ConservationLawsParticles.jl

abstract type AbstractModel{N}

Abstract type representing a particle model.

DiffusiveIntegratedModel((V₁, ...), ((W₁₁, ...), ...), (mob₁, ...), (D₁, ...))

Represents a particles system with:

• external velocities Vᵢ,
• integrated interactions Wᵢⱼ (this is the effect of the species j on the species i),
• mobilities mobᵢ,
• diffusions Dᵢ.

See also DiffusiveSampledModel.

Examples

julia> using ConservationLawsParticles.Examples, RecursiveArrayTools

julia> model = DiffusiveIntegratedModel(
           (V, V),
           ((W_attr, W_rep),
            (W_rep, W_attr)),
           (mobρ, mobσ),
           (Diffusion(1), nothing));

julia> x = ArrayPartition(gaussian_particles(2, 4), collect(range(-3, 4, length=5)))
([-2.000000000000003, -0.30305473018369145, 0.30305473018369145, 2.000000000000003], [-3.0, -1.25, 0.5, 2.25, 4.0])

julia> velocities_diff(x, model, 0.)
([6.032353228461558, -0.7551163082821832, 0.6338610075015758, -6.3250083138486275], [23.261098611816987, 0.9023583690557855, 0.7055593913708742, -8.774526834146023, -55.23308442021724])

DiffusiveSampledModel((V₁, ...), ((W′₁₁, ...), ...), (mob₁, ...), (D₁, ...))

Represents a particles system with:

• external velocities Vᵢ,
• sampled interactions W′ᵢⱼ (this is the effect of the species j on the species i),
• mobilities mobᵢ,
• diffusions Dᵢ.

See also DiffusiveIntegratedModel.

Examples

julia> using ConservationLawsParticles.Examples, RecursiveArrayTools

julia> model = DiffusiveSampledModel(
           (V, V),
           ((Wprime_attr, Wprime_rep),
            (Wprime_rep, Wprime_attr)),
           (mobρ, mobσ),
           (Diffusion(1), nothing));

julia> x = ArrayPartition(gaussian_particles(2, 4), collect(range(-3, 4, length=5)))
([-2.000000000000003, -0.30305473018369145, 0.30305473018369145, 2.000000000000003], [-3.0, -1.25, 0.5, 2.25, 4.0])

julia> velocities_diff(x, model, 0.)
([5.701842050142524, -0.8139064918316994, 0.3386810988127882, -6.637012508114956], [22.405129478914613, 1.1249366684885518, 1.5188519354999799, -7.87111869358889, -54.536397957423915])

HyperbolicModel((V₁, ...), ((I₁₁, ...), ...), (mob₁, ...))

Represents a particles system with:

• external velocities Vᵢ,
• integrated interactions Iᵢⱼ (this is the effect of the species j on the species i),
• mobilities mobᵢ.

See also ParabolicModel.

Examples

julia> using ConservationLawsParticles.Examples, RecursiveArrayTools

julia> model = HyperbolicModel(
           (V, V),
           ((SampledInteraction(Wprime_attr), IntegratedInteraction(W_rep)),
            (IntegratedInteraction(W_rep), SampledInteraction(Wprime_attr))),
           (mobρ, mobσ));

julia> x = ArrayPartition(gaussian_particles(2, 4), collect(range(-3, 4, length=5)))
([-2.000000000000003, -0.30305473018369145, 0.30305473018369145, 2.000000000000003], [-3.0, -1.25, 0.5, 2.25, 4.0])

julia> abstract_velocities(x, model, 0.)
([6.267901451064502, 0.2629261335925302, -0.3841814343731375, -6.560556536451573], [22.921033907881895, 0.8199108735979148, 0.7055593913708742, -8.681409489341364, -54.89301971628215])

IntegratedInteraction(W)

Represents the integrated interaction induced by the function W.

The interaction can be evaluated with compute_interaction.

See also SampledInteraction.

IntegratedModel((V₁, ...), ((W₁₁, ...), ...), (mob₁, ...))

Represents a particles system with:

• external velocities Vᵢ,
• integrated interactions Wᵢⱼ (this is the effect of the species j on the species i),
• mobilities mobᵢ.

