We describe three numerical approaches to the construction of three-dimensional initial data for the collision of two black holes. The first of our approaches involves finite differencing the 3 + 1 Hamiltonian constraint equation on a Cadei coordinate grid. The difference equations are then solved via the multigrid algorithm. The second approach also uses finite-difference techniques, but this time on a regular Cartesian coordinate grid. A Cartesian grid has the advantage of having no coordinate singularities. However, constant coordinate lines are not coincident with the throats of the black holes and, therefore, special treatment of the difference equations at these boundaries is required. The resulting equations are solved using a variant of line-successive overrelaxation. The third and final approach we use is a global, spectral-like method known as the multiquadric approximation scheme. In this case functions are approximated by a finite sum of weighted quadric basis functions which are continuously differentiable. We discuss particular advantages and disadvantages of each method and compare their performances on a set of test problems.

Recently an inherently mass-conserving semi-Lagrangian transport scheme has been successfully coupled to an iterative semi-implicit scheme in a global shallowwater-equation (

This paper continues a series investigating methods of combining Cauchy and characteristic techniques in numerical relativity. We present a technique for passing gravitational wave information between an inner Cauchy region and an outer, compactified, characteristic region via a numerical interface. In this developmental work the method is applied to vacuum cylindrical spacetime with two degrees of gravitational freedom. A comparison of our numerical results with an analytic solution, and a number of numerical experiments involving the propagation of Gaussian wave packets, indicate that the techniques presented here can produce very satisfactory results. PACS number(s): 04.

No abstract

This paper is part of a long term program to Cauchy-characteristic matching (CCM) codes as investigative tools in numerical relativity. The approach has two distinct features: (i) it dispenses with an outer boundary condition and replaces this with matching conditions at an interface between the Cauchy and characteristic regions, and (ii) by employing a compactified coordinate, it proves possible to generate global solutions. In this paper CCM is applied to an exact two-parameter family of cylindrically symmetric vacuum solutions possessing both gravitational degrees of freedom due to Piran, Safier and Katz. This requires a modification of the previously constructed CCM cylindrical code because, even after using Geroch decomposition to factor out the z-direction, the family is not asymptotically flat. The key equations in the characteristic regime turn out to be regular singular in nature.

No abstract

Split schemes for time-stepping physical parameterizations in numerical weather prediction and climate models are investigated within the context of simplified model equations. A symmetrized-splitting technique is applied to various parameterized systems containing fast and slow physics processes. The physics processes are represented by time-dependent forcing terms and linear damping/oscillatory terms. Finitedifference schemes, obtained from the splitting procedures, are examined to determine their stability properties, degree of splitting error, and truncation error. This analysis provides insight into the advantages and disadvantages of different splitting procedures across a range of possible parameterization scenarios. Many schemes obtained via splitting have time-step-dependent splitting errors, which can lead to inaccurate solutions when fast processes are present and the time step is large. Some splitting combinations, however, are more useful than others. The symmetrized-splitting procedure considered in this paper can produce stable first-and second-order accurate schemes, which have either no significant splitting errors or acceptably small errors relative to a steady-state solution. The consequences of this analysis for physics coupling strategies in realistic numerical weather prediction and climate models are noted.

SUMMARYAt the present time there exist a number of different approaches to the problem of coupling parametrized physical processes to the dynamical core in operational numerical weather-prediction (NWP) and climate models. Motivated by the various strategies in use, some idealized representative coupling schemes are constructed and subsequently analysed using a methodology in which the physics and dynamics terms are represented in a simplified way. Particular numerical properties of the idealized schemes which are of interest are the ability to capture correct steady-state solutions and to be second-order accurate in time. In general, the schemes require specific choices for the time-differencing of certain coupled processes if correct steady-state solutions are to be obtained. This has implications for the overall numerical stability of a coupling strategy. An alternative physicsdynamics coupling approach is then described and analysed. A multiple-sweep predictor-corrector coupling scheme is shown to capture the correct steady-state solution and to allow for second-order accuracy, provided that the convective process is coupled explicitly. This approach has a number of advantages over those currently used in operational NWP models.