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ideal_mhd_multiion_2d.jl
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# By default, Julia/LLVM does not use fused multiply-add operations (FMAs).
# Since these FMAs can increase the performance of many numerical algorithms,
# we need to opt-in explicitly.
# See https://ranocha.de/blog/Optimizing_EC_Trixi for further details.
@muladd begin
#! format: noindent
@doc raw"""
IdealMhdMultiIonEquations2D
The ideal compressible multi-ion MHD equations in two space dimensions.
"""
mutable struct IdealMhdMultiIonEquations2D{NVARS, NCOMP, RealT <: Real,
ElectronPressure} <:
AbstractIdealMhdMultiIonEquations{2, NVARS, NCOMP}
gammas::SVector{NCOMP, RealT} # Heat capacity ratios
charge_to_mass::SVector{NCOMP, RealT} # Charge to mass ratios
electron_pressure::ElectronPressure # Function to compute the electron pressure
c_h::RealT # GLM cleaning speed
function IdealMhdMultiIonEquations2D{NVARS, NCOMP, RealT,
ElectronPressure}(gammas
::SVector{NCOMP, RealT},
charge_to_mass
::SVector{NCOMP, RealT},
electron_pressure
::ElectronPressure,
c_h::RealT) where
{NVARS, NCOMP, RealT <: Real, ElectronPressure}
NCOMP >= 1 ||
throw(DimensionMismatch("`gammas` and `charge_to_mass` have to be filled with at least one value"))
new(gammas, charge_to_mass, electron_pressure, c_h)
end
end
function IdealMhdMultiIonEquations2D(; gammas, charge_to_mass,
electron_pressure = electron_pressure_zero,
initial_c_h = convert(eltype(gammas), NaN))
_gammas = promote(gammas...)
_charge_to_mass = promote(charge_to_mass...)
RealT = promote_type(eltype(_gammas), eltype(_charge_to_mass))
NVARS = length(_gammas) * 5 + 4
NCOMP = length(_gammas)
__gammas = SVector(map(RealT, _gammas))
__charge_to_mass = SVector(map(RealT, _charge_to_mass))
return IdealMhdMultiIonEquations2D{NVARS, NCOMP, RealT, typeof(electron_pressure)}(__gammas,
__charge_to_mass,
electron_pressure,
initial_c_h)
end
@inline function Base.real(::IdealMhdMultiIonEquations2D{NVARS, NCOMP, RealT}) where {
NVARS,
NCOMP,
RealT
}
RealT
end
have_nonconservative_terms(::IdealMhdMultiIonEquations2D) = True()
function varnames(::typeof(cons2cons), equations::IdealMhdMultiIonEquations2D)
cons = ("B1", "B2", "B3")
for i in eachcomponent(equations)
cons = (cons...,
tuple("rho_" * string(i), "rho_v1_" * string(i), "rho_v2_" * string(i),
"rho_v3_" * string(i), "rho_e_" * string(i))...)
end
cons = (cons..., "psi")
return cons
end
function varnames(::typeof(cons2prim), equations::IdealMhdMultiIonEquations2D)
prim = ("B1", "B2", "B3")
for i in eachcomponent(equations)
prim = (prim...,
tuple("rho_" * string(i), "v1_" * string(i), "v2_" * string(i),
"v3_" * string(i), "p_" * string(i))...)
end
prim = (prim..., "psi")
return prim
end
function default_analysis_integrals(::IdealMhdMultiIonEquations2D)
(entropy_timederivative, Val(:l2_divb), Val(:linf_divb))
end
# """
# initial_condition_convergence_test(x, t, equations::IdealMhdMultiIonEquations2D)
# An Alfvén wave as smooth initial condition used for convergence tests.
# """
# function initial_condition_convergence_test(x, t, equations::IdealMhdMultiIonEquations2D)
# # smooth Alfvén wave test from Derigs et al. FLASH (2016)
# # domain must be set to [0, 1], γ = 5/3
# rho = 1.0
# prim_rho = SVector{ncomponents(equations), real(equations)}(2^(i-1) * (1-2)/(1-2^ncomponents(equations)) * rho for i in eachcomponent(equations))
# v1 = zero(real(equations))
# si, co = sincos(2 * pi * x[1])
# v2 = 0.1 * si
# v3 = 0.1 * co
# p = 0.1
# B1 = 1.0
# B2 = v2
# B3 = v3
# prim_other = SVector{7, real(equations)}(v1, v2, v3, p, B1, B2, B3)
# return prim2cons(vcat(prim_other, prim_rho), equations)
# end
"""
initial_condition_weak_blast_wave(x, t, equations::IdealMhdMultiIonEquations2D)
A weak blast wave adapted from
- Sebastian Hennemann, Gregor J. Gassner (2020)
A provably entropy stable subcell shock capturing approach for high order split form DG
