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Copy pathSROCK_perform_step.jl
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SROCK_perform_step.jl
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@muladd function perform_step!(integrator,cache::SROCK1ConstantCache)
@unpack t,dt,uprev,u,W,p,f = integrator
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
cache.mdeg = Int(floor(sqrt(2*abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t))+1)) # this is the spectral radius estimate to choose optimal stage
choose_deg!(integrator,cache)
mdeg = cache.mdeg
η = cache.optimal_η
ω₀ = 1 + (η/(mdeg^2))
ωSq = (ω₀^2) - 1
Sqrt_ω = sqrt(ωSq)
cosh_inv = log(ω₀ + Sqrt_ω) # arcosh(ω₀)
ω₁ = (Sqrt_ω*cosh(mdeg*cosh_inv))/(mdeg*sinh(mdeg*cosh_inv))
if alg_interpretation(integrator.alg) == :Stratonovich
α = cosh(mdeg*cosh_inv)/(2*ω₀*cosh((mdeg-1)*cosh_inv))
γ = 1/(2*α)
β = -γ
end
uᵢ₋₂ = copy(uprev)
k = integrator.f(uprev,p,t)
Tᵢ₋₂ = oneunit(t)
Tᵢ₋₁ = oftype(t, ω₀)
Tᵢ = Tᵢ₋₁
tᵢ₋₁ = t+dt*(ω₁/ω₀)
tᵢ = tᵢ₋₁
tᵢ₋₂ = t
gₘ₋₁ = zero(k)
gₘ₋₂ = zero(k)
#stage 1
uᵢ₋₁ = uprev + (dt*ω₁/ω₀)*k
for i in 2:mdeg
Tᵢ = 2*ω₀*Tᵢ₋₁ - Tᵢ₋₂
μ = 2*ω₁*(Tᵢ₋₁/Tᵢ)
ν = 2*ω₀*(Tᵢ₋₁/Tᵢ)
κ = (-Tᵢ₋₂/Tᵢ)
k = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
u = dt*μ*k + ν*uᵢ₋₁ + κ*uᵢ₋₂
if (i > mdeg - 2) && alg_interpretation(integrator.alg) == :Stratonovich
if i == mdeg - 1
gₘ₋₂ = integrator.g(uᵢ₋₁,p,tᵢ₋₁)
if typeof(W.dW) <: Number || !is_diagonal_noise(integrator.sol.prob)
u += α*(gₘ₋₂*W.dW)
else
u .+= α .* gₘ₋₂ .* W.dW
end
else
gₘ₋₁ = integrator.g(uᵢ₋₁,p,tᵢ₋₁)
if typeof(W.dW) <: Number || !is_diagonal_noise(integrator.sol.prob)
u += β*(gₘ₋₂*W.dW) + γ*(gₘ₋₁*W.dW)
else
u .+= (β .* gₘ₋₂ .+ γ .* gₘ₋₁) .* W.dW
end
end
elseif (i == mdeg) && alg_interpretation(integrator.alg) == :Ito
if typeof(W.dW) <: Number
gₘ₋₂ = integrator.g(uᵢ₋₁,p,tᵢ₋₁)
uᵢ₋₂ = uᵢ₋₁ + sqrt(abs(dt))*gₘ₋₂
gₘ₋₁ = integrator.g(uᵢ₋₂,p,tᵢ₋₁)
u += gₘ₋₂*W.dW + 1/(2*sqrt(abs(dt)))*(gₘ₋₁ - gₘ₋₂)*(W.dW^2 - abs(dt))
elseif is_diagonal_noise(integrator.sol.prob)
gₘ₋₂ = integrator.g(uᵢ₋₁,p,tᵢ₋₁)
uᵢ₋₂ .= uᵢ₋₁ .+ sqrt(abs(dt)) .* gₘ₋₂
gₘ₋₁ = integrator.g(uᵢ₋₂,p,tᵢ₋₁)
