# Cauchy-perturbative matching reexamined: Tests in spherical symmetry

## Abstract

During the last few years progress has been made on several fronts making it possible to revisit Cauchy-perturbative matching (CPM) in numerical relativity in a more robust and accurate way. This paper is the first in a series where we plan to analyze CPM in the light of these new results. One of the new developments is an understanding of how to impose constraint-preserving boundary conditions (CPBC); though most of the related research has been driven by outer boundaries, one can use them for matching interface boundaries as well. Another front is related to numerically stable evolutions using multiple patches, which in the context of CPM allows the matching to be performed on a spherical surface, thus avoiding interpolations between Cartesian and spherical grids. One way of achieving stability for such schemes of arbitrary high order is through the use of penalty techniques and discrete derivatives satisfying summation by parts (SBP). Recently, new, very efficient and high-order accurate derivatives satisfying SBP and associated dissipation operators have been constructed. Here we start by testing all these techniques applied to CPM in a setting that is simple enough to study all the ingredients in great detail: Einstein's equations in spherical symmetry, describingmore »

- Authors:

- Max-Planck-Institut fuer Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching (Germany)
- Department of Physics and Astronomy, 202 Nicholson Hall, Louisana State University, Baton Rouge, Louisiana 70803 (United States)
- (United States)
- (Guatemala)

- Publication Date:

- OSTI Identifier:
- 20782910

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 73; Journal Issue: 8; Other Information: DOI: 10.1103/PhysRevD.73.084011; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BLACK HOLES; BOUNDARY CONDITIONS; COSMIC GAMMA BURSTS; COSMOLOGY; EINSTEIN FIELD EQUATIONS; INTERPOLATION; SCALAR FIELDS; SCHWARZSCHILD METRIC; SPATIAL RESOLUTION; SPHERICAL CONFIGURATION; SYMMETRY

### Citation Formats

```
Zink, Burkhard, Pazos, Enrique, Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803, Departamento de Matematica, Universidad de San Carlos de Guatemala, Edificio T4, Facultad de Ingenieria, Ciudad Universitaria Z. 12, Diener, Peter, Tiglio, Manuel, and Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803.
```*Cauchy-perturbative matching reexamined: Tests in spherical symmetry*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVD.73.084011.

```
Zink, Burkhard, Pazos, Enrique, Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803, Departamento de Matematica, Universidad de San Carlos de Guatemala, Edificio T4, Facultad de Ingenieria, Ciudad Universitaria Z. 12, Diener, Peter, Tiglio, Manuel, & Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803.
```*Cauchy-perturbative matching reexamined: Tests in spherical symmetry*. United States. doi:10.1103/PHYSREVD.73.084011.

```
Zink, Burkhard, Pazos, Enrique, Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803, Departamento de Matematica, Universidad de San Carlos de Guatemala, Edificio T4, Facultad de Ingenieria, Ciudad Universitaria Z. 12, Diener, Peter, Tiglio, Manuel, and Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803. Sat .
"Cauchy-perturbative matching reexamined: Tests in spherical symmetry". United States.
doi:10.1103/PHYSREVD.73.084011.
```

```
@article{osti_20782910,
```

title = {Cauchy-perturbative matching reexamined: Tests in spherical symmetry},

author = {Zink, Burkhard and Pazos, Enrique and Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803 and Departamento de Matematica, Universidad de San Carlos de Guatemala, Edificio T4, Facultad de Ingenieria, Ciudad Universitaria Z. 12 and Diener, Peter and Tiglio, Manuel and Center for Computation and Technology, 302 Johnston Hall, Louisana State University, Baton Rouge, Louisiana 70803},

abstractNote = {During the last few years progress has been made on several fronts making it possible to revisit Cauchy-perturbative matching (CPM) in numerical relativity in a more robust and accurate way. This paper is the first in a series where we plan to analyze CPM in the light of these new results. One of the new developments is an understanding of how to impose constraint-preserving boundary conditions (CPBC); though most of the related research has been driven by outer boundaries, one can use them for matching interface boundaries as well. Another front is related to numerically stable evolutions using multiple patches, which in the context of CPM allows the matching to be performed on a spherical surface, thus avoiding interpolations between Cartesian and spherical grids. One way of achieving stability for such schemes of arbitrary high order is through the use of penalty techniques and discrete derivatives satisfying summation by parts (SBP). Recently, new, very efficient and high-order accurate derivatives satisfying SBP and associated dissipation operators have been constructed. Here we start by testing all these techniques applied to CPM in a setting that is simple enough to study all the ingredients in great detail: Einstein's equations in spherical symmetry, describing a black hole coupled to a massless scalar field. We show that with the techniques described above, the errors introduced by Cauchy-perturbative matching are very small, and that very long-term and accurate CPM evolutions can be achieved. Our tests include the accretion and ring-down phase of a Schwarzschild black hole with CPM, where we find that the discrete evolution introduces, with a low spatial resolution of {delta}r=M/10, an error of 0.3% after an evolution time of 1,000,000M. For a black hole of solar mass, this corresponds to approximately 5s, and is therefore at the lower end of timescales discussed e.g. in the collapsar model of gamma-ray burst engines.},

doi = {10.1103/PHYSREVD.73.084011},

journal = {Physical Review. D, Particles Fields},

number = 8,

volume = 73,

place = {United States},

year = {Sat Apr 15 00:00:00 EDT 2006},

month = {Sat Apr 15 00:00:00 EDT 2006}

}