Validation results

We now present the results of applying our algorithm to the exact solution of Petrichet al.presented above. When taking cross sections of the solutions, we do so along the x-axis, since we have the wind coming in fromx=−∞by default. We do not plot actual values of the solution, however, but interpolate 800 equally spaced points between x = ±50M. Therefore, our plots show the correct shape of our evolved solution, but not exact numerical values. We note that for the low resolution grid (see Table 7.2), 100M/∆rmin ≈1700, so this is certainly not too many points to use for the interpolation, at least near the horizon.

Evolving the solution of Petrichet al.will test all aspects of our algorithm, as it requires the capturing of effects that do not have spherical or Cartesian symmetry, so that all flux components are tested, along with our implementation of evolution on curvilinear grids. The suitability of both the inner and outer boundary conditions will also be tested.

Unless otherwise stated, the following tests are all carried out using a low resolution grid (see Table 7.2) with an inner boundary at r=M and an outer boundary atr= 50M. We use a fluid velocity at infinity ofv_∞= 0.6, so as to compare to Petrichet al.’s plot.

−500 −25 0 25 50 0.02

0.04 0.06 0.08 0.1 0.12 0.14 0.16

x/M (ρnum-ρexact)/ρexact

(a)Relative difference betweenρexact andρnum

−50 −25 0 25 50

−20

−15

−10

−5 0 5x 10⁻³

x/M vxnum–vxexact

(b)Difference betweenvxexact andvxnum

Figure 7.3: The exact solution of Petrich et al. [83] is used as initial data for the simulation and evolved to timet= 200M. We plot the difference between the exact solution and the evolved solution at this time.

We see that the solutions are very close, except near to the excision boundary.

Our first test of the algorithm is to begin with the exact solution and to evolve it for some time. In Figure 7.3 we show a comparison of the simulated solution with the exact solution.

The density is within about 0.2% of the exact solution outside r = 5M and within 2% outside the horizon at r = 2M. The x-velocity is everywhere within about 0.015 of the exact solution. We also compare the mass accretion rates for the evolved exact solution with the analytic value, and this is shown in Figure 7.4. We see that the accretion rate stabilises slightly above the exact solution. However, the limiting value is within 1.2% of the exact solution.

We now perform the same evolution with the constant density and velocity initial data that we intend to use for later problems. A comparison of the exact and numerical solutions is shown in Figure 7.5. The differences between the exact solution and the numerical solution is shown in Figure 7.6. We see that although the solutions appear fairly close except near the outer boundaries, the relative difference between the exact and numerical solutions are somewhat larger than those from evolving the exact solution as initial data.

The mass accretion rate, shown in Figure 7.7, stabilises to a constant limit after a time of about 130M. The limit is again slightly in excess of the analytic value, exceeding it by about 4.3%.

We have repeated these plots for different resolutions with results as in Table 7.3, and we observe that the numerical solution does not converge to the analytic solution. We have eliminated some possibilities for this discrepancy as follows:

1. Generating the exact solution using our numerical code and then calculating the mass accretion rate from that yields a value within 10⁻⁴% of the exact value. The low resolution of the grid is therefore not an issue, and we are calculating the mass accretion rate correctly.

0 50 100 150 200 62.6

62.8 63 63.2 63.4 63.6 63.8 64

t M ˙

Figure 7.4: Mass accretion rate resulting from evolving Petrich et al.’s exact solution to timet= 200M, evaluated at radius r = 5M. The solid line shows the exact solution, and the crosses the numerical solution.

−50 −25 0 25 50

10⁰ 10¹

x/M

logρ

(a) Comparison of the numerical and exact solutions for logρ.

−50 −25 0 25 50

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

x/M v_x

(b)Comparison of the numerical and exact solutions for vx.

Figure 7.5: Comparison of the numerical and exact solutions forlogρandvx. The numerical solution is given by the blue+s and the exact solution is given by the green∇s.

2. Varying the radius at which we calculate the mass accretion rate allows for some variation, as shown in Table 7.3. However, they do not vary sufficiently to encompass the exact solution.

3. Varying the outer radius of the grid does not alter the limiting value significantly.

−50 −25 0 25 50

−0.1

−0.08

−0.06

−0.04

−0.02 0 0.02

x/M (ρnum–ρexact)/ρexact

(a)Relative difference betweenρexact andρnum

−50 −25 0 25 50

−0.025

−0.02

−0.015

−0.01

−0.005 0 0.005 0.01

x/M vxnum–vxexact

(b)Difference betweenvxexact andvxnum

Figure 7.6: A stiff fluid with an initial state ofρ= 1and √

vivⁱ= 0.6 is evolved to timet= 197M. We plot the difference between the exact and evolved solutions at this time.

0 50 100 150 200

40 60 80 100 120 140 160 180

t/M M ˙

Figure 7.7: Mass accretion rate resulting from evolving constant intial conditions to time t = 200M, evaluated at radius r= 5M. The solid line shows the exact solution, and the +s the numerical solution.

Further, plotting√

v²/u⁰in Boyer-Lindquist coordinates for a high resolution run gives a plot visually very close to that given in [83], as shown in Figure 7.8.

Applying our code to a stiff fluid accretion onto a black hole with spin a= 0.5 results in the mass accretion rate shown in Figure 7.9. We again see that the limiting value resulting from the evolved solution is slightly in excess of the analytic solution, this time by about 4.1%.

Resolution Radius M˙num/M˙exact

n= 35

2.50 1.029

3.00 1.033

3.80 1.037

5.00 1.043

6.00 1.043

n= 50

2.50 1.028

3.00 1.031

3.80 1.033

5.00 1.036

6.00 1.037

n= 60

2.50 1.028

3.00 1.030

3.80 1.032

5.00 1.034

6.00 1.035

Table 7.3: Final steady state accretion rates for various resolutions and extraction radii. The mass accretion rates are calculated at a timeT = 200M, where the solution has settled down to a steady state for all simulations.

Figure 7.8: Contours of √

v²/u⁰ in Boyer-Lindquist coordinates of the accretion of a stiff fluid onto a non-rotating black hole with v_∞ = 0.6. The high resolution grid was used (with inner radius r =M), and the simulation ran to a time t= 400M. The contour levels are at 0.08,0.17,0.26,0.35, with figure limits at±7M, the same as in the plot in [83]. The stream-lines are calculated using velocities in Boyer- Lindquist coordinates, which have a coordinate singularity at the horizon, which is why the streamlines do not continue to the excision boundary. The grid lines for the z= 0 plane are shown, and the regions of overlap can be seen.

0 50 100 150 200 40

60 80 100 120 140 160 180

t/M M ˙

Figure 7.9: Variation in time of the mass accretion rate of a stiff fluid with velocity v_∞ = 0.6 onto a Kerr black hole with spina= 0.5. The analytic solution as derived by Petrich et al. is shown by the line, and the numerical solution is shown by the+s.