u .+= gₘ₋₂ .* W.dW .+ (1/(2*sqrt(abs(dt)))) .* (gₘ₋₁ .- gₘ₋₂) .* (W.dW .^ 2 .- abs(dt))
else
gₘ₋₂ = integrator.g(uᵢ₋₁,p,tᵢ₋₁)
u += gₘ₋₂*W.dW
end
end
if i < mdeg
tᵢ = μ*dt + ν*tᵢ₋₁ + κ*tᵢ₋₂
uᵢ₋₂ = uᵢ₋₁
uᵢ₋₁ = u
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
Tᵢ₋₂ = Tᵢ₋₁
Tᵢ₋₁ = Tᵢ
end
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::SROCK1Cache)
@unpack uᵢ₋₁,uᵢ₋₂,k, gₘ₋₁, gₘ₋₂ = cache
@unpack t,dt,uprev,u,W,p,f = integrator
ccache = cache.constantcache
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
ccache.mdeg = Int(floor(sqrt(2*abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t))+1)) # this is the spectral radius estimate to choose optimal stage
choose_deg!(integrator,cache)
mdeg = ccache.mdeg
η = ccache.optimal_η
ω₀ = 1 + η/(mdeg^2)
ωSq = ω₀^2 - 1
Sqrt_ω = sqrt(ωSq)
cosh_inv = log(ω₀ + Sqrt_ω) # arcosh(ω₀)
ω₁ = (Sqrt_ω*cosh(mdeg*cosh_inv))/(mdeg*sinh(mdeg*cosh_inv))
if alg_interpretation(integrator.alg) == :Stratonovich
α = cosh(mdeg*cosh_inv)/(2*ω₀*cosh((mdeg-1)*cosh_inv))
γ = 1/(2*α)
β = -γ
end
@.. uᵢ₋₂ = uprev
Tᵢ₋₂ = oneunit(t)
Tᵢ₋₁ = oftype(t,ω₀)
Tᵢ = Tᵢ₋₁
tᵢ₋₁ = t + dt*(ω₁/ω₀)
tᵢ = tᵢ₋₁
tᵢ₋₂ = t
#stage 1
#this take advantage of the fact that cache.k === cache.fsalfirst
#and this has already been done i maxeig! i.e. integrator.f(fsalfirst, uprev, p, t)
# integrator.f(k,uprev,p,t)
@.. uᵢ₋₁ = uprev + (dt*ω₁/ω₀)*k
for i in 2:mdeg
Tᵢ = 2*ω₀*Tᵢ₋₁ - Tᵢ₋₂
μ = 2*ω₁*Tᵢ₋₁/Tᵢ
ν = 2*ω₀*Tᵢ₋₁/Tᵢ
κ = - Tᵢ₋₂/Tᵢ
integrator.f(k,uᵢ₋₁,p,tᵢ₋₁)
@.. u = dt*μ*k + ν*uᵢ₋₁ + κ*uᵢ₋₂
if (i > mdeg - 2) && alg_interpretation(integrator.alg) == :Stratonovich
if i == mdeg - 1
integrator.g(gₘ₋₂,uᵢ₋₁,p,tᵢ₋₁)
if typeof(W.dW) <: Number || is_diagonal_noise(integrator.sol.prob)
@.. u += α*gₘ₋₂*W.dW
else
mul!(k,gₘ₋₂,W.dW)
@.. u += α*k
end
else
integrator.g(gₘ₋₁,uᵢ₋₁,p,tᵢ₋₁)
if typeof(W.dW) <: Number || is_diagonal_noise(integrator.sol.prob)
@.. u += (β*gₘ₋₂ + γ*gₘ₋₁)*W.dW
else
mul!(k,gₘ₋₂,W.dW)
@.. u += β*k
mul!(k,gₘ₋₁,W.dW)
@.. u += γ*k
end
end
elseif (i == mdeg) && alg_interpretation(integrator.alg) == :Ito
if typeof(W.dW) <: Number || is_diagonal_noise(integrator.sol.prob)
integrator.g(gₘ₋₂,uᵢ₋₁,p,tᵢ₋₁)
