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Linearized solutions of the Einstein equations within a Bondi-Sachs framework, and implications for boundary conditions in numerical simulations  [PDF]
Nigel T. Bishop
Physics , 2004, DOI: 10.1088/0264-9381/22/12/006
Abstract: We linearize the Einstein equations when the metric is Bondi-Sachs, when the background is Schwarzschild or Minkowski, and when there is a matter source in the form of a thin shell whose density varies with time and angular position. By performing an eigenfunction decomposition, we reduce the problem to a system of linear ordinary differential equations which we are able to solve. The solutions are relevant to the characteristic formulation of numerical relativity: (a) as exact solutions against which computations of gravitational radiation can be compared; and (b) in formulating boundary conditions on the $r=2M$ Schwarzschild horizon.
Sliding periodic boundary conditions for lattice Boltzmann and lattice kinetic equations  [PDF]
R. Adhikari,J. -C. Desplat,K. Stratford
Physics , 2005,
Abstract: We present a method to impose linear shear flow in discrete-velocity kinetic models of hydrodynamics through the use of sliding periodic boundary conditions. Our method is derived by an explicit coarse-graining of the Lees-Edwards boundary conditions for Couette flow in molecular dynamics, followed by a projection of the resulting equations onto the subspace spanned by the discrete velocities of the lattice Boltzmann method. The boundary conditions are obtained without resort to perturbative expansions or modifications of the discrete velocity equilibria, allowing our method to be applied to a wide class of lattice Boltzmann models. Our numerical results for the sheared hydrodynamics of a one-component isothermal fluid show excellent agreement with analytical results, while for a two-component fluid the results show a clear improvement over previous methods for introducing Lees-Edwards boundary conditions into lattice Boltzmann. Using our method, we obtain a dynamical steady state in a sheared spinodally decomposing two-dimensional fluid, under conditions where previous methods give spurious finite-size artifacts.
Constraint preserving boundary conditions for the Ideal Newtonian MHD equations  [PDF]
Mariana Cecere,Luis Lehner,Oscar Reula
Physics , 2008, DOI: 10.1016/j.cpc.2008.04.015
Abstract: We study and develop constraint preserving boundary conditions for the Newtonian magnetohydrodynamic equations and analyze the behavior of the numerical solution upon considering different possible options.
Numerical implementation of isolated horizon boundary conditions  [PDF]
J. L. Jaramillo,M. Ansorg,F. Limousin
Physics , 2006, DOI: 10.1103/PhysRevD.75.024019
Abstract: We study the numerical implementation of a set of boundary conditions derived from the isolated horizon formalism, and which characterize a black hole whose horizon is in quasi-equilibrium. More precisely, we enforce these geometrical prescriptions as inner boundary conditions on an excised sphere, in the numerical resolution of the Conformal Thin Sandwich equations. As main results, we firstly establish the consistency of including in the set of boundary conditions a "constant surface gravity" prescription, interpretable as a lapse boundary condition, and secondly we assess how the prescriptions presented recently by Dain et al. for guaranteeing the well-posedness of the Conformal Transverse Traceless equations with quasi-equilibrium horizon conditions extend to the Conformal Thin Sandwich elliptic system. As a consequence of the latter analysis, we discuss the freedom of prescribing the expansion associated with the ingoing null normal at the horizon.
Numerical experiment with upstream boundary conditions  [cached]
Huml Jaroslav,Kozel Karel
EPJ Web of Conferences , 2012, DOI: 10.1051/epjconf/20122501027
Abstract: The work deals with a numerical solution of subsonic and transonic ows described by the system of Euler equations in 2D and 3D ows in a channel. Authors used Lax-Wendroff and the multistage Runge-Kutta scheme to numerically solve the ows in a 2D G AMM channel and an extended 3D channel. Authors compare the results achieved by two different upstream boundary conditions in 2D and also in 3D transonic channel ows.
A "Trefftz Machine" for Absorbing Boundary Conditions  [PDF]
Igor Tsukerman
Mathematics , 2014,
Abstract: The paper presents an automatic generator of approximate nonreflecting boundary conditions, analytical and numerical, for scalar wave equations. This generator has two main ingredients. The first one is a set of local Trefftz functions -- outgoing waves approximating the solution in the vicinity of a given point of the exterior boundary of the computational domain. The second ingredient is a set of linear test functionals (degrees of freedom). One example of such functionals is the nodal values of the solution at a set of grid points; in that case, one obtains a numerical condition -- a finite difference scheme at the boundary. Alternatively, the functionals may involve derivatives or integrals of the solution, in which case the proposed "Trefftz machine" yields analytical nonreflecting conditions. Corners and edges are treated algorithmically the same way as straight boundaries. With specific choices of bases and degrees of freedom, the machine produces classical conditions such as Engquist-Majda and Bayliss-Turkel. For other choices, one obtains a variety of analytical and numerical conditions, a few of which are presented as illustrative examples.
Finite element computation of absorbing boundary conditions for time-harmonic wave problems  [PDF]
Denis Duhamel,Tien-Minh Nguyen
Physics , 2009, DOI: 10.1016/j.cma.2009.05.002
Abstract: This paper proposes a new method, in the frequency domain, to define absorbing boundary conditions for general two-dimensional problems. The main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general. It is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software. Boundary conditions at various orders of accuracy can be obtained in a simple way. This is then applied to study some examples for which analytical or numerical results are available. Good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method.
Numerical Treatment of Initial-Boundary Value Problems with Mixed Boundary Conditions  [PDF]
Nawal Abdullah Alzaid, Huda Omar Bakodah
American Journal of Computational Mathematics (AJCM) , 2018, DOI: 10.4236/ajcm.2018.82012
Abstract: In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary conditions for linear and nonlinear partial differential equations. The method is applied to different forms of heat and wave equations as illustrative examples to exhibit the effectiveness of the method. The method provides the solution in a rapidly convergent series with components that can be computed iteratively. The numerical results for the illustrative examples obtained show remarkable agreement with the exact solutions. We also provide some graphical representations for clear-cut comparisons between the solutions using Maple software.
Nahm Equations and Boundary Conditions  [PDF]
Dimitrios Tsimpis
Physics , 1998, DOI: 10.1016/S0370-2693(98)00704-7
Abstract: We derive certain boundary conditions in Nahm's equations by considering a system of N parallel D1-branes perpendicular to a D3-brane in type IIB string theory.
Numerical relativity with the conformal field equations  [PDF]
Sascha Husa
Physics , 2002,
Abstract: I discuss the conformal approach to the numerical simulation of radiating isolated systems in general relativity. The method is based on conformal compactification and a reformulation of the Einstein equations in terms of rescaled variables, the so-called ``conformal field equations'' developed by Friedrich. These equations allow to include ``infinity'' on a finite grid, solving regular equations, whose solutions give rise to solutions of the Einstein equations of (vacuum) general relativity. The conformal approach promises certain advantages, in particular with respect to the treatment of radiation extraction and boundary conditions. I will discuss the essential features of the analytical approach to the problem, previous work on the problem - in particular a code for simulations in 3+1 dimensions, some new results, open problems and strategies for future work.
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