Implicit.jl

using LinearAlgebra
using SparseArrays
using DifferentialEquations
using DiffEqOperators
using BlockBandedMatrices
using DiffEqOperators
using RecursiveFactorization

mutable struct Single_Stack
    L::Float64
    N::Int64
    t::Float64

    ρ::Array{Float64,1}
    w::Array{Float64,1}
    ρθ::Array{Float64,1}

    ρ_res::Array{Float64,1}
    w_res::Array{Float64,1}
    ρθ_res::Array{Float64,1}

    _grav::Float64
    γ::Float64
    _R_m::Float64

    Π::Function

end


function combine!(ρ, w, ρθ, u)
    N = length(ρ)
    u[1:N] .=  ρ
    u[N+1:2N] .= ρθ
    u[2N+1:3N+1] .= w
end

function split!(u, ρ, w, ρθ)

    N = length(ρ)
    ρ      .=  u[1:N]
    ρθ     .=  u[N+1:2N]
    w      .=  u[2N+1:3N+1]
end


function compute_wave_speed(ρ, w, ρθ, _R_m, _C_p, γ, _MSLP)
    p =  ( ρθ * _R_m / _MSLP^(_R_m/_C_p) ).^(γ)
    c = sqrt.(γ*p./ρ)

    return maximum(w) + maximum(c)
end



function DecayingTemperatureProfile(z::Float64; T_virt_surf, T_min_ref, _R_d, _grav, _MSLP)
    # Scale height for surface temperature
    H_sfc = _R_d * T_virt_surf / _grav
    H_t = H_sfc

    z′ = z / H_t
    tanh_z′ = tanh(z′)

    ΔTv =  T_virt_surf -  T_min_ref
    Tv =  T_virt_surf - ΔTv * tanh_z′

    ΔTv′ = ΔTv / T_virt_surf
    p = -H_t * (z′ + ΔTv′ * (log(1 - ΔTv′ * tanh_z′) - log(1 + tanh_z′) + z′))
    p /= H_sfc * (1 - ΔTv′^2)
    p = _MSLP * exp(p)
    ρ = p/(_R_d*Tv)
    return (Tv, p, ρ)
end


function spatial_residual(du, u, ss::Single_Stack, t)

    @info "Tendency computation!!!!" t
    # u is a N+(N-1)+N vector
    N, L = ss.N, ss.L
    Δz = L/N
    _grav = ss._grav
    Π = ss.Π

    ρ, w, ρθ = ss.ρ, ss.w, ss.ρθ
    ρ_res, w_res, ρθ_res = ss.ρ_res, ss.w_res, ss.ρθ_res

    if eltype(u) != eltype(w)
        ρ = similar(ρ, eltype(u))
        w = similar(w, eltype(u))
        ρθ = similar(ρθ, eltype(u))
        ρ_res = similar(ρ_res, eltype(u))
        w_res = similar(w_res, eltype(u))
        ρθ_res = similar(ρθ_res, eltype(u))
    end

    split!(u, ρ, w, ρθ)


    for i = 1:N
        ρ₊ = (i == N ? 0.0 : (ρ[i] + ρ[i+1])/2.0)
        ρ₋ = (i == 1 ? 0.0 : (ρ[i] + ρ[i-1])/2.0)

        ρ_res[i] = -(w[i+1]ρ₊ - w[i]ρ₋)/Δz

        ρθ₊ = (i == N ? 0.0 : (ρθ[i] + ρθ[i+1])/2.0)
        ρθ₋ = (i == 1 ? 0.0 : (ρθ[i] + ρθ[i-1])/2.0)

        ρθ_res[i] = -(w[i+1]ρθ₊ - w[i]ρθ₋)/Δz
    end

    w_res[1] = w_res[N+1] = 0
    for i = 2:N
        w_res[i] = -1/2*(ρθ[i-1]/ρ[i-1] + ρθ[i]/ρ[i]) *(Π(ρθ[i]) - Π(ρθ[i-1]))/Δz - _grav
    end


    combine!(ρ_res, w_res, ρθ_res, du)