@.. uᵢ₋₂ = uᵢ₋₁ + sqrt(abs(dt))*gₘ₋₂
integrator.g(gₘ₋₁,uᵢ₋₂,p,tᵢ₋₁)
@.. u += gₘ₋₂*W.dW + 1/(2*sqrt(abs(dt)))*(gₘ₋₁ - gₘ₋₂)*(W.dW^2 - abs(dt))
else
integrator.g(gₘ₋₂,uᵢ₋₁,p,tᵢ₋₁)
mul!(uᵢ₋₁,gₘ₋₂,W.dW)
@.. u += uᵢ₋₁
end
end
if i < mdeg
tᵢ = dt*μ + ν*tᵢ₋₁ + κ*tᵢ₋₂
@.. uᵢ₋₂ = uᵢ₋₁
@.. uᵢ₋₁ = u
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
Tᵢ₋₂ = Tᵢ₋₁
Tᵢ₋₁ = Tᵢ
end
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::SROCK2ConstantCache)
@unpack t,dt,uprev,u,W,p,f = integrator
@unpack recf, recf2, mα, mσ, mτ = cache
gen_prob = !((is_diagonal_noise(integrator.sol.prob)) || (typeof(W.dW) <: Number) || (length(W.dW) == 1))
if gen_prob
vec_χ = similar(W.dW)
init_χ!(vec_χ, W)
end
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
cache.mdeg = Int(floor(sqrt((2*abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)+1.5)/0.811)+1))
cache.mdeg = max(3,min(cache.mdeg,200))-2
choose_deg!(integrator,cache)
mdeg = cache.mdeg
start = cache.start
deg_index = cache.deg_index
α = mα[deg_index]
σ = (1-α)*1//2 + α*mσ[deg_index]
τ = 1//2*((1-α)^2) + 2*α*(1-α)*mσ[deg_index] + (α^2)*(mσ[deg_index]*(mσ[deg_index]+mτ[deg_index]))
sqrt_dt = sqrt(abs(dt))
μ = recf[start] # here κ = 0
tᵢ = t + α*dt*μ
tᵢ₋₁ = tᵢ
tᵢ₋₂ = t
# stage 1
uᵢ₋₂ = uprev
uᵢ = integrator.f(uprev,p,t)
uᵢ₋₁ = uprev + α*dt*μ*uᵢ
# stages 2 upto s-2
for i in 2:mdeg
μ, κ = recf[start + 2*(i-2) + 1], recf[start + 2*(i-2) + 2]
ν = 1 + κ
uᵢ = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
uᵢ = α*dt*μ*uᵢ + ν*uᵢ₋₁ - κ*uᵢ₋₂
uᵢ₋₂ = uᵢ₋₁
uᵢ₋₁ = uᵢ
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
end
#2 stage-finishing procedure
#stage s-1
μ, κ = recf2[(deg_index-1)*4 + 1], recf2[(deg_index-1)*4 + 2]
ν = 1 + κ
uᵢ = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
tₓ = tᵢ₋₁ + 2*dt*τ # So that we don't have to calculate f(uₛ₋₂) again
uₓ = uᵢ₋₁ + 2*dt*τ*uᵢ # uₓ and tₓ represent u_star
u = uᵢ₋₁ + (2*σ - 1//2)*dt*uᵢ
uᵢ = α*dt*μ*uᵢ + ν*uᵢ₋₁ - κ*uᵢ₋₂
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
uᵢ₋₂ = uᵢ₋₁
uᵢ₋₁ = uᵢ
#stage s
μ, κ = recf2[(deg_index-1)*4 + 3], recf2[(deg_index-1)*4 + 4]
ν = 1 + κ
uᵢ = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