See also SampledModel.

Examples

julia> using ConservationLawsParticles.Examples, RecursiveArrayTools

julia> model = IntegratedModel(
           (V, V),
           ((W_attr, W_rep),
            (W_rep, W_attr)),
           (mobρ, mobσ));

julia> x = ArrayPartition(gaussian_particles(2, 4), collect(range(-3, 4, length=5)))
([-2.000000000000003, -0.30305473018369145, 0.30305473018369145, 2.000000000000003], [-3.0, -1.25, 0.5, 2.25, 4.0])

julia> velocities(x, model, 0.)
([6.621647425385332, 0.30545649966452776, -0.42671180044513507, -6.914302510772401], [23.261098611816987, 0.9023583690557855, 0.7055593913708742, -8.774526834146023, -55.23308442021724])

ParabolicModel((V₁, ...), ((I₁₁, ...), ...), (mob₁, ...), (D₁, ...))

Represents a particles system with:

• external velocities Vᵢ,
• interactions Iᵢⱼ (this is the effect of the species j on the species i),
• mobilities mobᵢ,
• diffusions Dᵢ.

See also HyperbolicModel.

Examples

julia> using ConservationLawsParticles.Examples, RecursiveArrayTools

julia> model = ParabolicModel(
           (V, V),
           ((SampledInteraction(Wprime_attr), IntegratedInteraction(W_rep)),
            (IntegratedInteraction(W_rep), SampledInteraction(Wprime_attr))),
           (mobρ, mobσ),
           (Diffusion(1), nothing));

julia> x = ArrayPartition(gaussian_particles(2, 4), collect(range(-3, 4, length=5)))
([-2.000000000000003, -0.30305473018369145, 0.30305473018369145, 2.000000000000003], [-3.0, -1.25, 0.5, 2.25, 4.0])

julia> abstract_velocities(x, model, 0.)
([5.678607254140728, -0.7976466743541808, 0.6763913735735734, -5.971262339527799], [22.921033907881895, 0.8199108735979148, 0.7055593913708742, -8.681409489341364, -54.89301971628215])

SampledInteraction(W′)

Represents the sampled interaction induced by the function W′.

The interaction can be evaluated with compute_interaction.

See also IntegratedInteraction.

SampledModel((V₁, ...), ((W′₁₁, ...), ...), (mob₁, ...))

Represents a particles system with:

• external velocities Vᵢ,
• sampled interactions W′ᵢⱼ (this is the effect of the species j on the species i),
• mobilities mobᵢ.

See also IntegratedModel.

Examples

julia> using ConservationLawsParticles.Examples, RecursiveArrayTools

julia> model = SampledModel(
           (V, V),
           ((Wprime_attr, Wprime_rep),
            (Wprime_rep, Wprime_attr)),
           (mobρ, mobσ));

julia> x = ArrayPartition(gaussian_particles(2, 4), collect(range(-3, 4, length=5)))
([-2.000000000000003, -0.30305473018369145, 0.30305473018369145, 2.000000000000003], [-3.0, -1.25, 0.5, 2.25, 4.0])

julia> velocities(x, model, 0.)
([6.291136247066298, 0.2466663161150116, -0.7218917091339228, -7.22630670503873], [22.405129478914613, 1.1249366684885518, 1.5188519354999799, -7.87111869358889, -54.536397957423915])

compute_interaction(t, x, interaction, ys[, dens_diff])

Evaluates the interaction generated by the particles ys at (t, x).

interaction can be a SampledInteraction, an IntegratedInteraction, or any other form of interaction that provides this interface.

See also SampledInteraction, IntegratedInteraction, sampled_interaction, integrated_interaction.

diffusion(mod::AbstractModel, i::Integer)

Returns the diffusion associated to the species i.

eachspecies(mod::AbstractModel) -> Any

Returns an iterator over the indices of the species of the model.

external_velocity(mod::AbstractModel, i::Integer) -> Any

Returns the external velocity field associated to the species i.

gaussian_particles(width::Real, number::Integer)

Generates number particles defining a Gaussian distribution with variance 1/2 truncated to the interval [-width, width].

integrated_interaction([t,] x, W, ys[, dens_diff])

Computes $-(W' * \rho)(t, x) = -(W * \rho')(t,x)$, where $\rho$ is the piecewise-constant density associated to the particles ys.