[arXiv: 2008.12044](https://arxiv.org/abs/2008.12044)
"""
function initial_condition_weak_blast_wave(x, t, equations::IdealMhdMultiIonEquations2D)
# Adapted MHD version of the weak blast wave from Hennemann & Gassner JCP paper 2020 (Sec. 6.3)
# Same discontinuity in the velocities but with magnetic fields
# Set up polar coordinates
inicenter = (0, 0)
x_norm = x[1] - inicenter[1]
y_norm = x[2] - inicenter[2]
r = sqrt(x_norm^2 + y_norm^2)
phi = atan(y_norm, x_norm)
# Calculate primitive variables
rho = zero(real(equations))
if r > 0.5
rho = 1.0
else
rho = 1.1691
end
v1 = r > 0.5 ? 0.0 : 0.1882 * cos(phi)
v2 = r > 0.5 ? 0.0 : 0.1882 * sin(phi)
p = r > 0.5 ? 1.0 : 1.245
prim = zero(MVector{nvariables(equations), real(equations)})
prim[1] = 1.0
prim[2] = 1.0
prim[3] = 1.0
for k in eachcomponent(equations)
set_component!(prim, k,
2^(k - 1) * (1 - 2) / (1 - 2^ncomponents(equations)) * rho, v1,
v2, 0, p, equations)
end
return prim2cons(SVector(prim), equations)
end
# TODO: Add initial condition equilibrium
# Calculate 1D flux in for a single point
@inline function flux(u, orientation::Integer, equations::IdealMhdMultiIonEquations2D)
B1, B2, B3 = magnetic_field(u, equations)
psi = divergence_cleaning_field(u, equations)
v1_plus, v2_plus, v3_plus, vk1_plus, vk2_plus, vk3_plus = charge_averaged_velocities(u,
equations)
mag_en = 0.5 * (B1^2 + B2^2 + B3^2)
div_clean_energy = 0.5 * psi^2
f = zero(MVector{nvariables(equations), eltype(u)})
if orientation == 1
f[1] = equations.c_h * psi
f[2] = v1_plus * B2 - v2_plus * B1
f[3] = v1_plus * B3 - v3_plus * B1
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, rho_v3, rho_e = get_component(k, u, equations)
v1 = rho_v1 / rho
v2 = rho_v2 / rho
v3 = rho_v3 / rho
kin_en = 0.5 * rho * (v1^2 + v2^2 + v3^2)
gamma = equations.gammas[k]
p = (gamma - 1) * (rho_e - kin_en - mag_en - div_clean_energy)
f1 = rho_v1
f2 = rho_v1 * v1 + p
f3 = rho_v1 * v2
f4 = rho_v1 * v3
f5 = (kin_en + gamma * p / (gamma - 1)) * v1 + 2 * mag_en * vk1_plus[k] -
B1 * (vk1_plus[k] * B1 + vk2_plus[k] * B2 + vk3_plus[k] * B3) + equations.c_h * psi * B1
set_component!(f, k, f1, f2, f3, f4, f5, equations)
end
f[end] = equations.c_h * B1
else #if orientation == 2
f[1] = v2_plus * B1 - v1_plus * B2
f[2] = equations.c_h * psi
f[3] = v2_plus * B3 - v3_plus * B2
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, rho_v3, rho_e = get_component(k, u, equations)
v1 = rho_v1 / rho
v2 = rho_v2 / rho
v3 = rho_v3 / rho
kin_en = 0.5 * rho * (v1^2 + v2^2 + v3^2)
gamma = equations.gammas[k]
p = (gamma - 1) * (rho_e - kin_en - mag_en - div_clean_energy)
f1 = rho_v2
f2 = rho_v2 * v1
f3 = rho_v2 * v2 + p
f4 = rho_v2 * v3
f5 = (kin_en + gamma * p / (gamma - 1)) * v2 + 2 * mag_en * vk2_plus[k] -
B2 * (vk1_plus[k] * B1 + vk2_plus[k] * B2 + vk3_plus[k] * B3) + equations.c_h * psi * B2
set_component!(f, k, f1, f2, f3, f4, f5, equations)
end
f[end] = equations.c_h * B2
end
return SVector(f)
end
"""
Standard source terms of the multi-ion MHD equations
"""
function source_terms_standard(u, x, t, equations::IdealMhdMultiIonEquations2D)
@unpack charge_to_mass = equations
B1, B2, B3 = magnetic_field(u, equations)
v1_plus, v2_plus, v3_plus, vk1_plus, vk2_plus, vk3_plus = charge_averaged_velocities(u,
equations)
s = zero(MVector{nvariables(equations), eltype(u)})
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, rho_v3, rho_e = get_component(k, u, equations)
v1 = rho_v1 / rho
v2 = rho_v2 / rho
v3 = rho_v3 / rho
v1_diff = v1_plus - v1
v2_diff = v2_plus - v2
v3_diff = v3_plus - v3
r_rho = charge_to_mass[k] * rho
s2 = r_rho * (v2_diff * B3 - v3_diff * B2)
s3 = r_rho * (v3_diff * B1 - v1_diff * B3)
s4 = r_rho * (v1_diff * B2 - v2_diff * B1)
s5 = v1 * s2 + v2 * s3 + v3 * s4
set_component!(s, k, 0, s2, s3, s4, s5, equations)
end
return SVector(s)
end
"""
electron_pressure_zero(u, equations::IdealMhdMultiIonEquations2D)
Returns the value of zero for the electron pressure. Consistent with the single-fluid MHD equations.
