MacroMicroSimulator

Macro simulator

This package provides a numerical solution to the viscous / inviscid quasilinear equation

$\displaystyle \frac{\partial \rho}{\partial t}(t, x) + \frac{\partial f(\rho)}{\partial x}(t, x) = \gamma \frac{\partial^2 \rho}{\partial x^2} + G(t, x, \rho(t, x))$

for $(t, x) \in [0, T] \times [0, L]$ and the dissipation coefficient $\gamma$ is positive or null. The distribution $G$ is defined as $G(t, x, \rho(t, x)) = \sum_i g_i(t, \rho(t, x)) \delta_{x_i}(x)$ where $g_i$ is an inflow / outflow at $x_i$. The Dirac at $u$ is represented by $\delta_u$.

The unknown is the density function $\rho: [0, T] \times [0, L] \to [0, 1]$. Initial condition and boundary conditions at $x = 0$ and $x = L$ must be provided. The flux function $f$ is defined such that

• $f$ is concave with a maximum at $\rho_c$;
• $f(\rho) = V_f\rho + o(\rho)$.

These conditions are coming from [1].

using MacroMicroSimulator

# Definition of the equation
flux = MacroMicroSimulator.Flux(ρ -> ρ * (1 - ρ), ρ_c=0.5f0, V_f=1.0f0)
g1 = (0.5f0, (t, ρ) -> (t >= 1.0f0) ? 0.05f0 * (1 + cos((t - 1) * 6)) : 0.0f0)
equation = MacroMicroSimulator.Equation(L=1.0f0, T=2.0f0, flux=flux, gs=[g1])

# Definition of the macro-simulator
simulator = MacroMicroSimulator.Simulator(equation, N_L=500)
initial_condition(simulator, x -> 0.8 * x)
top_boundary_condition(simulator, identity)
bottom_boundary_condition(simulator, x -> 0.9)
compute(simulator) # We compute the solution

MacroMicroSimulator.plot(simulator) |> display

For more information, please see the documentation.

Micro simulator

Based on the density function, it is possible to generate trajectories of particles. Their speed in the flow is defined as $V(\rho) = f(\rho) / \rho$ where $\rho$ is the density at the particle position.

# Definition of the micro-simulator
probe_vehicles_initial_condition = [
    (t=0.0f0, x=0.15f0),
    (t=0.0f0, x=0.8f0),
    (t=0.45f0, x=0.0f0),
    (t=1.5f0, x=0.5f0)
]
probe_vehicles = MacroMicroSimulator.Sensors(probe_vehicles_initial_condition, simulator)
compute(probe_vehicles) # We compute positions of the sensors

MacroMicroSimulator.plot(probe_vehicles) |> display

The output of this example is given below.

For more information, please see the documentation.

Aknowledgement

Please cite [1] if you use this script.

[1] M. Barreau, J. Liu, K. H. Johansson. Learning-based State Reconstruction for a Scalar Hyperbolic PDE under noisy Lagrangian Sensing, Proceedings of the 3rd Conference on Learning for Dynamics and Control, PMLR 144:34-46, 2021