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lhs φ , rhs φ
lhs β 1, rhs β 1
lhs β 3, rhs β 3
[arbitrary units]
[arbitrary units]
[arbitrary units]
8.4 Code tests and applications 279
∆ rel φ
0 1 2 3 4 5 6
φ
∆ rel β 1
10 -3
10 -4
10
0 1 2 3 4 5 6
-5
10 -1
10 -2
10
0 0.5 1 1.5
-3
0 0.5 1 1.5
θ
∆ rel β 3
10 0
10 -2
10
0 5 10
-4
0 5 10
r [km]
Figure 8.10: Left (solid line) and right (dashed line) hand sides (computed on the finite difference
grid) of the equation for the metric components φ along the azimuthal direction ϕ (upper
panel), β 1 along the meridional direction θ (center panel), and β 3 along the radial direction
(lower panel). Even for strong nonaxisymmetric perturbations of the rotating neutron star
model RNS, the metric solver 3 yields a highly accurate matching, such that the lines almost
lie on top of one another. The insets show the relative difference ∆rel u between the left and
right hand sides of the equation for the same metric components. The relative differences are
10 −2 , except where they exhibit a pole.
of the relative differences ∆rel u between the left and right hand sides of the equation for the various
metric components u. Correspondingly, the metric solution evaluated on the finite difference grid
exhibits second order convergence with grid resolution for a fixed (and high) spectral grid resolution.
Furthermore, the (at least) second order accurate time integration scheme of the code in combination
with the PPM reconstruction of the Riemann solver also guarantees second order convergence during
time evolution. For fixed time steps we actually observe this theoretical convergence order globally
and even locally (except close to the grid boundaries, where symmetry conditions and ghost zone
extrapolation spoil local convergence).
In the three-dimensional case the computational load of the interpolation from the spectral grid to
the finite difference grid after every metric calculation on the spectral grid becomes significant. The
time spent in the interpolation between grids can, in fact, even surpass the computational costs of the