"""
function electron_pressure_zero(u, equations::IdealMhdMultiIonEquations2D)
return zero(u[1])
end
"""
Total entropy-conserving non-conservative two-point "flux"" as described in
- Rueda-Ramírez et al. (2023)
The term is composed of three parts
* The Powell term: Implemented
* The MHD term: Implemented
* The "term 3": Implemented
"""
@inline function flux_nonconservative_ruedaramirez_etal(u_ll, u_rr,
orientation::Integer,
equations::IdealMhdMultiIonEquations2D)
@unpack charge_to_mass = equations
# Unpack left and right states to get the magnetic field
B1_ll, B2_ll, B3_ll = magnetic_field(u_ll, equations)
B1_rr, B2_rr, B3_rr = magnetic_field(u_rr, equations)
psi_ll = divergence_cleaning_field(u_ll, equations)
psi_rr = divergence_cleaning_field(u_rr, equations)
# Compute important averages
B1_avg = 0.5 * (B1_ll + B1_rr)
B2_avg = 0.5 * (B2_ll + B2_rr)
B3_avg = 0.5 * (B3_ll + B3_rr)
mag_norm_ll = B1_ll^2 + B2_ll^2 + B3_ll^2
mag_norm_rr = B1_rr^2 + B2_rr^2 + B3_rr^2
mag_norm_avg = 0.5 * (mag_norm_ll + mag_norm_rr)
# Mean electron pressure
pe_mean = 0.5 * (equations.electron_pressure(u_ll, equations) +
equations.electron_pressure(u_rr, equations))
# Compute charge ratio of u_ll
charge_ratio_ll = zero(MVector{ncomponents(equations), eltype(u_ll)})
total_electron_charge = zero(eltype(u_ll))
for k in eachcomponent(equations)
rho_k = u_ll[3 + (k - 1) * 5 + 1]
charge_ratio_ll[k] = rho_k * charge_to_mass[k]
total_electron_charge += charge_ratio_ll[k]
end
charge_ratio_ll ./= total_electron_charge
# Compute auxiliary variables
v1_plus_ll, v2_plus_ll, v3_plus_ll, vk1_plus_ll, vk2_plus_ll, vk3_plus_ll = charge_averaged_velocities(u_ll,
equations)
v1_plus_rr, v2_plus_rr, v3_plus_rr, vk1_plus_rr, vk2_plus_rr, vk3_plus_rr = charge_averaged_velocities(u_rr,
equations)
f = zero(MVector{nvariables(equations), eltype(u_ll)})
if orientation == 1
# Entries of Powell term for induction equation (already in Trixi's non-conservative form)
f[1] = v1_plus_ll * B1_rr
f[2] = v2_plus_ll * B1_rr
f[3] = v3_plus_ll * B1_rr
for k in eachcomponent(equations)
# Compute term 2 (MHD)
f2 = charge_ratio_ll[k] * (0.5 * mag_norm_avg - B1_avg * B1_avg + pe_mean)
f3 = charge_ratio_ll[k] * (-B1_avg * B2_avg)
f4 = charge_ratio_ll[k] * (-B1_avg * B3_avg)
f5 = vk1_plus_ll[k] * pe_mean
# Compute term 3 (only needed for NCOMP>1)
vk1_minus_ll = v1_plus_ll - vk1_plus_ll[k]
vk2_minus_ll = v2_plus_ll - vk2_plus_ll[k]
vk3_minus_ll = v3_plus_ll - vk3_plus_ll[k]
vk1_minus_rr = v1_plus_rr - vk1_plus_rr[k]
vk2_minus_rr = v2_plus_rr - vk2_plus_rr[k]
vk3_minus_rr = v3_plus_rr - vk3_plus_rr[k]
vk1_minus_avg = 0.5 * (vk1_minus_ll + vk1_minus_rr)
vk2_minus_avg = 0.5 * (vk2_minus_ll + vk2_minus_rr)
vk3_minus_avg = 0.5 * (vk3_minus_ll + vk3_minus_rr)
f5 += (B2_ll * (vk1_minus_avg * B2_avg - vk2_minus_avg * B1_avg) +
B3_ll * (vk1_minus_avg * B3_avg - vk3_minus_avg * B1_avg))
# Adjust non-conservative terms 2 and 3 to Trixi discretization: CHANGE!?!
f2 = 2 * f2 - charge_ratio_ll[k] * (0.5 * mag_norm_ll - B1_ll * B1_ll)
f3 = 2 * f3 + charge_ratio_ll[k] * B1_ll * B2_ll
f4 = 2 * f4 + charge_ratio_ll[k] * B1_ll * B3_ll
f5 = (2 * f5 - B2_ll * (vk1_minus_ll * B2_ll - vk2_minus_ll * B1_ll)
-
B3_ll * (vk1_minus_ll * B3_ll - vk3_minus_ll * B1_ll))
# Compute Powell term (already consistent with Trixi's non-conservative discretization)
f2 += charge_ratio_ll[k] * B1_ll * B1_rr
f3 += charge_ratio_ll[k] * B2_ll * B1_rr
f4 += charge_ratio_ll[k] * B3_ll * B1_rr
f5 += (v1_plus_ll * B1_ll + v2_plus_ll * B2_ll + v3_plus_ll * B3_ll) * B1_rr
# Compute GLM term for the energy
f5 += v1_plus_ll * psi_ll * psi_rr
# Append to the flux vector
set_component!(f, k, 0, f2, f3, f4, f5, equations)
end
# Compute GLM term for psi
f[end] = v1_plus_ll * psi_rr
else #if orientation == 2
# Entries of Powell term for induction equation (already in Trixi's non-conservative form)
f[1] = v1_plus_ll * B2_rr
f[2] = v2_plus_ll * B2_rr
f[3] = v3_plus_ll * B2_rr
for k in eachcomponent(equations)
# Compute term 2 (MHD)
f2 = charge_ratio_ll[k] * (-B2_avg * B1_avg)
f3 = charge_ratio_ll[k] * (-B2_avg * B2_avg + 0.5 * mag_norm_avg + pe_mean)
f4 = charge_ratio_ll[k] * (-B2_avg * B3_avg)
f5 = vk2_plus_ll[k] * pe_mean
# Compute term 3 (only needed for NCOMP>1)
vk1_minus_ll = v1_plus_ll - vk1_plus_ll[k]
vk2_minus_ll = v2_plus_ll - vk2_plus_ll[k]
vk3_minus_ll = v3_plus_ll - vk3_plus_ll[k]
vk1_minus_rr = v1_plus_rr - vk1_plus_rr[k]
vk2_minus_rr = v2_plus_rr - vk2_plus_rr[k]
vk3_minus_rr = v3_plus_rr - vk3_plus_rr[k]
vk1_minus_avg = 0.5 * (vk1_minus_ll + vk1_minus_rr)
vk2_minus_avg = 0.5 * (vk2_minus_ll + vk2_minus_rr)
vk3_minus_avg = 0.5 * (vk3_minus_ll + vk3_minus_rr)
f5 += (B1_ll * (vk2_minus_avg * B1_avg - vk1_minus_avg * B2_avg) +
B3_ll * (vk2_minus_avg * B3_avg - vk3_minus_avg * B2_avg))
