User: eschnett Date: 2011/02/14 10:36 AM
Modified: / ET.tex
Log: Describe simulation domains etc.
File Changes:
Directory: / ============
File [modified]: ET.tex Delta lines: +57 -9 =================================================================== --- ET.tex 2011-02-14 15:35:14 UTC (rev 26) +++ ET.tex 2011-02-14 16:36:41 UTC (rev 27) @@ -331,8 +331,11 @@ CarpetCode:web} provides multi-block methods and adaptive mesh refinement (AMR@). Multi-block methods cover the domain with a set of distorted Cartesian blocks that exchange information e.g.\ via -interpolation or penalty methods. The AMR capabilities employ the -standard Berger-Oliger algorithm \todo{cite} with subcycling in time. +interpolation or penalty methods.\footnote{Although multi-block + methods are supported by Carpet, the Einstein Toolkit does not yet + contain any multi-block coordinate systems.} The AMR capabilities +employ the standard Berger-Oliger algorithm \todo{cite} with +subcycling in time.
Carpet is the main driver used today for Cactus-based astrophysical simulations. Carpet offers hybrid MPI/OpenMP parallelisation and is @@ -340,10 +343,10 @@ in 2010, about 7,000 core years of computing time (45 million core hours) were used via Carpet by more than a dozen research groups world-wide. To date, more than 90 peer-reviewed publication and more -than 15 student theses are based on Carpet \todo{update these numbers} +than 15 student theses are based on Carpet \todo{ES: update these numbers} \cite{CarpetCode:web}. Carpet's continued development is overseen by -E. Schnetter \todo{do we want names here?} and is funded via several -NSF awards \todo{list them somewhere}. +E. Schnetter \todo{ES: do we want names here?} and is funded via several +NSF awards \todo{ES: list them somewhere}.
\subsection{Simulation Factory}
@@ -598,15 +601,60 @@ Equation~(\ref{eq:puncturetracking}) is implemented with a simple first order Euler scheme, which seems to be accurate enough for controlling the mesh refinement grids. +\todo{ES: I think this paragraph provides too much detail for this paper.} +\todo{ES: Can also track objects via their maximum density or centre + of mass; I believe GRHydro does something like this}
-\subsection{Infrastructure and Numerical Methods} -\todo{1/2 page Erik}
-\paragraph{Domains and Coordinates}
+%\subsection{Infrastructure and Numerical Methods} +\subsection{Simulation Domain, Symmetries, Boundaries} + +\paragraph{Domains and Coordinates.} +The simulation domain is specified via the parameter file \todo{ES: + ensure this has been defined above}. Cactus distinguishes between +the \emph{physical} domain, which lives in the continuum, and +\emph{discrete} domain, which consists of a discrete set of grid +points. The physical domain is defined via its coordinate extent and +is independent of the numerical resolution; in particular, the +boundary of the physical domain has a width of zero (and is thus a set +of measure zero). The discrete domain is defined indirectly via a +discretisation procecure. This procedure specifies the number of +boundary points, their location in respect to the physical boundary, +and either the grid spacing or the number of grid points spanning the +domain. This defines the discrete domain, i.e.\ the number and +location of the discrete grid points. In particular, the discrete +domain can have grid points outside of the physical domain, and can +have a non-zero boundary width. This mechanism ensures that changes in +the numerical resolution do not affect the extent of the physical +domain, i.e.\ that the discrete domains converge to the physical +domain in the limint of infinite resolution. + +\paragraph{Symmetries, Boundary Conditions.} +The Einstein Toolkit offers a set of reflecting or rotating symmetry +conditions that can be used to reduce the size of the simulation +domain. These symmetries include periodicity, reflections about the +coordinate planes, and $90^{\circ}$ and $180^{\circ}$ rotational +symmetries about the $z$ axis. In applying these symmetries, the +transformation behaviour of tensorial quantities (including tensor +densities and non-tensors such as Christoffel symbols) is correctly +taken into account. + +The Einstein Toolkit also offers a set of boundary conditions, +including Dirichlet, von Neumann, and Robin (fall-off) boundary +conditions, as well as extrapolation and radiative boundary +conditions. + \paragraph{Adaptive Mesh Refinement} +Carpet supports feature-based mesh refinement, which is based on +extracting e.g.\ the locations of black holes or neutron stars, and +then constructing a mesh hierarchy (stacks of refined regions) based +on the locations, sizes, and speeds of these objects. This allows +tracking objects as they move through the domain. One can also add or +remove stacks if e.g.\ the number of objects changes. If initial +conditions are constructed outside of Carpet (which is often the +case), then the initial mesh hierarchy has to be defined manually.
-\paragraph{Boundary Conditions}
\section{Examples}