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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] =- 1)*Πh[i]./(ρh[i]*Δzh[i])
J[i+2N, (i-1)+1+N] = -- 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 /- 1)
const _C_v = _R_m /- 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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