Cauchy-Characteristic Matching: A New Approach to Radiation Boundary Conditions

Nigel T. Bishop, Roberto Gomez, Paulo R. Holvorcem, Richard A. Matzner, Philippos Papadopoulos, Jeffrey Winicour
1996 Physical Review Letters  
We investigate a new methodology for computing wave generation, using Cauchy evolution in a bounded interior region and characteristic evolution in the exterior. Matching the two schemes eliminates usual difficulties such as backreflection from the outer computational boundary. Mapping radiative infinity into a finite grid domain allows a global solution. The matching interface can be close to the sources, the wave fronts can have arbitrary geometry, and strong nonlinearity can be present. The
more » ... an be present. The matching algorithm dramatically outperforms traditional radiation boundary conditions. [S0031-9007 (96) 00365-1] PACS numbers: 04.25.Dm, 04.30.Db We present a new computational approach to radiation boundary conditions, based on discretizing an exact treatment of the radiation from source to infinity. The numerical solution therefore converges to the exact analytic solution of the radiating system, as we show. This is in contrast to usual radiation boundary conditions [1-4], which involve an approximation that does not converge to the exact solution. Our algorithm converges to second order so that any desired accuracy can be achieved by refining the grid; no amount of grid refinement can lower the error of the traditional approach beyond a certain level. This is manifest even at moderate resolutions, where our method has an error 2 orders of magnitude below that of traditional schemes. Here we treat scalar waves but the method is being used in the binary black hole grand challenge to calculate gravitational radiation. It has applicability to a wide range of hyperbolic systems, e.g., acoustic wave generation in nonlinear hydrodynamics and light emission in a nonlinear medium. Traditional Cauchy methods impose artificial conditions at the computational boundary [1-4], typified by the well known Sommerfeld outgoing radiation condition which is strictly valid only at an infinite distance from the sources. This introduces an error of analytic origin, which persists even in high resolution simulations. Improvement by moving the boundary to a larger radius is computationally very expensive in three-dimensional simulations. An exact treatment of the boundary is possible if the retarded Green's function is known [5] but in a nonlinear problem this approach can be carried out only by a perturbation approximation. In contrast, a characteristic formulation [6] can be compactified [7], mapping radiative infinity (the asymptotic limit of ougoing characteristics) to a finite coordinate ra-dius. This provides a finite grid boundary, with no loss of accuracy because of the very simple asymptotic behavior of outgoing waves along characteristics. Only in a characteristic approach is there no need of an artificial outer boundary condition, because we can discretize the whole physical domain, with the outer boundary where the true radiation zone wave form can be identified. A global characteristic approach must deal with caustics where the characteristics focus [8] . In hydrodynamics or general relativity, the caustic structure is dynamic and would have to be computed along with the evolution. These problems make it difficult to use the characteristic formulation in the near-field region but it proves to be both accurate and computationally efficient in the treatment of an exterior, caustic-free region [9] . Our procedure is a matched Cauchy-characteristic evolution [10] [11] [12] [13] [14] . A Cauchy formulation evolves a 3space of field values step by step forward in time t; a characteristic formulation [6,9] evolves three-dimensional characteristic "cones" forward in retarded time u; see Fig. 1 . The characteristic algorithm then provides an outer boundary condition for the interior Cauchy evolution, with infinity rigorously included in a compactified grid. The Cauchy algorithm supplies the inner boundary condition for the characteristic evolution. For nonlinear systems, such Cauchy-characteristic matching is much more computationally efficient than alternative methods of obtaining highly accurate wave forms [15] . Our work is the first systematic study of the stability and accuracy of this method. Consider the scalar wave equation for f͑x, y, z, t͒, ≠ tt f = 2 f 1 F͑f͒ 1 S͑x, y, z, t͒ , with nonlinear self-coupling F͑f͒ and external source S. 0031-9007͞96͞76(23)͞4303(4)$10.00
doi:10.1103/physrevlett.76.4303 pmid:10061256 fatcat:xqpsvj5ebfcmroq5eyfhq7qrve