# Adjust non-conservative terms 2 and 3 to Trixi discretization: CHANGE!?!
f2 = 2 * f2 + charge_ratio_ll[k] * B2_ll * B1_ll
f3 = 2 * f3 - charge_ratio_ll[k] * (0.5 * mag_norm_ll - B2_ll * B2_ll)
f4 = 2 * f4 + charge_ratio_ll[k] * B2_ll * B3_ll
f5 = (2 * f5 - B1_ll * (vk2_minus_ll * B1_ll - vk1_minus_ll * B2_ll)
-
B3_ll * (vk2_minus_ll * B3_ll - vk3_minus_ll * B2_ll))
# Compute Powell term (already consistent with Trixi's non-conservative discretization)
f2 += charge_ratio_ll[k] * B1_ll * B2_rr
f3 += charge_ratio_ll[k] * B2_ll * B2_rr
f4 += charge_ratio_ll[k] * B3_ll * B2_rr
f5 += (v1_plus_ll * B1_ll + v2_plus_ll * B2_ll + v3_plus_ll * B3_ll) * B2_rr
# Compute GLM term for the energy
f5 += v2_plus_ll * psi_ll * psi_rr
# Append to the flux vector
set_component!(f, k, 0, f2, f3, f4, f5, equations)
end
# Compute GLM term for psi
f[end] = v2_plus_ll * psi_rr
end
return SVector(f)
end
"""
Total central non-conservative two-point "flux"", where the symmetric parts are computed with standard averages
The term is composed of three parts
* The Powell term: Implemented. The central Powell "flux" is equivalent to the EC Powell "flux".
* The MHD term: Implemented
* The "term 3": Implemented
"""
@inline function flux_nonconservative_central(u_ll, u_rr, orientation::Integer,
equations::IdealMhdMultiIonEquations2D)
@unpack charge_to_mass = equations
# Unpack left and right states to get the magnetic field
B1_ll, B2_ll, B3_ll = magnetic_field(u_ll, equations)
B1_rr, B2_rr, B3_rr = magnetic_field(u_rr, equations)
psi_ll = divergence_cleaning_field(u_ll, equations)
psi_rr = divergence_cleaning_field(u_rr, equations)
# Compute important averages
mag_norm_rr = B1_rr^2 + B2_rr^2 + B3_rr^2
# Electron pressure
pe_rr = equations.electron_pressure(u_rr, equations)
# Compute charge ratio of u_ll
charge_ratio_ll = zero(MVector{ncomponents(equations), eltype(u_ll)})
total_electron_charge = zero(real(equations))
for k in eachcomponent(equations)
rho_k = u_ll[3 + (k - 1) * 5 + 1]
charge_ratio_ll[k] = rho_k * charge_to_mass[k]
total_electron_charge += charge_ratio_ll[k]
end
charge_ratio_ll ./= total_electron_charge
# Compute auxiliary variables
v1_plus_ll, v2_plus_ll, v3_plus_ll, vk1_plus_ll, vk2_plus_ll, vk3_plus_ll = charge_averaged_velocities(u_ll,
equations)
v1_plus_rr, v2_plus_rr, v3_plus_rr, vk1_plus_rr, vk2_plus_rr, vk3_plus_rr = charge_averaged_velocities(u_rr,
equations)
f = zero(MVector{nvariables(equations), eltype(u_ll)})
if orientation == 1
# Entries of Powell term for induction equation (already in Trixi's non-conservative form)
f[1] = v1_plus_ll * B1_rr
f[2] = v2_plus_ll * B1_rr
f[3] = v3_plus_ll * B1_rr
for k in eachcomponent(equations)
# Compute term 2 (MHD)
f2 = charge_ratio_ll[k] * (0.5 * mag_norm_rr - B1_rr * B1_rr + pe_rr)
f3 = charge_ratio_ll[k] * (-B1_rr * B2_rr)
f4 = charge_ratio_ll[k] * (-B1_rr * B3_rr)
f5 = vk1_plus_ll[k] * pe_rr
# Compute term 3 (only needed for NCOMP>1)
vk1_minus_rr = v1_plus_rr - vk1_plus_rr[k]
vk2_minus_rr = v2_plus_rr - vk2_plus_rr[k]
vk3_minus_rr = v3_plus_rr - vk3_plus_rr[k]
f5 += (B2_ll * (vk1_minus_rr * B2_rr - vk2_minus_rr * B1_rr) +
B3_ll * (vk1_minus_rr * B3_rr - vk3_minus_rr * B1_rr))
# Compute Powell term (already consistent with Trixi's non-conservative discretization)
f2 += charge_ratio_ll[k] * B1_ll * B1_rr
f3 += charge_ratio_ll[k] * B2_ll * B1_rr
f4 += charge_ratio_ll[k] * B3_ll * B1_rr
f5 += (v1_plus_ll * B1_ll + v2_plus_ll * B2_ll + v3_plus_ll * B3_ll) * B1_rr
# Compute GLM term for the energy
f5 += v1_plus_ll * psi_ll * psi_rr
# It's not needed to adjust to Trixi's non-conservative form
# Append to the flux vector
set_component!(f, k, 0, f2, f3, f4, f5, equations)
end
# Compute GLM term for psi
f[end] = v1_plus_ll * psi_rr
else #if orientation == 2
# Entries of Powell term for induction equation (already in Trixi's non-conservative form)
f[1] = v1_plus_ll * B2_rr
f[2] = v2_plus_ll * B2_rr
f[3] = v3_plus_ll * B2_rr
for k in eachcomponent(equations)
# Compute term 2 (MHD)
f2 = charge_ratio_ll[k] * (-B2_rr * B1_rr)
f3 = charge_ratio_ll[k] * (-B2_rr * B2_rr + 0.5 * mag_norm_rr + pe_rr)
