Self-organized hydrodynamics with density-dependent velocity

Figure 1.  The accuracy test at $t=1$ shows that the splitting scheme is of first order. The initial data is given by $\rho_0 = \rho_s(1+0.1\sin(\pi x)), \theta_0 = \theta_s(1+0.1\sin(\pi x))$ with $(\rho_s, \theta_s) = (0.01, \frac{\pi}{4})$ and the mesh sizes are iteratively $N_x = N_y = 32, 64,128,256$

Figure 2.  Comparison between the particle (left) and SOH models (right). For the particle model, $N = 10^5, \nu = 100, D = 10, R_1 = R_2 = 0.1$. The result is the average of 40 simulations. For the SOH model, $N_x = N_y = 100$

Figure 3.  Stability test of the SOH model with $\gamma = 0$ and $\sigma = 0.1$. The profiles show the RMSF of $\rho$ and $\theta$ in $\log$-scale with respect to time $t$. The numerical solutions evolves from the steady states, and locally high concentrations develop

Figure 4.  Stability test of the SOH model with $d = 0.5$, $\sigma = 0.01$ and $\rho^* = 0.02$. RMSF of the density and angle are decreasing with time

Figure 5.  Stability test of the SOH model with $d = 0.5$, $\sigma = 0.01$ and $\rho^* = 0.005$. The RMSF of the numerical solutions grows gradually and high local concentrations develop. The linear scaling of the $\log$ of RMSF implies an exponential growth of the perturbation as a function of time $t$

Figure 6.  Growth rate of the perturbation $\rho_\sigma$. The parameters are $c_1 = 0.975, c_2 = 0.925, d = 0.05$ and $k_0 = 0.125$. The three parameters for $v(\rho)$ are chosen as $(\rho^*, \alpha, \beta) = (0.005, 2, 5)$. The steady state for the density is fixed at $\rho_s = 0.01$ and the final time is $t = 1$. (a) is computed using the fomula (11). In order to obtain (b), we compute the numerical solutions of the SOH model and perform a simple linear regression on the Discrete Fourier transform of the perturbed part, i.e. $\rho_\sigma = \rho - \rho_s$. The growth rate is interpreted as the slope of the function $t \to \hat\rho_\sigma(\xi, t)$

Figure 7.  Growth rate of the perturbation $\rho_\sigma$ given by the particle model with $N = 10^5$. The parameters are $\nu = 100, D = 5, R_1 = R_2 = 0.1$. The three parameters for $v(\rho)$ are chosen as $(\rho^*, \alpha, \beta) = (0.005, 2, 5)$. For each $\theta_s$, the growth rate is computed using the average of $10$ simulations, in order to reduce the effects of noise