Theory

prim_conserve(prim, γ)

Transform primitive -> conservative variables

prim_conserve(ρ, U, λ, γ)

prim_conserve(ρ, U, V, λ, γ)

prim_conserve(ρ, U, V, W, λ, γ)

prim_conserve(prim, γ, Kr)

Polyatomic gas

conserve_prim(u)

Transform conservative -> primitive variables

scalar: pseudo primitive vector for scalar conservation laws

conserve_prim(u, a)

conserve_prim(W, γ)

vector: primitive vector for Euler, Navier-Stokes and extended equations

conserve_prim(ρ, M, E, γ)

conserve_prim(ρ, MX, MY, E, γ)

conserve_prim(ρ, MX, MY, MZ, E, γ)

conserve_prim(w, K, Kr)

Rykov model

mixture_prim_conserve(prim, γ)

Transform multi-component primitive -> conservative variables

mixture_prim_conserve(prim, γ, Kr)

Transform multi-component primitive -> conservative variables

mixture_conserve_prim(W, γ)

Transform multi-component conservative -> primitive variables

mixture_conserve_prim(W, K, Kr)

Transform multi-component polyatomic conservative -> primitive variables

em_coefficients(prim, E, B, mr, lD, rL, dt)

Calculate electromagnetic coeffcients in hyperbolic Maxwell's equations

advection_flux(u, a)

Theoretical flux of linear advection equation

burgers_flux(u)

Theoretical flux of Burgers' equation

euler_flux(w, γ; frame)

Theoretical fluxes of Euler Equations

euler_jacobi(w, γ)

Flux Jacobian of Euler Equations

gauss_moments(prim)

Calculate moments of Gaussian distribution G = (λ / π)^(D / 2) * exp[-λ(c^2 + ξ^2)]

gauss_moments(prim, inK)

Calculate moments of Gaussian distribution with internal energy

mixture_gauss_moments(prim, inK)

Calculate moments of Gaussian distribution in multi-component gas

moments_conserve(Mu, alpha)

Calculate conservative moments of particle distribution

moments_conserve(Mu, Mxi, alpha, delta)

moments_conserve(Mu, Mv, Mw, alpha, beta, delta)

moments_conserve(f, u, ω, _)

Discrete moments of conservative variables

1F1V

moments_conserve(f, u, v, ω, _)

1F2V

moments_conserve(h, b, u, ω, _)

2F1V

moments_conserve(h, b, u, v, ω, _)

2F2V

moments_conserve(h0, h1, h2, u, v, ω, _)

3F2V

moments_conserve(f, u, v, w, ω, _)

1F3V

moments_conserve(h0, h1, h2, h3, u, ω, _)

4F1V

moments_conserve(f, u, ω)

Shortcut methods

1F1V

moments_conserve(a1, a2, a3, a4)

2F1V & 1F2V

moments_conserve(a1, a2, a3, a4, a5)

2F2V & 1F3V

moments_conserve(a1, a2, a3, a4, a5, a6)

4F1V & 3F2V

mixture_moments_conserve(Mu, Mxi, alpha, delta)

Calculate conservative moments of particle distribution in multi-component gas

mixture_moments_conserve(Mu, Mv, Mw, alpha, beta, delta)

mixture_moments_conserve(f, u, ω)

Discrete moments of conservative variables

1F1V

mixture_moments_conserve(h, b, u, ω)

2F1V

mixture_moments_conserve(h0, h1, h2, h3, u, ω)

4F1V

mixture_moments_conserve(f, u, v, ω)

1F2V

mixture_moments_conserve(h, b, u, v, ω)

2F2V

mixture_moments_conserve(h0, h1, h2, u, v, ω)

3F2V

mixture_moments_conserve(f, u, v, w, ω)

1F3V

pdf_slope(u, Δ)

Calculate slope of particle distribution function a = a1 + u * a2 + 0.5 * u^2 * a3

pdf_slope(prim, sw, inK)

mixture_pdf_slope(prim, sw, inK)

Calculate slope of multi-component particle distribution function a = a1 + u * a2 + 0.5 * u^2 * a3

moments_conserve_slope(a, Mu, alpha)

Calculate slope-related conservative moments a = a1 + u * a2 + 0.5 * u^2 * a3

moments_conserve_slope(a, Mu, Mxi, alpha)

moments_conserve_slope(a, Mu, Mv, Mw, alpha, beta)
moments_conserve_slope(a, Mu, Mv, Mw, alpha, beta, delta)

mixture_moments_conserve_slope(a, Mu, Mxi, alpha)

Calculate slope-related conservative moments under the assumption a = a1 + u * a2 + 0.5 * u^2 * a3

mixture_moments_conserve_slope(a, Mu, Mv, Mxi, alpha, beta)

mixture_moments_conserve_slope(
    a,
    Mu,
    Mv,
    Mw,
    alpha,
    beta,
    delta
)

discrete_moments(f, ω)

Discrete moments of particle distribution

discrete_moments(f, u, ω, n)

stress(f, prim, u, ω)

Calculate stress tensor from particle distribution function

stress(f, prim, u, v, ω)

stress(f, prim, u, v, w, ω)

heat_flux(f, prim, u, ω)

Calculate heat flux from particle distribution function

Multiple dispatch doesn't consider unstructured multi-dimensional velocity space. In that case a new method needs to be defined.

