Reducing reflections from mesh refinement interfaces in numerical relativity

Full interpretation of data from gravitational wave observations will require accurate numerical simulations of source systems, particularly binary black hole mergers. A leading approach to improving accuracy in numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a manifestation of numerical interface truncation error which appears as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh refinement implementations, potentially compromising the effectiveness of mesh refinement techniques for some numerical relativity applications if left untreated. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite differencing stencil modifications, termed mesh-adapted differencing (MAD), which eliminate this pathology in both our model problem and in numerical relativity examples.Comment: 7 page

Thermodynamics of Extended Bodies in Special Relativity

Relativistic thermodynamics is generalized to accommodate four dimensional rotation in a flat spacetime. An extended body can be in equilibrium when its each element moves along a Killing flow. There are three types of basic Killing flows in a flat spacetime, each of which corresponds to translational motion, spatial rotation, and constant linear acceleration; spatial rotation and constant linear acceleration are regarded as four dimensional rotation. Translational motion has been mainly investigated in the past literature of relativistic thermodynamics. Thermodynamics of the other two is derived in the present paper.Comment: 8 pages, no figur

Phase Field Model for Dynamics of Sweeping Interface

Motivated by the drying pattern experiment by Yamazaki and Mizuguchi[J. Phys. Soc. Jpn. {\bf 69} (2000) 2387], we propose the dynamics of sweeping interface, in which material distributed over a region is swept by a moving interface. A model based on a phase field is constructed and results of numerical simulations are presented for one and two dimensions. Relevance of the present model to the drying experiment is discussed.Comment: 4 pages, 7 figure

Identification of nonlinearity in conductivity equation via Dirichlet-to-Neumann map

We prove that the linear term and quadratic nonlinear term entering a nonlinear elliptic equation of divergence type can be uniquely identified by the Dirichlet to Neuman map. The unique identifiability is proved using the complex geometrical optics solutions and singular solutions

Mechanism of CDW-SDW Transition in One Dimension

The phase transition between charge- and spin-density-wave (CDW, SDW) phases is studied in the one-dimensional extended Hubbard model at half-filling. We discuss whether the transition can be described by the Gaussian and the spin-gap transitions under charge-spin separation, or by a direct CDW-SDW transition. We determine these phase boundaries by level crossings of excitation spectra which are identified according to discrete symmetries of wave functions. We conclude that the Gaussian and the spin-gap transitions take place separately from weak- to intermediate-coupling region. This means that the third phase exists between the CDW and the SDW states. Our results are also consistent with those of the strong-coupling perturbative expansion and of the direct evaluation of order parameters.Comment: 5 pages(REVTeX), 5 figures(EPS), 1 table, also available from http://wwwsoc.nacsis.ac.jp/jps/jpsj/1999/p68a/p68a42/p68a42h/p68a42h.htm

Nonaxisymmetric Evolution of Magnetically Subcritical Clouds: Bar Growth, Core Elongation, and Binary Formation

We have begun a systematic numerical study of the nonlinear growth of nonaxisymmetric perturbations during the ambipolar diffusion-driven evolution of initially magnetically subcritical molecular clouds, with an eye on the formation of binaries, multiple stellar systems and small clusters. In this initial study, we focus on the $m=2$ (or bar) mode, which is shown to be unstable during the dynamic collapse phase of cloud evolution after the central region has become magnetically supercritical. We find that, despite the presence of a strong magnetic field, the bar can grow fast enough that for a modest initial perturbation (at 5% level) a large aspect ratio is obtained during the isothermal phase of cloud collapse. The highly elongated bar is expected to fragment into small pieces during the subsequent adiabatic phase. Our calculations suggest that the strong magnetic fields observed in some star-forming clouds and envisioned in the standard picture of single star formation do not necessarily suppress bar growth and fragmentation; on the contrary, they may actually promote these processes, by allowing the clouds to have more than one (thermal) Jeans mass to begin with without collapsing promptly. Nonlinear growth of the bar mode in a direction perpendicular to the magnetic field, coupled with flattening along field lines, leads to the formation of supercritical cores that are triaxial in general. It removes a longstanding objection to the standard scenario of isolated star formation involving subcritical magnetic field and ambipolar diffusion based on the likely prolate shape inferred for dense cores. Continuted growth of the bar mode in already elongated starless cores, such as L1544, may lead to future binary and multiple star formation.Comment: 5 pages, 2 figures, accepted by ApJ