uᵢ = α*dt*μ*uᵢ + ν*uᵢ₋₁ - κ*uᵢ₋₂
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
# Now uᵢ₋₂ = uₛ₋₂, uᵢ₋₁ = uₛ₋₁, uᵢ = uₛ
# Similarly tᵢ₋₂ = tₛ₋₂, tᵢ₋₁ = tₛ₋₁, tᵢ = tₛ
if (typeof(W.dW) <: Number) || (length(W.dW) == 1) || is_diagonal_noise(integrator.sol.prob)
Gₛ = integrator.g(uᵢ₋₁,p,tᵢ₋₁)
u += Gₛ .* W.dW
Gₛ = integrator.g(uᵢ,p,tᵢ)
uₓ += Gₛ .* W.dW
uₓ = integrator.f(uₓ,p,tₓ)
u += (1//2*dt)*uₓ
uₓ = 1//2 .* Gₛ .* (W.dW .^ 2 .- abs(dt))
uᵢ₋₂ = uᵢ + uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u += (1//2)*Gₛ₁
uᵢ₋₂ = uᵢ - uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u -= (1//2)*Gₛ₁
uₓ = Gₛ*sqrt_dt
uᵢ₋₂ = uᵢ + uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u += 1//4 .* W.dW .* (Gₛ₁ .- 2 .* Gₛ)
uᵢ₋₂ = uᵢ - uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u += 1//4 .* W.dW .* Gₛ₁
else
Gₛ = integrator.g(uᵢ₋₁,p,tᵢ₋₁)
u += Gₛ*W.dW
Gₛ = integrator.g(uᵢ,p,tᵢ)
uₓ += Gₛ*W.dW
uₓ = integrator.f(uₓ,p,tₓ)
u += (1//2)*dt*uₓ
for i in 1:length(W.dW)
WikJ = W.dW[i]; WikJ2 = vec_χ[i]
WikRange = 1//2 .* (W.dW .* WikJ .- (1:length(W.dW) .== i) .* abs(dt)) #.- (1:length(W.dW) .> i) .* dt .* vec_χ .+ (1:length(W.dW) .< i) .* dt .* WikJ2)
uₓ = Gₛ*WikRange
WikRange = 1//2 .* (1:length(W.dW) .== i)
uᵢ₋₂ = uᵢ + uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u += (Gₛ₁*WikRange)
uᵢ₋₂ = uᵢ - uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u -= (Gₛ₁*WikRange)
end
uₓ = sqrt_dt*(Gₛ*vec_χ)
uᵢ₋₂ = uᵢ + uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u += 1//4*(Gₛ₁*W.dW) - 1//2*(Gₛ*W.dW)
uᵢ₋₂ = uᵢ - uₓ
Gₛ₁ = integrator.g(uᵢ₋₂,p,tᵢ)
u += 1//4*(Gₛ₁*W.dW)
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::SROCK2Cache)
@unpack uᵢ, uₓ, uᵢ₋₁, uᵢ₋₂, k, Gₛ, Gₛ₁, vec_χ, WikRange = cache
@unpack t,dt,uprev,u,W,p,f = integrator
@unpack recf, recf2, mα, mσ, mτ = cache.constantcache
ccache = cache.constantcache
gen_prob = !((is_diagonal_noise(integrator.sol.prob)) || (typeof(W.dW) <: Number) || (length(W.dW) == 1))
if gen_prob
init_χ!(vec_χ, W)
end
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
ccache.mdeg = Int(floor(sqrt((2*abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)+1.5)/0.811)+1))