f4 = charge_ratio_ll[k] * (-B2_rr * B3_rr)
f5 = vk2_plus_ll[k] * pe_rr
# Compute term 3 (only needed for NCOMP>1)
vk1_minus_rr = v1_plus_rr - vk1_plus_rr[k]
vk2_minus_rr = v2_plus_rr - vk2_plus_rr[k]
vk3_minus_rr = v3_plus_rr - vk3_plus_rr[k]
f5 += (B1_ll * (vk2_minus_rr * B1_rr - vk1_minus_rr * B2_rr) +
B3_ll * (vk2_minus_rr * B3_rr - vk3_minus_rr * B2_rr))
# Compute Powell term (already consistent with Trixi's non-conservative discretization)
f2 += charge_ratio_ll[k] * B1_ll * B2_rr
f3 += charge_ratio_ll[k] * B2_ll * B2_rr
f4 += charge_ratio_ll[k] * B3_ll * B2_rr
f5 += (v1_plus_ll * B1_ll + v2_plus_ll * B2_ll + v3_plus_ll * B3_ll) * B2_rr
# Compute GLM term for the energy
f5 += v2_plus_ll * psi_ll * psi_rr
# It's not needed to adjust to Trixi's non-conservative form
# Append to the flux vector
set_component!(f, k, 0, f2, f3, f4, f5, equations)
end
# Compute GLM term for psi
f[end] = v2_plus_ll * psi_rr
end
return SVector(f)
end
"""
flux_ruedaramirez_etal(u_ll, u_rr, orientation, equations::IdealMhdMultiIonEquations2D)
Entropy conserving two-point flux adapted by:
- Rueda-Ramírez et al. (2023)
This flux (together with the MHD non-conservative term) is consistent in the case of one species with the flux of:
- Derigs et al. (2018)
Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field
divergence diminishing ideal magnetohydrodynamics equations for multi-ion
[DOI: 10.1016/j.jcp.2018.03.002](https://doi.org/10.1016/j.jcp.2018.03.002)
"""
function flux_ruedaramirez_etal(u_ll, u_rr, orientation::Integer,
equations::IdealMhdMultiIonEquations2D)
@unpack gammas = equations
# Unpack left and right states to get the magnetic field
B1_ll, B2_ll, B3_ll = magnetic_field(u_ll, equations)
B1_rr, B2_rr, B3_rr = magnetic_field(u_rr, equations)
psi_ll = divergence_cleaning_field(u_ll, equations)
psi_rr = divergence_cleaning_field(u_rr, equations)
v1_plus_ll, v2_plus_ll, v3_plus_ll, vk1_plus_ll, vk2_plus_ll, vk3_plus_ll = charge_averaged_velocities(u_ll,
equations)
v1_plus_rr, v2_plus_rr, v3_plus_rr, vk1_plus_rr, vk2_plus_rr, vk3_plus_rr = charge_averaged_velocities(u_rr,
equations)
f = zero(MVector{nvariables(equations), eltype(u_ll)})
# Compute averages for global variables
v1_plus_avg = 0.5 * (v1_plus_ll + v1_plus_rr)
v2_plus_avg = 0.5 * (v2_plus_ll + v2_plus_rr)
v3_plus_avg = 0.5 * (v3_plus_ll + v3_plus_rr)
B1_avg = 0.5 * (B1_ll + B1_rr)
B2_avg = 0.5 * (B2_ll + B2_rr)
B3_avg = 0.5 * (B3_ll + B3_rr)
mag_norm_ll = B1_ll^2 + B2_ll^2 + B3_ll^2
mag_norm_rr = B1_rr^2 + B2_rr^2 + B3_rr^2
mag_norm_avg = 0.5 * (mag_norm_ll + mag_norm_rr)
psi_avg = 0.5 * (psi_ll + psi_rr)
if orientation == 1
psi_B1_avg = 0.5 * (B1_ll * psi_ll + B1_rr * psi_rr)
# Magnetic field components from f^MHD
f6 = equations.c_h * psi_avg
f7 = v1_plus_avg * B2_avg - v2_plus_avg * B1_avg
f8 = v1_plus_avg * B3_avg - v3_plus_avg * B1_avg
f9 = equations.c_h * B1_avg
# Start building the flux
f[1] = f6
f[2] = f7
f[3] = f8
f[end] = f9
# Iterate over all components
for k in eachcomponent(equations)
# Unpack left and right states
rho_ll, rho_v1_ll, rho_v2_ll, rho_v3_ll, rho_e_ll = get_component(k, u_ll,
equations)
rho_rr, rho_v1_rr, rho_v2_rr, rho_v3_rr, rho_e_rr = get_component(k, u_rr,
equations)
v1_ll = rho_v1_ll / rho_ll
v2_ll = rho_v2_ll / rho_ll
v3_ll = rho_v3_ll / rho_ll
v1_rr = rho_v1_rr / rho_rr
v2_rr = rho_v2_rr / rho_rr
v3_rr = rho_v3_rr / rho_rr
vel_norm_ll = v1_ll^2 + v2_ll^2 + v3_ll^2
vel_norm_rr = v1_rr^2 + v2_rr^2 + v3_rr^2
p_ll = (gammas[k] - 1) *
(rho_e_ll - 0.5 * rho_ll * vel_norm_ll - 0.5 * mag_norm_ll - 0.5 * psi_ll^2)
p_rr = (gammas[k] - 1) *
(rho_e_rr - 0.5 * rho_rr * vel_norm_rr - 0.5 * mag_norm_rr - 0.5 * psi_rr^2)
beta_ll = 0.5 * rho_ll / p_ll
beta_rr = 0.5 * rho_rr / p_rr
# for convenience store vk_plus⋅B
vel_dot_mag_ll = vk1_plus_ll[k] * B1_ll + vk2_plus_ll[k] * B2_ll +
vk3_plus_ll[k] * B3_ll
vel_dot_mag_rr = vk1_plus_rr[k] * B1_rr + vk2_plus_rr[k] * B2_rr +
vk3_plus_rr[k] * B3_rr
# Compute the necessary mean values needed for either direction
rho_avg = 0.5 * (rho_ll + rho_rr)
rho_mean = ln_mean(rho_ll, rho_rr)
beta_mean = ln_mean(beta_ll, beta_rr)