1F1V

heat_flux(h, b, prim, u, ω)

2F1V

heat_flux(h, b, r, prim, u, ω)

3F1V Rykov model

heat_flux(h, prim, u, v, ω)

1F2V

heat_flux(h, b, prim, u, v, ω)

2F2V

heat_flux(h, b, r, prim, u, v, ω)

3F2V Rykov model

heat_flux(f, prim, u, v, w, ω)

1F3V

maxwellian(u, ρ, U, λ)

Maxwellian in discrete form

1V

maxwellian(u, prim)

maxwellian(u, v, ρ, U, V, λ)

2V

maxwellian(u, v, prim)

maxwellian(u, v, w, ρ, U, V, W, λ)

3V

maxwellian(u, v, w, prim)

mixture_maxwellian(u, prim)

Multi-component Maxwellian in discrete form

1V

mixture_maxwellian(u, v, prim)

2V

mixture_maxwellian(u, v, w, prim)

3V

shakhov(u, M, q, prim, Pr)

Shakhov non-equilibrium part

1F1V

shakhov(u, H, B, q, prim, Pr, K)

2F1V

shakhov(u, v, M, q, prim, Pr)

1F2V

shakhov(u, v, H, B, q, prim, Pr, K)

2F2V

shakhov(u, v, w, M, q, prim, Pr)

1F3V

reduce_distribution(f, weights)
reduce_distribution(f, weights, dim)

Reduced distribution function

Arguments

• f: particle distribution function with full velocity space
• weights: quadrature weights with reduced velocity setting
• dim: dimension of the reduced distribution function (1 by default)

For 3D -> 1D, the quadrature weights can be obtained from VSpace2D.

reduce_distribution(f, weights)
reduce_distribution(f, weights, dim)

reduce_distribution(f, v, w, weights)
reduce_distribution(f, v, w, weights, dim)

full_distribution(h, b, u, weights, v, w, ρ)
full_distribution(h, b, u, weights, v, w, ρ, γ)

Recover full distribution function from reduced ones

Arguments

• h, b: reduced particle distribution function with 1D velocity space
• u: quadrature nodes in 1D velocity space
• weights: quadrature weights in 1D velocity space
• v, w: quadrature nodes in the rest velocity space (with 3D setting)

Outputs

• f: particle distribution function with 3D velocity space

full_distribution(h, b, u, weights, v, w, prim)
full_distribution(h, b, u, weights, v, w, prim, γ)

ref_vhs_vis(Kn, alpha, omega)

Calculate reference viscosity with variable hard sphere (VHS) model

vhs_collision_time(ρ, λ, μᵣ, ω)

Calculate collision time with variable hard sphere (VHS) model

vhs_collision_time(prim, muRef, omega)

For Rykov model prim[end] should be λₜ

aap_hs_collision_time(prim, mi, ni, me, ne, kn)

Calculate mixture collision time from AAP model

aap_hs_prim(prim, tau, mi, ni, me, ne, kn)

Calculate mixture primitive variables from AAP model

aap_hs_diffeq!(du, u, p, t)

Source term of AAP model in DifferentialEquations.jl

shift_pdf!(f, a, du, dt)

Shift distribution function by external force

shift_pdf!(f, a, du, dt)

Multi-component gas

hs_boltz_kn(mu_ref, alpha)

Calculate effective Knudsen number for fast spectral method with hard sphere (HS) model

kernel_mode(
    M::Integer,
    umax,
    vmax,
    wmax,
    du,
    dv,
    dw,
    unum::Integer,
    vnum::Integer,
    wnum::Integer,
    alpha;
    quad_num
) -> Tuple{Any, Any, Array{Float64, 3}}

Calculate collision kernel for fast spectral method...

kernel_mode(
    M::Integer,
    umax,
    vmax,
    wmax,
    unum::Integer,
    vnum::Integer,
    wnum::Integer,
    alpha;
    quad_num
) -> Tuple{Any, Any, Array{Float64, 3}}

kernel_mode(
    M::Integer,
    umax,
    vmax,
    du,
    dv,
    unum::Integer,
    vnum::Integer;
    quad_num
) -> Tuple{Array{Float64, 3}, Array{Float64, 3}, Matrix{Float64}}

boltzmann_fft(
    f0::AbstractArray{<:Real, 3},
    Kn,
    M::Integer,
    ϕ::AbstractArray{<:Real, 4},
    ψ::AbstractArray{<:Real, 4},
    phipsi::AbstractArray{<:Real, 3}
) -> Any

Calculate collision operator with FFT-based fast spectral method...

boltzmann_fft(f, p)

boltzmann_fft!(
    Q::AbstractArray{<:Real, 3},
    f0::AbstractArray{<:Real, 3},
    Kn::Real,
    M::Integer,
    ϕ::AbstractArray{<:Real, 4},
    ψ::AbstractArray{<:Real, 4},
    phipsi::AbstractArray{<:Real, 3}
) -> Any

Calculate collision operator with FFT-based fast spectral method...

boltzmann_fft!(Q, f, p)

heat_capacity_ratio(K, D)

Calculate heat capacity ratio (monatomic gas)

heat_capacity_ratio(K, Nr, D)

Calculate heat capacity ratio (diatomic gas)

sound_speed(λ, γ)

Calculate speed of sound

sound_speed(prim, γ)

sound_speed(prim, γ)

Calculate sound speed in mixture