ccache.mdeg = max(3,min(ccache.mdeg,200))-2
choose_deg!(integrator,cache)
mdeg = ccache.mdeg
start = ccache.start
deg_index = ccache.deg_index
α = mα[deg_index]
σ = (1-α)*1//2 + α*mσ[deg_index]
τ = 1//2*((1-α)^2) + 2*α*(1-α)*mσ[deg_index] + (α^2)*(mσ[deg_index]*(mσ[deg_index]+mτ[deg_index]))
sqrt_dt = sqrt(abs(dt))
μ = recf[start] # here κ = 0
tᵢ = t + α*dt*μ
tᵢ₋₁ = tᵢ
tᵢ₋₂ = t
# stage 1
@.. uᵢ₋₂ = uprev
#this take advantage of the fact that cache.k === cache.fsalfirst
#and this has already been done i maxeig! i.e. integrator.f(fsalfirst, uprev, p, t)
# integrator.f(k,uprev,p,t)
@.. uᵢ₋₁ = uprev + α*dt*μ*k
# stages 2 upto s-2
for i in 2:mdeg
μ, κ = recf[start + 2*(i-2) + 1], recf[start + 2*(i-2) + 2]
ν = 1 + κ
integrator.f(k,uᵢ₋₁,p,t)
@.. uᵢ = α*dt*μ*k + ν*uᵢ₋₁ - κ*uᵢ₋₂
@.. uᵢ₋₂ = uᵢ₋₁
@.. uᵢ₋₁ = uᵢ
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
end
#2 stage-finishing procedure
#stage s-1
μ, κ = recf2[(deg_index-1)*4 + 1], recf2[(deg_index-1)*4 + 2]
ν = 1 + κ
integrator.f(k,uᵢ₋₁,p,tᵢ₋₁)
tₓ = tᵢ₋₁ + 2*dt*τ # So that we don't have to calculate f(uₛ₋₂) again
@.. uₓ = uᵢ₋₁ + 2*dt*τ*k # uₓ and tₓ represent u_star
@.. u = uᵢ₋₁ + (2*σ - 1//2)*dt*k
@.. uᵢ = α*dt*μ*k + ν*uᵢ₋₁ - κ*uᵢ₋₂
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
@.. uᵢ₋₂ = uᵢ₋₁
@.. uᵢ₋₁ = uᵢ
#stage s
μ, κ = recf2[(deg_index-1)*4 + 3], recf2[(deg_index-1)*4 + 4]
ν = 1 + κ
integrator.f(k,uᵢ₋₁,p,tᵢ₋₁)
@.. uᵢ = α*dt*μ*k + ν*uᵢ₋₁ - κ*uᵢ₋₂
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
# Now uᵢ₋₂ = uₛ₋₂, uᵢ₋₁ = uₛ₋₁, uᵢ = uₛ
# Similarly tᵢ₋₂ = tₛ₋₂, tᵢ₋₁ = tₛ₋₁, tᵢ = tₛ
if (typeof(W.dW) <: Number) || (length(W.dW) == 1) || is_diagonal_noise(integrator.sol.prob)
integrator.g(Gₛ,uᵢ₋₁,p,tᵢ₋₁)
@.. u += Gₛ*W.dW
integrator.g(Gₛ,uᵢ,p,tᵢ)
@.. uₓ += Gₛ*W.dW
integrator.f(k,uₓ,p,tₓ)
@.. u += (1//2)*dt*k
@.. uₓ = Gₛ*((W.dW^2 - abs(dt))/2)
@.. uᵢ₋₂ = uᵢ + uₓ
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
@.. u += (1//2)*Gₛ₁
@.. uᵢ₋₂ = uᵢ - uₓ
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
@.. u -= (1//2)*Gₛ₁
@.. uₓ = sqrt_dt*Gₛ
@.. uᵢ₋₂ = uᵢ + uₓ
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
@.. u += (1//4*W.dW)*(Gₛ₁ - 2*Gₛ)
