ner_tags: O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O B-Model O O O O O B-Model O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O B-Model B-Model O B-Model O O O O O O
tokens: that we also tested the Cartesian grid initial distribution, similar to Hopkins ( 2015 ). The main mechanism for the decay of the disk is the same. In this paper, therefore, we do not show the results for the Cartesian grid case. In order to carry out a quantitative comparison of the results, we introduce a rough estimate of the disk lifetime. Let us consider the root mean square of the radius of the particles initially located at the inner edge of the disk. When this value exceeds 10% of the radius of the inner edge, we regard the disk as disrupted. Figures 1 and 2 show the lifetime and snapshots of the Keplerian disk for each AV implementation. The differences among the lifetimes of the disks obtained with different implementations are very large. In the case of vNRL-B95-CD10-G+12, the disk survived for a time more than 10 times longer than with the standard method used today (M97-B95-R+00-SPH). Note that the longest case, vNRL-B95-CD10-G+12′, indicates the results with the vNRL-B95-CD10-G+12 scheme, but with and . These parameters in the AV switch would also play important role for the long-time evolution of the disk. Figure 1. Lifetime of the disk for each AV implementation in units of Kepler time at r = 0.5 (inner edge of the disk). Note that the disk lifetime is shown in the log scale. Figure 2. Snapshots from the Keplerian disk test at ∼28 orbits at r = 0.5 (inner edge of the disk) with each AV implementation. The color-coding of the density is given at the bottom. Since no disk lifetime with M97 AV exceeds ∼28 orbits at r = 0.5, we only show the results of four cases. To clarify the effects of each option on the disk lifetime, we show the dependences of the disk lifetime for each option in Figure 3 . Here, we show how the lifetime of the disk changes when we change the method in one type, while keeping the methods for the other three types unchanged. We can clearly see the following tendencies. (i) The disk lifetime obtained with vNRL AV is much longer than that with M97 AV. (ii) Both of the two shear switches extend the disk lifetime. The CD10 switch works better than the B95 switch for M97 AV, while the B95 switch works better than the CD10 switch for the vNRL AV. (iii) Both shock indicators extend the disk lifetime. The CD10 shock indicator works better than R+00 does. (iv) The G+12 derivative operator works better than the SSPH derivative operator. This might be due to the fact that the SPH derivative operator omits the surface term (for details see Section 3.1 in Price 2008 ). Figure 3. The difference of the lifetime of the disk when (a) the AV scheme, (b) the shear switch, (c) the shock indicator, and (d) the derivative estimator are changed. Points connected by lines indicate the runs performed with the same method, except for that shown in panel. Figures 4 – 6 show the evolution of the inner part of the Keplerian disks for three representative runs (M97-No-No-SPH, M97-B95-R+00-SPH, and vNRL-B95-R+00-G+12). In Figures 7 – 9 ,