It t is omitted, then W(x) is assumed independent of time.

See also sampled_interaction, compute_interaction.

interaction(
    mod::AbstractModel,
    i::Integer,
    j::Integer
) -> Any

Returns the interaction exerted on the species i by the species j.

mobility(mod::AbstractModel, i::Integer) -> Any

Returns the mobility associated to the species i.

num_species(mod::AbstractModel) -> Any

Returns the number of species of the model.

pwc_densities(xs::AbstractVector{<:Real}...)
pwc_densities(xs::Tuple{Vararg{AbstractVector{<:Real}}})

Given a list of n families of particles of lengths l₁, …, lₙ, returns an n-tuple whose s-th entry is an n×2×lₛ array densₛ.

This array contains the information about the densities of all the species around the particles of the s-th species. This array is indexed as [o,side,i], where o∈{1, …, n} is the index of the other species whose density we want to know, side is either 1 if we want to know the left density or 2 for the right density, and i∈{1, …, lₛ} is the index of the particle.

In other words, densₛ[o,side,i] is the density of the o-th species on the left (side=1) or right (side=2) of the i-th particle of the s-th species.

See also pwc_densities! for an inplace version.

Examples

julia> pwc_densities([0,1,2], [0,2,3])
([0.0 0.5; 0.0 0.25;;; 0.5 0.5; 0.25 0.25;;; 0.5 0.0; 0.25 0.5], [0.0 0.5; 0.0 0.25;;; 0.5 0.0; 0.25 0.5;;; 0.0 0.0; 0.5 0.0])

pwc_densities!(dens, xs::AbstractVector{<:Real}...)
pwc_densities!(dens, xs::Tuple{Vararg{AbstractVector{<:Real}}})

This is an inplace version of pwc_densities that writes the result in dens.

See the documentation of pwc_densities for an explanation.

pwc_density(x::AbstractVector, y::AbstractVector)

The returned x_dens is indexed as x_dens[x_or_y, left_or_right, i].

pwc_density(x::AbstractVector)

Returns the piecewise constant probability density from the quantile particle positions.

Let the quantile particles be at positions x₀, x₁, …, xₙ (remember that in Julia they are actually x[1], x[2], ..., x[n], x[n+1]). The returned array R is such that R[1] = R[n+2] = 0 and R[i] = 1 / (N * (x[i] - x[i-1])) for the intermediate indices (this formula holds also for the first and last entry if we assume x[0] = -∞ and x[n+2] = ∞).

Examples

julia> pwc_density([0, 1, 3])
4-element Vector{Float64}:
 0.0
 0.5
 0.25
 0.0

sampled_interaction([t,] x, W′, ys)

Computes $-(W' * \dot\rho)(t, x)$, which is the sampled approximation of $-(W' * \rho)(t, x)$, where $\rho$ is the piecewise-constant density associated to the particles ys.

It t is omitted, then W′(x) is assumed independent of time.

See also integrated_interaction, compute_interaction.

species(state, i::Integer) -> Any

Returns the particles associated to the species i.

Automatically define a method which takes time as first argument and discards it.

Examples

The definition

@time_independent V(x) = -x^3

is equivalent to

V(x) = -x^3
V(t, x) = V(x)

This also works with more than one argument, for instance

@time_independent V(x₁, x₂) = x₁ * x₂

is equivalent to

V(x₁, x₂) = x₁ * x₂
V(t, x₁, x₂) = V(x₁, x₂)

make_velocities((V₁, ...), ((W′₁₁, ...), ...), (mob₁, ...))

Creates a function velocities(dx, x, p, t) that computes the velocity of the particles under the influence of the external velocities Vᵢ, mutual interactions W′ᵢⱼ and congestions given by mobᵢ.

See also make_velocity.

make_velocity(V::Function, Wprime::Function, mobility::Function)

Creates a function velocity(dx, x, p, t) that computes the velocity of the particles under the influence of an external velocity V, a mutual interaction Wprime and the congestion given by mobility.

See also make_velocities.