@.. uᵢ₋₂ = uᵢ - uₓ
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
@.. u += (1//4*W.dW)*Gₛ₁
else
integrator.g(Gₛ,uᵢ₋₁,p,tᵢ₋₁)
mul!(uᵢ₋₂,Gₛ,W.dW)
@.. u += uᵢ₋₂
integrator.g(Gₛ,uᵢ,p,tᵢ)
mul!(uᵢ₋₁,Gₛ,W.dW)
@.. uₓ += uᵢ₋₁
integrator.f(k,uₓ,p,tₓ)
@.. u += (1//2)*dt*k
for i in 1:length(W.dW)
WikJ = W.dW[i]; WikJ2 = vec_χ[i]
dwrange = 1:length(W.dW)
abs_dt = abs(dt)
@.. WikRange = 1//2 * (W.dW * WikJ - (dwrange == i) * abs_dt) #+ (dwrange < i) * dt * WikJ2 - (dwrange > i) * dt * vec_χ)
mul!(uₓ,Gₛ,WikRange)
@.. uᵢ₋₂ = uᵢ + uₓ
@.. WikRange = 1//2 * (dwrange == i)
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
mul!(uᵢ₋₂,Gₛ₁,WikRange)
@.. u += uᵢ₋₂
@.. uᵢ₋₂ = uᵢ - uₓ
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
mul!(uᵢ₋₂,Gₛ₁,WikRange)
@.. u -= uᵢ₋₂
end
@.. WikRange = vec_χ*sqrt_dt
mul!(uₓ,Gₛ,WikRange)
@.. uᵢ₋₂ = uᵢ + uₓ
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
mul!(uᵢ₋₂,Gₛ₁,W.dW)
# mul!(uᵢ₋₁,Gₛ,W.dW) This is already been calculated
@.. u += 1//4*(uᵢ₋₂ - 2*uᵢ₋₁)
@.. uᵢ₋₂ = uᵢ - uₓ
integrator.g(Gₛ₁,uᵢ₋₂,p,tᵢ)
mul!(uᵢ₋₂,Gₛ₁,W.dW)
@.. u += 1//4*uᵢ₋₂
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::SROCKEMConstantCache)
@unpack t,dt,uprev,u,W,p,f = integrator
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
if integrator.alg.strong_order_1
cache.mdeg = Int(floor(sqrt(abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)/0.19)+1))
else
cache.mdeg = Int(floor(sqrt(abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)/0.33)+1))
end
cache.mdeg = max(3,min(cache.mdeg,200))
choose_deg!(integrator,cache)
mdeg = cache.mdeg
η = cache.optimal_η
ω₀ = oneunit(t) + (η/(mdeg^2))
ωSq = ω₀^2 - oneunit(t)
Sqrt_ω = sqrt(ωSq)
cosh_inv = log(ω₀ + Sqrt_ω) # arcosh(ω₀)
ω₁ = (Sqrt_ω*cosh(mdeg*cosh_inv))/(mdeg*sinh(mdeg*cosh_inv))
uᵢ₋₂ = copy(uprev)
k = integrator.f(uprev,p,t)
Tᵢ₋₂ = oneunit(t)
Tᵢ₋₁ = oftype(t,ω₀)
Tᵢ = Tᵢ₋₁
tᵢ₋₁ = t+dt*(ω₁/ω₀)
tᵢ = tᵢ₋₁
tᵢ₋₂ = t
#stage 1
uᵢ₋₁ = uprev + dt*(ω₁/ω₀)*k
for i in 2:mdeg
Tᵢ = 2*ω₀*Tᵢ₋₁ - Tᵢ₋₂
μ = 2*ω₁*(Tᵢ₋₁/Tᵢ)
ν = 2*ω₀*(Tᵢ₋₁/Tᵢ)
κ = -Tᵢ₋₂/Tᵢ
k = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
u = @. dt*μ*k + ν*uᵢ₋₁ + κ*uᵢ₋₂