    # @info "norm(du) = ", norm(du)

end

function RK4!(f, u0, ss::Single_Stack, T, Δt)
    u = copy(u0)
    k1, k2, k3, k4 = similar(u0), similar(u0), similar(u0), similar(u0)
    Nt = Int64(round(T / Δt))

    t = 0
    for i = 1:Nt
        f(k1, u, ss, t)
        f(k2, u + Δt/2*k1, ss, t+Δt/2)
        f(k3, u + Δt/2*k2, ss, t+Δt/2)
        f(k4, u + Δt*k3, ss, t+Δt)
        u  .+=  Δt/6*(k1  + 2k2  + 2k3  + k4)
        t += Δt
    end
    return u
end


function jacobian!(J,u,ss,t)

    @info "Jacobian computation!!!!" t
    # N cells
    N = div(length(u) - 1 , 3)

    ρ, ρθ, w = u[1:N], u[N+1:2N], u[2N+1:3N+1]

    # construct cell center
    ρh =  [ρ[1] ; (ρ[1:N-1] + ρ[2:N])/2.0;  ρ[N]]
    ρθh = [ρθ[1]; (ρθ[1:N-1] + ρθ[2:N])/2.0; ρθ[N]]


    Πc = Π.(ρθ)
    Πh = [NaN64; (Πc[1:N-1] + Πc[2:N])/2.0;  NaN64]
    Δzh = [NaN64; (Δz[1:N-1] + Δz[2:N])/2.0; NaN64]


    for i = 1:N
        J[i,   i+2N]     =   ρh[i]/Δz[i]
        J[i,   i+2N+1]   =  -ρh[i+1]/Δz[i]
    end

    for i = 1:N
        J[i+N, i+2N]     = ρθh[i]/Δz[i]
        J[i+N, i+2N+1]   = -ρθh[i+1]/Δz[i]
    end

    # 0 for i = 1, N+1
    for i = 2:N
        J[i+2N, (i-1)]     =  -_grav/(2*ρh[i])
        J[i+2N, (i-1)+1]   =  -_grav/(2*ρh[i])

        J[i+2N, (i-1)+N]         =  (γ - 1)*Πh[i]./(ρh[i]*Δzh[i])
        J[i+2N, (i-1)+1+N]       = -(γ - 1)*Πh[i]./(ρh[i]*Δzh[i])
    end


    # D_ρ = diagm(0=>-ρh/Δz, -1=>ρh/Δz)[1:N, 1:N-1]
    # D_Θ = diagm(0=>-ρθh/Δz, -1=>ρθh/Δz)[1:N, 1:N-1]
    # G_W = (γ - 1) * diagm(0=>Πh./ρh/Δz, 1=>-Πh./ρh/Δz)[1:N-1, 1:N]
    # A_W = diagm(0=>-ones(N-1)./ρh/2, 1=>-ones(N-1)./ρh/2)[1:N-1, 1:N]

    # P = ([zeros(N,N)     D_ρ       zeros(N,N);
    #      A_W*_grav       zeros(N-1,N-1)      G_W
    #      zeros(N,N)     D_Θ              zeros(N,N)])
end

function discrete_hydrostatic_balance!(ρ, w, ρθ, Δz::Float64, _grav::Float64, Π::Function)
    for i = 1:length(ρ)-1
        ρ[i+1] = ρθ[i+1]/(-2Δz*_grav/(Π(ρθ[i+1]) - Π(ρθ[i])) - ρθ[i]/ρ[i])
    end
end

function linsolve!(::Type{Val{:init}},f,u0;kwargs...)
    function _linsolve!(x,A,b,update_matrix=false;kwargs...)
        _A = RecursiveFactorization.lu!(A)
      ldiv!(x,_A,b)
    end
end


const DHB = false
const ndays = 2
const N = 30
const L = 30e3
const Δz0 = L/N

const _MSLP = 1e5
const _grav = 9.8
const _R_m = 287.058
const γ = 1.4
const _C_p = _R_m * γ / (γ - 1)
const _C_v = _R_m / (γ - 1)
const _R_d = _R_m


function Π(ρθ)
    ρθ > 0 || return NaN
    _C_p*(_R_d*ρθ/_MSLP)^(_R_m/_C_v)
end
ρ = zeros(Float64, N)
ρθ = zeros(Float64, N)
w = zeros(Float64, N+1)


