In [398], the authors present a numerical spectrum stability analysis for block-diagonal–based SBP operators in the presence of curvilinear coordinates. However, the case of non-diagonal SBP norms and the full Einstein equations in multi-domain scenarios for orders higher than four in the interior needs further development and analysis.

Efficient algorithms for computing the weights for generic FDs operators (though not necessarily satisfying SBP or with proven stability) are given in [166].

Discretizing second-order time-dependent problems without reducing them to first order leads to a similar concept of SBP for operators approximating second derivatives. There is steady progress in an effort to combine SBP with penalty interface and outer boundary conditions for high-order multi-domain simulations of second-order-in-space systems. At present though these tools have not yet reached the state of those for first-order systems, and they have not been used within numerical relativity except for the test case of a ‘shifted advection equation’ [302]. The difficulties appear in the variable coefficient case. We discuss some of these difficulties and the state of the art in Section 10. In short, unlike the first-order case, SBP by itself does not imply an energy estimate in the variable coefficient case, even if using diagonal norms, unless the operators are built taking into account the PDE as well. In [300] the authors explicitly constructed minimal-width diagonal norms SBP difference operators approximating up to eighth order in the interior, and in [118] non-minimal width operators up to sixth order using full norms are given.

[440] presents a stability analysis around flat spacetime for a family of generalized BSSN-type formulations, along with numerical experiments, which include binary black-hole inspirals.

SBP operators have also been constructed to deal with coordinate singularities in specific systems of equations [105, 375, 225]. Since a sharp semi-discrete energy estimate is explicitly derived in these references, (strict) stability is guaranteed. In particular, in [225] schemes for which the truncation error converges pointwise everywhere – including the origin – are derived for wave equations on arbitrary space dimensions decomposed in spherical harmonics. Interestingly enough, popular schemes [158] to deal with the singularity at the origin, which had not been explicitly designed to satisfy SBP, were found a posteriori to do so at the origin and closed at the outer boundary, see [225] for more details. In these cases the SBP operators are tailored to deal with specific equations and coordinate singularities; therefore, they are problem dependent. For this reason their explicit construction has so far been restricted to second and fourth-order operators (with diagonal scalar products), though the procedure conceptually extends to arbitrary orders. For higher-order operators, optimization of at least the spectral radius might become necessary to address.

In [166] the authors use SBP operators to design high-order quadratures. The reference also includes a detailed description of many properties of SBP operators.

Superconvergence of some estimates in the case of diagonal SBP operators is discussed in [239].

Living Rev. Relativity 15, (2012), 9
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