Multipatch methods in general relativistic astrophysics: Hydrodynamical flows on fixed backgrounds

Burkhard Zink, Erik Schnetter, and Manuel Tiglio

Phys. Rev. D 77, 103015 – Published 30 May 2008

Abstract

Many systems of interest in general relativistic astrophysics, including neutron stars, accreting compact objects in x-ray binaries and active galactic nuclei, core collapse, and collapsars, are assumed to be approximately spherically symmetric or axisymmetric. In Newtonian or fixed-background relativistic approximations it is common practice to use spherical polar coordinates for computational grids; however, these coordinates have singularities and are difficult to use in fully relativistic models. We present, in this series of papers, a numerical technique which is able to use effectively spherical grids by employing multiple patches. We provide detailed instructions on how to implement such a scheme, and present a number of code tests for the fixed-background case, including an accretion torus around a black hole.

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Received 4 December 2007

DOI:https://doi.org/10.1103/PhysRevD.77.103015

Authors & Affiliations

Burkhard Zink^1,2, Erik Schnetter^1,2, and Manuel Tiglio^1,2,3,4

¹Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
²Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
³Department of Physics, University of Maryland, College Park Maryland 20742, USA
⁴Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park Maryland 20742, USA

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Vol. 77, Iss. 10 — 15 May 2008

