Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to extend such simulations to null infinity where the waveform from the binary inspiral and merger can be unambiguously computed. This has now been accomplished by Cauchy-characteristic extraction, where data for the characteristic evolution is supplied by Cauchy data on an extraction worldtube inside the artificial outer boundary. The ultimate application of characteristic evolution is to eliminate the role of this outer boundary by constructing a global solution via Cauchy-characteristic matching. Progress in this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note: updated version of arXiv:gr-qc/050809

Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black spacetime. A prime application of characteristic evolution is to compute waveforms via Cauchy-characteristic matching, which is also reviewed.Comment: Published version http://www.livingreviews.org/lrr-2005-1

ON THE DEFINITION OF SURFACEPOTENTIALS FOR FINITE-DIFFERENCE

under Contract NAS 1-97046 Available from the following

Acknowledgments

constitute the foundation of our joint research in numerical methods for external problems and who influenced much of the content of this paper. S. Abarbanel of the School of Mathematical Sciences, Tel-Aviv University, is acknowledged for his valuable help in formulating the problem and for numerous useful discussions. I would also like to thank V. Matsaev and E. Shustin of the School of Mathematical Sciences, Tel-Aviv University, and D. Gottlieb of the Division of Applied Mathematics, Brown University, for their helpful comments. The suggestions of H. Atkins, T. Roberts, and C. Swanson of the NASA Langley Research Center have contributed much to the improvement of this paper. Available electronically at the followin

Active Shielding and Control of Environmental Noise

In the framework of the research project supported by NASA under grant # NAG-1-01064, we have studied the mathematical aspects of the problem of active control of sound, i.e., time-harmonic acoustic disturbances. The foundations of the methodology are described in our paper [1]. Unlike. many other existing techniques, the approach of [1] provides for the exact volumetric cancellation of the unwanted noise on a given predetermined region airspace, while leaving unaltered those components of the total acoustic field that are deemed as friendly. The key finding of the work is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, maybe, in a narrow area of space near the perimeter of the protected region. The controls are built based solely on the measurements performed on the perimeter of the domain to be shielded; moreover, the controls themselves (i.e., additional sources) are concentrated also only on or near this perimeter. Perhaps as important, the measured quantities can refer to the total acoustic field rather than to its unwanted component only, and the methodology can automatically distinguish between the two. In [1], we have constructed the general solution for controls. The apparatus used for deriving this general solution is closely connected to the concepts of generalized potentials and boundary projections of Calderon's type. For a given total wave field, the application of a Calderon's projection allows one to definitively tell between its incoming and outgoing components with respect to a particular domain of interest, which may have arbitrary shape. Then, the controls are designed so that they suppress the incoming component for the domain to be shielded or alternatively, the outgoing component for the domain, which is complementary to the one to be shielded

Artificial Boundary Conditions for Computation of Oscillating External Flows

In this paper, we propose a new technique for the numerical treatment of external flow problems with oscillatory behavior of the solution in time. Specifically, we consider the case of unbounded compressible viscous plane flow past a finite body (airfoil). Oscillations of the flow in time may be caused by the time-periodic injection of fluid into the boundary layer, which in accordance with experimental data, may essentially increase the performance of the airfoil. To conduct the actual computations, we have to somehow restrict the original unbounded domain, that is, to introduce an artificial (external) boundary and to further consider only a finite computational domain. Consequently, we will need to formulate some artificial boundary conditions (ABC's) at the introduced external boundary. The ABC's we are aiming to obtain must meet a fundamental requirement. One should be able to uniquely complement the solution calculated inside the finite computational domain to its infinite exteri..

Inverse source problem and active shielding for composite domains

AbstractThe problem of active shielding (AS) for a multiply connected domain consists of constructing additional sources of the field (e.g., acoustic) so that all individual subdomains can either communicate freely with one another or otherwise be shielded from their peers. This problem can be interpreted as a special inverse source problem for the differential equation (or system) that governs the field. In the paper, we obtain general solution for a discretized composite AS problem and show that it reduces to solving a collection of auxiliary problems for simply connected domains

Active Shielding and Control of Environmental Noise

We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed as friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources needs to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, maybe, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are concentrated also only near this perimeter. Perhaps as important, the measured quantities can refer to the total acoustic field rather than to its unwanted component only, and the methodology can automatically distinguish between the two. In the paper, we construct a general solution to the aforementioned noise control problem. The apparatus used for deriving the general solution is closely connected to the concepts of generalized potentials and boundary projections of Calderon's type. For a given total wave field, the application of Calderon's projections allows us to definitively split between its incoming and outgoing components with respect to a particular domain of interest, which may have arbitrary shape. Then, the controls are designed so that they suppress the inc..

LONG-TIME NUMERICAL INTEGRATION OF THE THREE-DIMENSIONAL WAVE EQUATION IN THE VICINITY OFAMOVING SOURCE

We propose a family of algorithms for solving numerically a Cauchy problem for the three-dimensional wave equation. The sources that drive the equation (i.e., the right-hand side) are compactly supported in space for any given time; they, however, may actually move inspace with a subsonic speed. The solution is calculated inside a finite domain (e.g., sphere) that also moves with a subsonic speed and always contains the support of the right-hand side. The algorithms employ a standard consistent and stable explicit finite-difference scheme for the wave equation. They allow one to calculate the solution for arbitrarily long time intervals without error accumulation and with the fixed non-growing amount of the CPU time and memory required for advancing one time step. The algorithms are inherently three-dimensional; they rely on the presence of lacunae in the solutions of the wave equation in oddly dimensional spaces. The methodology presented in the paper is, in fact, a building block for constructing the nonlocal highly accurate unsteady artificial boundary conditions to be used for the numerical simulation of waves propagating with finite speed over unbounded domains