beta_avg = 0.5 * (beta_ll + beta_rr)
p_mean = 0.5 * rho_avg / beta_avg
v1_avg = 0.5 * (v1_ll + v1_rr)
v2_avg = 0.5 * (v2_ll + v2_rr)
v3_avg = 0.5 * (v3_ll + v3_rr)
vel_norm_avg = 0.5 * (vel_norm_ll + vel_norm_rr)
vel_dot_mag_avg = 0.5 * (vel_dot_mag_ll + vel_dot_mag_rr)
vk1_plus_avg = 0.5 * (vk1_plus_ll[k] + vk1_plus_rr[k])
vk2_plus_avg = 0.5 * (vk2_plus_ll[k] + vk2_plus_rr[k])
vk3_plus_avg = 0.5 * (vk3_plus_ll[k] + vk3_plus_rr[k])
# v_minus
vk1_minus_ll = v1_plus_ll - vk1_plus_ll[k]
vk2_minus_ll = v2_plus_ll - vk2_plus_ll[k]
vk3_minus_ll = v3_plus_ll - vk3_plus_ll[k]
vk1_minus_rr = v1_plus_rr - vk1_plus_rr[k]
vk2_minus_rr = v2_plus_rr - vk2_plus_rr[k]
vk3_minus_rr = v3_plus_rr - vk3_plus_rr[k]
vk1_minus_avg = 0.5 * (vk1_minus_ll + vk1_minus_rr)
vk2_minus_avg = 0.5 * (vk2_minus_ll + vk2_minus_rr)
vk3_minus_avg = 0.5 * (vk3_minus_ll + vk3_minus_rr)
# Ignore orientation since it is always "1" in 1D
f1 = rho_mean * v1_avg
f2 = f1 * v1_avg + p_mean
f3 = f1 * v2_avg
f4 = f1 * v3_avg
# total energy flux is complicated and involves the previous eight components
v1_plus_mag_avg = 0.5 * (vk1_plus_ll[k] * mag_norm_ll +
vk1_plus_rr[k] * mag_norm_rr)
# Euler part
f5 = f1 * 0.5 * (1 / (gammas[k] - 1) / beta_mean - vel_norm_avg) +
f2 * v1_avg + f3 * v2_avg + f4 * v3_avg
# MHD part
f5 += (f6 * B1_avg + f7 * B2_avg + f8 * B3_avg - 0.5 * v1_plus_mag_avg +
B1_avg * vel_dot_mag_avg # Same terms as in Derigs (but with v_plus)
+ f9 * psi_avg - equations.c_h * psi_B1_avg # GLM term
+ 0.5 * vk1_plus_avg * mag_norm_avg -
vk1_plus_avg * B1_avg * B1_avg - vk2_plus_avg * B1_avg * B2_avg -
vk3_plus_avg * B1_avg * B3_avg # Additional terms coming from the MHD non-conservative term (momentum eqs)
-
B2_avg * (vk1_minus_avg * B2_avg - vk2_minus_avg * B1_avg) -
B3_avg * (vk1_minus_avg * B3_avg - vk3_minus_avg * B1_avg)) # Terms coming from the non-conservative term 3 (induction equation!)
set_component!(f, k, f1, f2, f3, f4, f5, equations)
end
else #if orientation == 2
psi_B2_avg = 0.5 * (B2_ll * psi_ll + B2_rr * psi_rr)
# Magnetic field components from f^MHD
f6 = v2_plus_avg * B1_avg - v1_plus_avg * B2_avg
f7 = equations.c_h * psi_avg
f8 = v2_plus_avg * B3_avg - v3_plus_avg * B2_avg
f9 = equations.c_h * B2_avg
# Start building the flux
f[1] = f6
f[2] = f7
f[3] = f8
f[end] = f9
# Iterate over all components
for k in eachcomponent(equations)
# Unpack left and right states
rho_ll, rho_v1_ll, rho_v2_ll, rho_v3_ll, rho_e_ll = get_component(k, u_ll,
equations)
rho_rr, rho_v1_rr, rho_v2_rr, rho_v3_rr, rho_e_rr = get_component(k, u_rr,
equations)
v1_ll = rho_v1_ll / rho_ll
v2_ll = rho_v2_ll / rho_ll
v3_ll = rho_v3_ll / rho_ll
v1_rr = rho_v1_rr / rho_rr
v2_rr = rho_v2_rr / rho_rr
v3_rr = rho_v3_rr / rho_rr
vel_norm_ll = v1_ll^2 + v2_ll^2 + v3_ll^2
vel_norm_rr = v1_rr^2 + v2_rr^2 + v3_rr^2
p_ll = (gammas[k] - 1) *
(rho_e_ll - 0.5 * rho_ll * vel_norm_ll - 0.5 * mag_norm_ll - 0.5 * psi_ll^2)
p_rr = (gammas[k] - 1) *
(rho_e_rr - 0.5 * rho_rr * vel_norm_rr - 0.5 * mag_norm_rr - 0.5 * psi_rr^2)
beta_ll = 0.5 * rho_ll / p_ll
beta_rr = 0.5 * rho_rr / p_rr
# for convenience store vk_plus⋅B
vel_dot_mag_ll = vk1_plus_ll[k] * B1_ll + vk2_plus_ll[k] * B2_ll +
vk3_plus_ll[k] * B3_ll
vel_dot_mag_rr = vk1_plus_rr[k] * B1_rr + vk2_plus_rr[k] * B2_rr +
vk3_plus_rr[k] * B3_rr
# Compute the necessary mean values needed for either direction
rho_avg = 0.5 * (rho_ll + rho_rr)
rho_mean = ln_mean(rho_ll, rho_rr)
beta_mean = ln_mean(beta_ll, beta_rr)
beta_avg = 0.5 * (beta_ll + beta_rr)
p_mean = 0.5 * rho_avg / beta_avg
v1_avg = 0.5 * (v1_ll + v1_rr)
v2_avg = 0.5 * (v2_ll + v2_rr)
v3_avg = 0.5 * (v3_ll + v3_rr)
vel_norm_avg = 0.5 * (vel_norm_ll + vel_norm_rr)
vel_dot_mag_avg = 0.5 * (vel_dot_mag_ll + vel_dot_mag_rr)
vk1_plus_avg = 0.5 * (vk1_plus_ll[k] + vk1_plus_rr[k])
vk2_plus_avg = 0.5 * (vk2_plus_ll[k] + vk2_plus_rr[k])
vk3_plus_avg = 0.5 * (vk3_plus_ll[k] + vk3_plus_rr[k])
# v_minus
vk1_minus_ll = v1_plus_ll - vk1_plus_ll[k]
vk2_minus_ll = v2_plus_ll - vk2_plus_ll[k]
vk3_minus_ll = v3_plus_ll - vk3_plus_ll[k]
vk1_minus_rr = v1_plus_rr - vk1_plus_rr[k]