tᵢ = @. μ*dt + ν*tᵢ₋₁ + κ*tᵢ₋₂
if i < mdeg
uᵢ₋₂ = uᵢ₋₁
uᵢ₋₁ = u
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
Tᵢ₋₂ = Tᵢ₋₁
Tᵢ₋₁ = Tᵢ
end
end
Gₛ = integrator.g(u,p,tᵢ)
if (typeof(W.dW) <: Number) || (length(W.dW) == 1) || is_diagonal_noise(integrator.sol.prob)
u += Gₛ .* W.dW
else
u += Gₛ * W.dW
end
if integrator.alg.strong_order_1
if (typeof(W.dW) <: Number) || (length(W.dW) == 1) || (is_diagonal_noise(integrator.sol.prob))
uᵢ₋₂ = @. 1//2 * Gₛ * (W.dW ^ 2 - abs(dt))
tmp = @. u + uᵢ₋₂
Gₛ = integrator.g(tmp,p,tᵢ)
uᵢ₋₁ = @. 1//2*Gₛ
tmp = @. u - uᵢ₋₂
Gₛ = integrator.g(tmp,p,tᵢ)
uᵢ₋₁ = @. uᵢ₋₁ - 1//2*Gₛ
u = @. u + uᵢ₋₁
else
for i in 1:length(W.dW)
WikJ = W.dW[i]
WikRange = 1//2 .* (W.dW .* WikJ - (1:length(W.dW) .== i) .* abs(dt))
uᵢ₋₂ = Gₛ*WikRange
WikRange = 1//2 .* (1:length(W.dW) .== i)
tmp = u + uᵢ₋₂
Gₛ₁ = integrator.g(tmp,p,tᵢ)
uᵢ₋₁ = Gₛ₁*WikRange
tmp = u - uᵢ₋₂
Gₛ₁ = integrator.g(tmp,p,tᵢ)
uᵢ₋₁ -= Gₛ₁*WikRange
u += uᵢ₋₁
end
end
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::SROCKEMCache)
@unpack uᵢ₋₁,uᵢ₋₂,tmp,k,Gₛ,Gₛ₁,WikRange = cache
@unpack t,dt,uprev,u,W,p,f = integrator
ccache = cache.constantcache
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
if integrator.alg.strong_order_1
ccache.mdeg = Int(floor(sqrt(abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)/0.19)+1))
else
ccache.mdeg = Int(floor(sqrt(abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)/0.33)+1))
end
ccache.mdeg = max(3,min(ccache.mdeg,200))
choose_deg!(integrator,cache)
mdeg = ccache.mdeg
η = ccache.optimal_η
ω₀ = oneunit(t) + η/(mdeg^2)
ωSq = ω₀^2 - oneunit(t)
Sqrt_ω = sqrt(ωSq)
cosh_inv = log(ω₀ + Sqrt_ω) # arcosh(ω₀)
ω₁ = (Sqrt_ω*cosh(mdeg*cosh_inv))/(mdeg*sinh(mdeg*cosh_inv))
@.. uᵢ₋₂ = uprev
integrator.f(k,uprev,p,t)
Tᵢ₋₂ = oneunit(t)
Tᵢ₋₁ = convert(eltype(u),ω₀)
Tᵢ = Tᵢ₋₁
tᵢ₋₁ = t + dt*(ω₁/ω₀)
tᵢ = tᵢ₋₁
tᵢ₋₂ = t
#stage 1
@.. uᵢ₋₁ = uprev + (dt*ω₁/ω₀)*k
for i in 2:mdeg
Tᵢ = 2*ω₀*Tᵢ₋₁ - Tᵢ₋₂
μ = 2*ω₁*Tᵢ₋₁/Tᵢ
ν = 2*ω₀*Tᵢ₋₁/Tᵢ
κ = - Tᵢ₋₂/Tᵢ
integrator.f(k,uᵢ₋₁,p,tᵢ₋₁)
@.. u = dt*μ*k + ν*uᵢ₋₁ + κ*uᵢ₋₂
tᵢ = dt*μ + ν*tᵢ₋₁ + κ*tᵢ₋₂
if i < mdeg
@.. uᵢ₋₂ = uᵢ₋₁
@.. uᵢ₋₁ = u
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
