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Bimetric 3+1 toolkit for spherical symmetry
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Bimetric 3+1 toolkit for spherical symmetry
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+-------------------------------------------------------+ +-----------------+
| Bimetric Evolution | | Bimetric Model |
+-------------------------------------------------------+ +-----------------+
| bim-solver.cpp eomEvolution.h eomGauge.h | | bimetricModel.h |
| maximalSlice.h eomMatter.h eomLapseRatios.h | +-----------------+
| eomSources.h eomMiscEquations.h |
| eomConstraints.h |
+-------------------------------------------------------+
+-----------------+ +-------------------+ +----------------+ +---------------+
| Grid Driver | | Initial Data | | MoL Integrator | | Output Data |
+-----------------+ +-------------------+ +----------------+ +---------------+
| gridDriver.h | | gridInitialData.h | | integrator.h | | gridOutput.h |
| gridFunctions.h | +-------------------+ +----------------+ +---------------+
+-----------------+
+---------------------+ +--------------+ +-----------------+
| Numerical Methods | | Data Types | | Utilities |
+---------------------+ +--------------+ +-----------------+
| finiteDifferences.h | | matrix.h | | trackUsedTime.h |
| methodOfLines.h | | dataTypes.h | | mpiWorld.h |
| cubicSpline.h | +--------------+ | mpiDummyWorld.h |
| bandSol.h | | paramsHolder.h |
+---------------------+ | slog.h |
+-----------------+ +---------------------------------+ +----------------------------+ | Mathematica Notebook |------------>>| Initial Data & Parameters | | bimetric-ss-in-3+1 (cpp solver) | Solve the | input.dat (initial data) | +---------------------------------+ constraint | config.ini (parameters) | | Export the equations +----------------------------+ | C++ code | V V +---------------------------------+ eom-std.h +----------------------------+ | eomEvolution.h eomMatter.h |------------>>| bim-solver.cpp | | eomSources.h eomConstraints.h | +----------------------------+ | eomGauge.h eomLapseRatios.h | | | eomMiscEquations.h | V +---------------------------------+ +----------------------------+ | output.dat | See also: maximalSlice.h +----------------------------+ Numerics: methodOfLines.h, finiteDifferences.h, dataTypes.h, matrix.h, bandSol.h, cubicSpline.h
r = 0 can be found in [11].At the core of finite difference approximation is a discretization of the spacetime, or a numerical grid (shown below). Then, a function f(t,r) is represented by values at a discrete set of points (m,n).
p in M <--> (t,r) <--> (m,n)
(point in a manifold) (in a chart) (discretized point)
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V V
f(t,r) f(m,n) The points (m,n) are called the grid points or nodes. The grid points are chosen to reside at the center of grid cells (the other approach would be to associate the grid points with the vertices of the grid cells). The data structure GridPoint is a wrapper for the set of fields {f,...} that arekept in the memory for the point (m,n).The grid consists of the interior points, and the virtual points at the boundary. The virtual points are used to impose the boundary conditions. A cell-centered uniform grid with nLen grid points in each grid row is shown below (see also [3], p.192).
left virtual grid cell grid point right virtual
point(s) .---^---. | point(s)
. | n-1 ' n ' n+1 | |
: +----|----+-------+-------+---------+-------+---|---+-------+----|----+
m+1 | O | | | (m+1,n) | | X | | O |
+---------+-------+-------.=========.-------+-------+-------+---------+
m | | |(m,n-1)| (m,n) |(m,n+1)| | | |
+---------+-------+-------'========='-------+-------+-------+---------+
m-1 | | | | | | | | |
. +---------'-------+-------+---------+-------+-------+-------'---------+
: 0 | 1 2 . . . nLen | nLen+1
^ ^
r_min r_max The cell-centered uniform grid implies:
r = r_min + ( n - 1/2 ) delta_r delta_r = ( r_max - r_min ) / nLen
nGhost = 1, n ranges between 1 and nLen; the boundary virtual points are grid[m][0] and grid[m][nLen+1]m ranges between 1 and mLen, and wraps around; the slices grid[mLen+1]... are used for intermediate integration steps.In parallel simulations, the grid is split across processes using MPI, allowing storage of large grid functions and parallel work on each part.
1.8.0