vk2_minus_rr = v2_plus_rr - vk2_plus_rr[k]
vk3_minus_rr = v3_plus_rr - vk3_plus_rr[k]
vk1_minus_avg = 0.5 * (vk1_minus_ll + vk1_minus_rr)
vk2_minus_avg = 0.5 * (vk2_minus_ll + vk2_minus_rr)
vk3_minus_avg = 0.5 * (vk3_minus_ll + vk3_minus_rr)
# Ignore orientation since it is always "1" in 1D
f1 = rho_mean * v2_avg
f2 = f1 * v1_avg
f3 = f1 * v2_avg + p_mean
f4 = f1 * v3_avg
# total energy flux is complicated and involves the previous eight components
v2_plus_mag_avg = 0.5 * (vk2_plus_ll[k] * mag_norm_ll +
vk2_plus_rr[k] * mag_norm_rr)
# Euler part
f5 = f1 * 0.5 * (1 / (gammas[k] - 1) / beta_mean - vel_norm_avg) +
f2 * v1_avg + f3 * v2_avg + f4 * v3_avg
# MHD part
f5 += (f6 * B1_avg + f7 * B2_avg + f8 * B3_avg - 0.5 * v2_plus_mag_avg +
B2_avg * vel_dot_mag_avg # Same terms as in Derigs (but with v_plus)
+ f9 * psi_avg - equations.c_h * psi_B2_avg # GLM term
+ 0.5 * vk2_plus_avg * mag_norm_avg -
vk1_plus_avg * B2_avg * B1_avg - vk2_plus_avg * B2_avg * B2_avg -
vk3_plus_avg * B2_avg * B3_avg # Additional terms coming from the MHD non-conservative term (momentum eqs)
-
B1_avg * (vk2_minus_avg * B1_avg - vk1_minus_avg * B2_avg) -
B3_avg * (vk2_minus_avg * B3_avg - vk3_minus_avg * B2_avg)) # Terms coming from the non-conservative term 3 (induction equation!)
set_component!(f, k, f1, f2, f3, f4, f5, equations)
end
end
return SVector(f)
end
"""
# Calculate maximum wave speed for local Lax-Friedrichs-type dissipation
!!!ATTENTION: This routine is provisional. TODO: Update with the right max_abs_speed
"""
@inline function max_abs_speed_naive(u_ll, u_rr, orientation::Integer,
equations::IdealMhdMultiIonEquations2D)
# Calculate fast magnetoacoustic wave speeds
# left
cf_ll = calc_fast_wavespeed(u_ll, orientation, equations)
# right
cf_rr = calc_fast_wavespeed(u_rr, orientation, equations)
# Calculate velocities
v_ll = zero(eltype(u_ll))
v_rr = zero(eltype(u_rr))
if orientation == 1
for k in eachcomponent(equations)
rho, rho_v1, _ = get_component(k, u_ll, equations)
v_ll = max(v_ll, abs(rho_v1 / rho))
rho, rho_v1, _ = get_component(k, u_rr, equations)
v_rr = max(v_rr, abs(rho_v1 / rho))
end
else #if orientation == 2
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, _ = get_component(k, u_ll, equations)
v_ll = max(v_ll, abs(rho_v2 / rho))
rho, rho_v1, rho_v2, _ = get_component(k, u_rr, equations)
v_rr = max(v_rr, abs(rho_v2 / rho))
end
end
λ_max = max(abs(v_ll), abs(v_rr)) + max(cf_ll, cf_rr)
end
@inline function max_abs_speeds(u, equations::IdealMhdMultiIonEquations2D)
v1 = zero(real(equations))
v2 = zero(real(equations))
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, _ = get_component(k, u, equations)
v1 = max(v1, abs(rho_v1 / rho))
v2 = max(v2, abs(rho_v2 / rho))
end
cf_x_direction = calc_fast_wavespeed(u, 1, equations)
cf_y_direction = calc_fast_wavespeed(u, 2, equations)
return (abs(v1) + cf_x_direction, abs(v2) + cf_y_direction)
end
"""
Convert conservative variables to primitive
"""
function cons2prim(u, equations::IdealMhdMultiIonEquations2D)
@unpack gammas = equations
B1, B2, B3 = magnetic_field(u, equations)
psi = divergence_cleaning_field(u, equations)
prim = zero(MVector{nvariables(equations), eltype(u)})
prim[1] = B1
prim[2] = B2
prim[3] = B3
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, rho_v3, rho_e = get_component(k, u, equations)
srho = 1 / rho
v1 = srho * rho_v1
v2 = srho * rho_v2
v3 = srho * rho_v3
p = (gammas[k] - 1) * (rho_e -
0.5 * (rho_v1 * v1 + rho_v2 * v2 + rho_v3 * v3
+ B1 * B1 + B2 * B2 + B3 * B3
+ psi * psi))
set_component!(prim, k, rho, v1, v2, v3, p, equations)
end
prim[end] = psi
return SVector(prim)
end
"""
Convert conservative variables to entropy
"""
@inline function cons2entropy(u, equations::IdealMhdMultiIonEquations2D)
@unpack gammas = equations
B1, B2, B3 = magnetic_field(u, equations)
psi = divergence_cleaning_field(u, equations)
prim = cons2prim(u, equations)
entropy = zero(MVector{nvariables(equations), eltype(u)})
rho_p_plus = zero(real(equations))
for k in eachcomponent(equations)
rho, v1, v2, v3, p = get_component(k, prim, equations)
s = log(p) - gammas[k] * log(rho)
rho_p = rho / p