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Figure 1
The general relativistic multipatch model used in this paper. The spacetime subdomain $M_{0} \subset M$ of interest is assumed to be covered by a single global coordinate system $ϕ_{G}$ . The target domain $ϕ_{G} (M_{0}) \subset R^{4}$ is then further covered by several patches associated with the charts $ϕ_{G L}^{n}$ , which give rise to the local coordinate systems $ϕ_{L}^{n}$ .Reuse & Permissions
Figure 2
The notions of interior, interface and boundary region associated with a patch. The global coordinate domain $ϕ_{G} (M_{0})$ is decomposed into the patch interiors as indicated in the figure, with interfaces adjoining two interiors. The charts $ϕ_{G L}^{n}$ therefore define an interior and a boundary region in $ϕ_{L}^{n} (M_{0})$ .Reuse & Permissions
Figure 3
Illustration of the patch interface boundary treatment. The interface, here represented by the vertical thick line in the center, separates the left patch (L) from the right one (R). The coordinate lines of both patches are represented in the local coordinates of patch R, which makes the lines of L appear curved in general. To evolve the system on patch R, boundary ghost zones, here represented by nonfilled circles, are introduced as a linear extrapolation of patch R’s coordinate lines, and, before the time update is performed, receive data from a data interpolation operation on patch L. Afterward, the data are transformed to the tensor basis defined by patch R’s coordinate system via the transformation map.Reuse & Permissions
Figure 4
An illustration of the distorted two patches system. The left patch is constructed using an identity map between local and global coordinates, whereas the right patch follows from Eq. (10). The common interface is located at $x = 1$ , which also coincides with the local coordinates $a_{1}^{1} = 1$ and $a_{2}^{1} = 1$ . Note the ghost zones extending beyond the interface (cf. Fig. 3).Reuse & Permissions
Figure 5
An illustration of the cubed-sphere six patches system. The diagram shows a cut of grid lines with the equatorial plane $z = 0$ . In the notation of Eq. (12), patch 0 is to the right, counterclockwise continuing with patch 1, patch 2 and patch 3. The inner spherical boundary can be used as an excision surface for modeling black holes.Reuse & Permissions
Figure 6
An illustration of the cubed-sphere seven patches system. The diagram shows a cut of grid lines with the equatorial plane $z = 0$ . In the notation of Eq. (13), patch 0 is to the right, counterclockwise continuing with patch 1, patch 2 and patch 3. The central, Cartesian patch is patch 6. The parameters for the patch system were chosen to be $r_{0} = 0.5$ and $r_{1} = 2$ , so the outer boundary is a sphere of radius 2 in this diagram. The outermost two grid lines, however, are ghost zones required for the MC reconstruction algorithm (see Sec. 2b2).Reuse & Permissions
Figure 7
An illustration of the cubed-sphere 13 patches system. This system consist of an outer six patches system, where all surfaces of constant coordinate $a^{3}$ are spheres, and an inner seven patches system matched to the inner boundary of the outer system to cover the origin in a regular manner.Reuse & Permissions
Figure 8
Sod test on the unipatch system. For each of the primitive variables $ρ$ , $u$ , and $u^{1}$ , the left plots show the comparison of the numerical result with the exact solution for different times (spaced in units of $Δ t = 0.1$ ), and right plots show the error function ( $l_{1}$ norm of the difference between the exact solution and the discrete result) over time. The error depends approximately linearly on the grid distance as expected from the asymptotic first-order nature of the algorithm.Reuse & Permissions
Figure 9
Simple shock test on the unipatch system. Same as in Fig. 8. The error depends approximately linearly on the grid distance as expected from the asymptotic first-order nature of the algorithm.Reuse & Permissions
Figure 10
Blast wave test on the unipatch system. Same as in Fig. 8. The error depends approximately linearly on the grid distance as expected from the asymptotic first-order nature of the algorithm.Reuse & Permissions
Figure 11
Sod test on the distorted two patches system. This system, defined in Sec. 4b, consists of a left patch (patch 0) which is undistorted, and a right patch (patch 1) which has a nontrivial Jacobian associated with the transformation from local to global coordinates. The left panel shows the primitive variables $ρ$ , $u$ and $u^{1}$ in dependence on the global coordinate $x$ , for different coordinate times ( $Δ t = 0.2$ ). The interface between the patches is located at $x = 1$ . In the lower left diagram, note how the velocity component $u^{1}$ appears discontinuous across the interface since it is represented in different tensor bases on each patch. The right panel shows global convergence with respect to the $l_{1}$ norm. (The left panel does not contain the graphs for $n = 200$ to enhance the visual clarity of the plots.)Reuse & Permissions
Figure 12
Sod test on the six patches system: evolution of the density for the case where each of the patches has resolution of $80 \times 80 \times 80$ . The white lines indicate the boundaries of the six patches which are used to cover the computational domain. The surface is the cut of the plane $z = 0$ with the grid boundaries, and the density function is shown at $z = 0$ . The left panels show a perspective view of the evolution for coordinate times 0, 1 and 3, whereas the right panels show an orthogonal projection from the negative $y$ axis at the same times.Reuse & Permissions
Figure 13
Sod test on the six patches system. These series of plots demonstrate the convergence of the primitive variables to the solution produced by the Riemann code. Each of the six patches has the same resolution defined by the parameter $n$ , which determines the number of cells by $n \times n \times n$ .Reuse & Permissions
Figure 14
Sod test on the six patches system. The plots show a close-up view of the transmission of the shock front across one of the interfaces, using a pseudo-Schlieren plot: The function displayed is the norm of the density gradient, in logarithmic scale. The interface is between the patch boundaries indicated by white lines. Note that the HLL scheme employed here causes the contact discontinuity (left of the shock wave) to be dissipated significantly. The sharp appearance near the inner boundary is partly due to the higher resolution there, but mostly an effect of imposing the exact solution on the boundary ghost zones.Reuse & Permissions
Figure 15
Sod test on the 13 patches system. The left panel shows the evolution of the density for the case where each of the patches has a resolution of $80 \times 80 \times 80$ . The white lines indicate the boundaries of the 13 patches which are used to cover the computational domain. The surface is the cut of the plane $z = 0$ with the grid boundaries, and the density function is shown at $z = 0$ . The right panel shows convergence in the primitive variables.Reuse & Permissions
Figure 16
Rotating neutron star on the cubed-sphere system of 13 patches. The initial data are produced with the rns code [58] then mapped to each patch and transformed to the local coordinate system. These plots show the logarithm of the density function in the equatorial plane $z = 0$ (left) and the plane $x = 0$ (right). For visual clarity, the density is cut below $10^{- 8}$ ; the actual initial density in the atmosphere is $10^{- 10}$ .Reuse & Permissions
Figure 17
Evolution of a rotating neutron star on the cubed-sphere system of 13 patches with different resolutions. The star is evolved up to a coordinate time of 350, which translates into about ten dynamical times ( $t_{D} = R_{e} \sqrt{R_{e} / M} \approx 35.428$ ). The top left panel shows the central density, the top right, middle left, and middle right panels show convergence of the conserved variables, and the last two plots show the errors in rest mass (bottom left panel) and angular momentum (bottom right panel). For a definition of these quantities see 57.Reuse & Permissions
Figure 18
Equilibrium accretion torus on a Schwarzschild background, using the cubed-sphere six patches system. The plots show, for the evolution with $60 \times 60 \times 160$ cells per patch, the decadic logarithm of the density for coordinate times $t = 0$ , 1000 and 2000 (the rotational time of the torus at the locus of highest density is about 200, i.e., we are evolving for about 10 rotational times). The left panels show the cross-section surface $x = 0$ in terms of the global, pseudo-Cartesian coordinates, and the right panels provide a view of the equatorial plane $z = 0$ . The system is stationary, so all deviations from the initial data are artifacts due to the finite resolution, finite precision of floating point numbers, or the use of an artificial atmosphere (though small numerical errors might initiate physical instabilities in the disk). A small amount of material gets unbound and is either accreted into the black hole or forms a corona. In disks where a real physical effect like turbulence causes redistribution of angular momentum [35], similar domains are present, though with much higher densities (note that the color map covers a range of 10 orders of magnitude in density).Reuse & Permissions
Figure 19
Equilibrium accretion torus on a Schwarzschild background, using the cubed-sphere six patches system. The disk is evolved up to a coordinate time of 2000, which translates into about ten rotational times at the location of maximal density. The top left panel shows the maximal density, the top right, middle left, and middle right panels show convergence of the conserved variables, and the last two plots show the errors in rest mass (bottom left panel) and angular momentum (bottom right panel). For a definition of these quantities see 57. (The reference solution mentioned in the text is the value corresponding to the initial data on the grid of $120 \times 120 \times 320$ cells per patch.) Note that the case $n = 15 \times 15 \times 40$ is clearly underresolved.Reuse & Permissions

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