8 Spatial Approximations: Finite Differences

As mentioned in Section 7.6, a general stability theory (referred to as GKS) for IBVPs was developed by Gustafsson, Kreiss and Sundström [229Jump To The Next Citation Point], and a simpler approach, when applicable, is the energy method. The latter is particularly and considerably simpler than a GKS analysis for complicated systems such as Einstein’s field equations, high-order schemes, spectral methods, and/or complex geometries. The Einstein vacuum equations can be written in linearly-degenerate form and are therefore expected to be free of physical shocks (see the discussion at the beginning of Section 3.3) and ideally suited for methods, which exploit the smoothness of the solution to achieve fast convergence, such as high-order finite-difference and spectral methods. In addition, an increasing number of approaches in numerical relativity use some kind of multi-domain or grid structure approach (see Section 11). There are multi-domain schemes for which numerical stability can relatively easily be established for a large class of linear symmetric hyperbolic problems and maximal dissipative boundary conditions through the energy method. In particular, such schemes could be applied to the symmetric hyperbolic formulations of Einstein’s equations discussed in Sections 4 and 6.

In this section we discuss spatial finite difference (FD) approximations of arbitrary high order for which the energy method can be applied, and in Section 10 boundary closures for them. We start by reviewing polynomial interpolation, followed by the systematic construction of FD approximations of arbitrary high order and stencils through interpolation. Next, we introduce the concept of operators satisfying SBP, present a semi-discrete stability analysis, and the construction of high-order operators optimized in terms of minimizing their boundary truncation error and their associated timestep (CFL) limits (more specifically, their spectral radius). Finally, we discuss numerical dissipation, with emphasis on the region near boundaries or grid interfaces.

 8.1 Polynomial interpolation
 8.2 Finite differences through interpolation
 8.3 Summation by parts
 8.4 Stability
 8.5 Numerical dissipation
 8.6 Going further

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