w1 = (gammas[k] - s) / (gammas[k] - 1) - 0.5 * rho_p * (v1^2 + v2^2 + v3^2)
w2 = rho_p * v1
w3 = rho_p * v2
w4 = rho_p * v3
w5 = -rho_p
rho_p_plus += rho_p
set_component!(entropy, k, w1, w2, w3, w4, w5, equations)
end
# Additional non-conservative variables
entropy[1] = rho_p_plus * B1
entropy[2] = rho_p_plus * B2
entropy[3] = rho_p_plus * B3
entropy[end] = rho_p_plus * psi
return SVector(entropy)
end
"""
Convert primitive to conservative variables
"""
@inline function prim2cons(prim, equations::IdealMhdMultiIonEquations2D)
@unpack gammas = equations
B1, B2, B3 = magnetic_field(prim, equations)
psi = divergence_cleaning_field(prim, equations)
cons = zero(MVector{nvariables(equations), eltype(prim)})
cons[1] = B1
cons[2] = B2
cons[3] = B3
for k in eachcomponent(equations)
rho, v1, v2, v3, p = get_component(k, prim, equations)
rho_v1 = rho * v1
rho_v2 = rho * v2
rho_v3 = rho * v3
rho_e = p / (gammas[k] - 1.0) +
0.5 * (rho_v1 * v1 + rho_v2 * v2 + rho_v3 * v3) +
0.5 * (B1^2 + B2^2 + B3^2) +
0.5 * psi^2
set_component!(cons, k, rho, rho_v1, rho_v2, rho_v3, rho_e, equations)
end
cons[end] = psi
return SVector(cons)
end
"""
Compute the fastest wave speed for ideal MHD equations: c_f, the fast magnetoacoustic eigenvalue
!!! ATTENTION: This routine is provisional.. Change once the fastest wave speed is known!!
"""
@inline function calc_fast_wavespeed(cons, orientation::Integer,
equations::IdealMhdMultiIonEquations2D)
B1, B2, B3 = magnetic_field(cons, equations)
psi = divergence_cleaning_field(cons, equations)
c_f = zero(real(equations))
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, rho_v3, rho_e = get_component(k, cons, equations)
v1 = rho_v1 / rho
v2 = rho_v2 / rho
v3 = rho_v3 / rho
v_mag = sqrt(v1^2 + v2^2 + v3^2)
gamma = equations.gammas[k]
p = (gamma - 1) * (rho_e - 0.5 * rho * v_mag^2 - 0.5 * (B1^2 + B2^2 + B3^2) - 0.5 * psi^2)
a_square = gamma * p / rho
sqrt_rho = sqrt(rho)
b1 = B1 / sqrt_rho
b2 = B2 / sqrt_rho
b3 = B3 / sqrt_rho
b_square = b1^2 + b2^2 + b3^2
if orientation == 1
c_f = max(c_f,
sqrt(0.5 * (a_square + b_square) +
0.5 * sqrt((a_square + b_square)^2 - 4.0 * a_square * b1^2)))
else #if orientation == 2
c_f = max(c_f,
sqrt(0.5 * (a_square + b_square) +
0.5 * sqrt((a_square + b_square)^2 - 4.0 * a_square * b2^2)))
end
end
return c_f
end
"""
Routine to compute the Charge-averaged velocities:
* v*_plus: Charge-averaged velocity
* vk*_plus: Contribution of each species to the charge-averaged velocity
"""
@inline function charge_averaged_velocities(u, equations::IdealMhdMultiIonEquations2D)
total_electron_charge = zero(real(equations))
vk1_plus = zero(MVector{ncomponents(equations), eltype(u)})
vk2_plus = zero(MVector{ncomponents(equations), eltype(u)})
vk3_plus = zero(MVector{ncomponents(equations), eltype(u)})
for k in eachcomponent(equations)
rho, rho_v1, rho_v2, rho_v3, _ = get_component(k, u,
equations::IdealMhdMultiIonEquations2D)
total_electron_charge += rho * equations.charge_to_mass[k]
vk1_plus[k] = rho_v1 * equations.charge_to_mass[k]
vk2_plus[k] = rho_v2 * equations.charge_to_mass[k]
vk3_plus[k] = rho_v3 * equations.charge_to_mass[k]
end
vk1_plus ./= total_electron_charge
vk2_plus ./= total_electron_charge
vk3_plus ./= total_electron_charge
v1_plus = sum(vk1_plus)
v2_plus = sum(vk2_plus)
v3_plus = sum(vk3_plus)
return v1_plus, v2_plus, v3_plus, SVector(vk1_plus), SVector(vk2_plus),
SVector(vk3_plus)
end
"""
Get the flow variables of component k
"""
@inline function get_component(k, u, equations::IdealMhdMultiIonEquations2D)
# The first 3 entries of u contain the magnetic field. The following entries contain the density, momentum (3 entries), and energy of each component.
return SVector(u[3 + (k - 1) * 5 + 1],
u[3 + (k - 1) * 5 + 2],
u[3 + (k - 1) * 5 + 3],
u[3 + (k - 1) * 5 + 4],
u[3 + (k - 1) * 5 + 5])
end
"""
Set the flow variables of component k
"""
@inline function set_component!(u, k, u1, u2, u3, u4, u5,
equations::IdealMhdMultiIonEquations2D)
# The first 3 entries of u contain the magnetic field. The following entries contain the density, momentum (3 entries), and energy of each component.