Abstract
We present a high-order scheme for solving the full non-linear Einstein equations on characteristic null hypersurfaces using the framework established by Bondi and Sachs. This formalism allows asymptotically flat spaces to be represented on a finite, compactified grid, and is thus ideal for far-field studies of gravitational radiation. We have designed an algorithm based on 4th-order radial integration and finite differencing, and a spectral representation of angular components. Consequently the scheme offers more accuracy at a given computational cost compared to previous methods which are second-order accurate. Based on a newly implemented code, we show that the new numerical scheme remains stable and is convergent at the expected order of accuracy.
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Notes
An explicit form for \(q^A\) will not be needed here, since we will represent angular dependence in terms of spin-weighted spherical harmonic basis functions.
We have found that lower values of \(\epsilon \) may trigger an instability at \({\mathcal{J }^{+}}\). This can be cured by either using larger \(\epsilon \) everywhere, or by just increasing \(\epsilon \) at \({\mathcal{J }^{+}}\).
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Acknowledgments
The authors would like to thank Peter Diener for providing dissipation operator stencil coefficients, and Harald Pfeiffer for comments on the manuscript. We thank the Erwin Schroedinger Institute, Austria, Universitas de les Illes Balears, Spain, and Rhodes University, South Africa, for hospitality. This work is supported by the National Science Foundation under grant numbers AST-0855535 and OCI-0905046. CR acknowledges support by NASA through Einstein Postdoctoral Fellowship grant number PF2-130099 awarded by the Chandra X-ray center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. NTB has been supported by the National Research Foundation, South Africa. Computations were performed on the LONI network (www.loni.org) under allocation loni_numrel06 and loni_numrel07, and the Caltech compute cluster “Zwicky” (NSF MRI award No. PHY-0960291).
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Christian Reisswig: Einstein Fellow.
Appendices
Appendix A: The nonlinear terms in the Einstein equations
The nonlinear terms \(N_\beta , N_Q, N_U, N_W\) and \(N_J\) in (16a) through (17) were first presented in [12]. We repeat them here, but with a mis-print in Eq. (A3) of [12] corrected.
where
Appendix B: Regularized equations at \({\mathcal{J }^{+}}\)
At \({\mathcal{J }^{+}}\), the equations simplify when inserting Eqs. (21) and (22) and taking the limit \(r\rightarrow \infty \). In particular, terms containing powers of \(1/r,...,1/r^n\) vanish. Below, we give the equations evaluated at \({\mathcal{J }^{+}}\).
where
Appendix C: Radial derivative operators
We use fourth-order sidewinded derivatives
where \(\Delta x\) is the grid spacing in the compactified radial coordinate direction.
Close to \({\mathcal{J }^{+}}\) when \(i > N_x-5\), we switch to 4th-order centred stencils
and when \(i > N_x-3\), we switch to side-winded stencils pointing towards the inner boundary
Appendix D: Dissipation operator stencil
We use a 5th-order Kreiss–Oliger dissipation operator of the form
where \(\epsilon _\mathrm{diss}\) controls the strength of the applied dissipation operator \(D\).
Close to the outer boundary where we do not have enough points to apply the standard centred-stencil Kreiss–Oliger dissipation operator Eq. (79), we use side-winded dissipation operator stencils derived for SBP operator (though we do not make explicit use of the SBP property) \(D_{4-2}\) of [37]. This particular dissipation operator is defined via coefficients \(a_{ij}\) and \(q_{i}\). In the interior of the domain the operator reads
where the \(q_i\) are the coefficients given in Eq. (79). Near the outer boundary (on points \(i=N_x-5,...,N_x-1\)), the operator can be written as
where the coefficients \(a_{ij}\) are taken from [37] reported below. We note that \(\epsilon _\mathrm{diss}\ge 0\) selects the amount of dissipation and is usually of order unity.
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Reisswig, C., Bishop, N. & Pollney, D. General relativistic null-cone evolutions with a high-order scheme. Gen Relativ Gravit 45, 1069–1094 (2013). https://doi.org/10.1007/s10714-013-1513-1
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DOI: https://doi.org/10.1007/s10714-013-1513-1