Δz = zeros(N+1) .+ L/N

for i = 1:N
    z = (i-0.5)*L/N
    Tvᵢ, pᵢ, ρᵢ = DecayingTemperatureProfile(z; T_virt_surf = 280.0, T_min_ref = 230.0, _R_d = _R_d, _grav = _grav, _MSLP = _MSLP)
    ρ[i]  = ρᵢ
    ρθ[i] = ρᵢ*Tvᵢ*(_MSLP/pᵢ)^(_R_m/_C_p)
end

if DHB
    discrete_hydrostatic_balance!(ρ, w, ρθ, Δz0, _grav, Π)
end



ss = Single_Stack(L, N, 0.0, copy(ρ), copy(w), copy(ρθ),copy(ρ), copy(w), copy(ρθ), _grav, γ, _R_m,  Π)
max_wave_speed = compute_wave_speed(ρ, w, ρθ, _R_m, _C_p, γ, _MSLP)

@info "maximum Δt (cfl = 1) = "  Δz0 / max_wave_speed


u0 = zeros(3*N+1)
combine!(ρ, w, ρθ, u0)


Δt = 600.0

jac_prototype = zeros(3N+1, 3N+1)
jacobian!(jac_prototype,u0,ss,0.0)
jac_prototype = sparse(jac_prototype)

f = ODEFunction(spatial_residual, jac=jacobian!); #, jac_prototype = jac_prototype);
prob_ode_hydrostatic_balance = ODEProblem(f,u0,(0.,86400  * ndays),ss);

u1 = solve(prob_ode_hydrostatic_balance, Rosenbrock23(), dt = Δt, reltol=0.1)
u2 = solve(prob_ode_hydrostatic_balance, Rosenbrock23(linsolve=linsolve!), dt = Δt, reltol=0.1)

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[[QuadGK]]
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[[Random123]]
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[[RecursiveFactorization]]
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[[Rmath_jll]]
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[[RuntimeGeneratedFunctions]]
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[[SHA]]
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[[SafeTestsets]]
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[[SciMLBase]]
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version = "1.18.7"

[[Serialization]]
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[[Setfield]]
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[[SharedArrays]]
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[[SimpleTraits]]
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[[Sockets]]
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[[SortingAlgorithms]]
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[[SparseArrays]]
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[[SparseDiffTools]]
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[[SpecialFunctions]]
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[[Statistics]]
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[[StatsAPI]]
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[[StatsBase]]
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[[StatsFuns]]
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[[SteadyStateDiffEq]]
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[[StochasticDiffEq]]
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[[SuiteSparse]]
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[[Sundials_jll]]
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[[SymbolicUtils]]
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[[Symbolics]]
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[[TOML]]
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[[Tar]]
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[[ThreadingUtilities]]
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[[TimerOutputs]]
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[[Tokenize]]
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[[TreeViews]]
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[[TriangularSolve]]
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[[URIs]]
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[[UUIDs]]
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[[UnPack]]
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[[Unicode]]
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[[Unitful]]
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[[VertexSafeGraphs]]
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[[Zlib_jll]]
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[[ZygoteRules]]
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[[nghttp2_jll]]
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[[p7zip_jll]]
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Project.toml

[deps]
BlockBandedMatrices = "ffab5731-97b5-5995-9138-79e8c1846df0"
DiffEqOperators = "9fdde737-9c7f-55bf-ade8-46b3f136cc48"
DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
RecursiveFactorization = "f2c3362d-daeb-58d1-803e-2bc74f2840b4"
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"