Equations
We consider the inverse refractor and the inverse reflector problem. The task
is to design a free-form lens or a free-form mirror that, when illuminated by a
point light source, produces a given illumination pattern on a target. Both
problems can be modeled by strongly nonlinear second-order partial differential
equations of Monge-Amp\`ere type. In [Math. Models Methods Appl. Sci. 25
(2015), pp. 803--837, DOI: 10.1142/S0218202515500190] the authors have proposed
a B-spline collocation method which has been applied to the inverse reflector
problem. Now this approach is extended to the inverse refractor problem. We
explain in depth the collocation method and how to handle boundary conditions
and constraints. The paper concludes with numerical results of refracting and
reflecting optical surfaces and their verification via ray tracing.Comment: 16 pages, 6 figures, 2 tables; Keywords: Inverse refractor problem,
inverse reflector problem, elliptic Monge-Amp\`ere equation, B-spline
collocation method, Picard-type iteration; OCIS: 000.4430, 080.1753,
080.4225, 080.4228, 080.4298, 100.3190. Minor revision: two typos have been
corrected and copyright note has been adde
Numerical evolution of multiple black holes with accurate initial data
We present numerical evolutions of three equal-mass black holes using the
moving puncture approach. We calculate puncture initial data for three black
holes solving the constraint equations by means of a high-order multigrid
elliptic solver. Using these initial data, we show the results for three black
hole evolutions with sixth-order waveform convergence. We compare results
obtained with the BAM and AMSS-NCKU codes with previous results. The
approximate analytic solution to the Hamiltonian constraint used in previous
simulations of three black holes leads to different dynamics and waveforms. We
present some numerical experiments showing the evolution of four black holes
and the resulting gravitational waveform.Comment: Published in PR
Lattice Boltzmann Methods for Partial Differential Equations
Lattice Boltzmann methods provide a robust and highly scalable numerical technique in modern computational fluid dynamics. Besides the discretization procedure, the relaxation principles form the basis of any lattice Boltzmann scheme and render the method a bottom-up approach, which obstructs its development for approximating broad classes of partial differential equations. This work introduces a novel coherent mathematical path to jointly approach the topics of constructability, stability, and limit consistency for lattice Boltzmann methods. A new constructive ansatz for lattice Boltzmann equations is introduced, which highlights the concept of relaxation in a top-down procedure starting at the targeted partial differential equation. Modular convergence proofs are used at each step to identify the key ingredients of relaxation frequencies, equilibria, and moment bases in the ansatz, which determine linear and nonlinear stability as well as consistency orders of relaxation and space-time discretization. For the latter, conventional techniques are employed and extended to determine the impact of the kinetic limit at the very foundation of lattice Boltzmann methods. To computationally analyze nonlinear stability, extensive numerical tests are enabled by combining the intrinsic parallelizability of lattice Boltzmann methods with the platform-agnostic and scalable open-source framework OpenLB. Through upscaling the number and quality of computations, large variations in the parameter spaces of classical benchmark problems are considered for the exploratory indication of methodological insights. Finally, the introduced mathematical and computational techniques are applied for the proposal and analysis of new lattice Boltzmann methods. Based on stabilized relaxation, limit consistent discretizations, and consistent temporal filters, novel numerical schemes are developed for approximating initial value problems and initial boundary value problems as well as coupled systems thereof. In particular, lattice Boltzmann methods are proposed and analyzed for temporal large eddy simulation, for simulating homogenized nonstationary fluid flow through porous media, for binary fluid flow simulations with higher order free energy models, and for the combination with Monte Carlo sampling to approximate statistical solutions of the incompressible Euler equations in three dimensions
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