Convergence and stability Sample Clauses

Convergence and stability. ‌ We use the following to measure the volume averaged Hamiltonian constraint vio- lation: V V L2(H) = . 1 ∫ |H2|dV , (2.25) where V is the box volume with the interior of the apparent horizon excised. As can be seen in Fig. 2.4, we have good control over the constraint violation throughout the simulation. We test the convergence of our simulations with the formation of an Axion Star with initial total mass of M = 1.34 MⓈ 10−10eV m−a 1, fa = 5.0 Mpl and L˜ = 16 ma−1. We use a fixed grid for the convergence test with resolutions of 0.25 m−a 1, 0.125 m−a 1 and 0.0625 ma−1. The results are shown in Fig. 2.5, where we obtain an order of convergence between 3rd and 4th order on average. The variation in the convergence test is due to the methodology, where we extract values of φ at the centre of the grid. φ passes through 0 during the evolution, that causes the spikes present in the convergence test. Figure 2.4: The plot shows the L2 norm Eqn. (3.5) of the Hamiltonian constraint violation over time for a simulation that forms an axion star, with an initial total mass of M = 1.34 MⓈ 10−10eV m−a 1, fa = 5.0 Mpl and L˜ = 16 ma−1. The spikes in the plot are due to the regridding in the simulation and are rapidly damped.