Tᵢ₋₂ = Tᵢ₋₁
Tᵢ₋₁ = Tᵢ
end
end
integrator.g(Gₛ,u,p,tᵢ)
if (typeof(W.dW) <: Number) || (length(W.dW) == 1) || is_diagonal_noise(integrator.sol.prob)
@.. u += Gₛ*W.dW
else
mul!(uᵢ₋₁,Gₛ,W.dW)
u += uᵢ₋₁
end
if integrator.alg.strong_order_1
if (typeof(W.dW) <: Number) || (length(W.dW) == 1) || (is_diagonal_noise(integrator.sol.prob))
@.. uᵢ₋₂ = 1//2*Gₛ*(W.dW^2 - abs(dt))
@.. tmp = u + uᵢ₋₂
integrator.g(Gₛ,tmp,p,tᵢ)
@.. uᵢ₋₁ = 1//2*Gₛ
@.. tmp = u - uᵢ₋₂
integrator.g(Gₛ,tmp,p,tᵢ)
@.. uᵢ₋₁ -= 1//2*Gₛ
u += uᵢ₋₁
else
for i in 1:length(W.dW)
WikJ = W.dW[i]
dwrange = 1:length(W.dW)
abs_dt = abs(dt)
@.. WikRange = 1//2 * (W.dW * WikJ - (dwrange == i) * abs_dt)
mul!(uᵢ₋₂,Gₛ,WikRange)
@.. WikRange = 1//2 * (dwrange == i)
@.. tmp = u + uᵢ₋₂
integrator.g(Gₛ₁,tmp,p,tᵢ)
mul!(uᵢ₋₁,Gₛ₁,WikRange)
@.. tmp = u - uᵢ₋₂
integrator.g(Gₛ₁,tmp,p,tᵢ)
@.. u += uᵢ₋₁
mul!(uᵢ₋₁,Gₛ₁,WikRange)
@.. u -= uᵢ₋₁
end
end
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::SKSROCKConstantCache)
@unpack t,dt,uprev,u,W,p,f = integrator
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
η = oftype(t,0.05)
mdeg = Int(floor(sqrt((abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t) + 1.5)/(2-η*4/3))+1))
mdeg = max(3,min(mdeg,200))
ω₀ = 1 + (η/(mdeg^2))
ωSq = (ω₀^2) - 1
Sqrt_ω = sqrt(ωSq)
cosh_inv = log(ω₀ + Sqrt_ω) # arcosh(ω₀)
ω₁ = (Sqrt_ω*cosh(mdeg*cosh_inv))/(mdeg*sinh(mdeg*cosh_inv))
μ, ν , κ = ω₁/ω₀, mdeg*ω₁/2, mdeg*ω₁/ω₀
Tᵢ₋₂ = oneunit(t)
Tᵢ₋₁ = convert(eltype(u),ω₀)
Tᵢ = Tᵢ₋₁
tᵢ₋₁ = t+dt*(ω₁/ω₀)
tᵢ = tᵢ₋₁
tᵢ₋₂ = t
#stage 1
Gₛ = integrator.g(uprev,p,t)
if (typeof(W.dW) <: Number) || !is_diagonal_noise(integrator.sol.prob)
u = Gₛ*W.dW
else
u = Gₛ .* W.dW
end
if integrator.alg.post_processing
uᵢ₋₁ = uprev + ν*u
uᵢ₋₂ = integrator.f(uprev,p,t)
uᵢ₋₁ = integrator.f(uᵢ₋₁,p,t)
uᵢ₋₁ = uprev + (μ*dt)*uᵢ₋₁ + κ*u + cache.mα[mdeg-1]*dt*(uᵢ₋₁ - 2*uᵢ₋₂)
uᵢ₋₂ = uprev - ν*u
uᵢ₋₂ = integrator.f(uᵢ₋₂,p,t)
uᵢ₋₁ += (cache.mα[mdeg-1]*dt)*uᵢ₋₂
else
uᵢ₋₁ = uprev + ν*u
uᵢ₋₂ = integrator.f(uᵢ₋₁,p,t)
uᵢ₋₁ = uprev + (μ*dt)*uᵢ₋₂ + κ*u
end
uᵢ₋₂ = uprev
for i in 2:mdeg
Tᵢ = 2*ω₀*Tᵢ₋₁ - Tᵢ₋₂
μ = 2*ω₁*(Tᵢ₋₁/Tᵢ)
ν = 2*ω₀*(Tᵢ₋₁/Tᵢ)
κ = (-Tᵢ₋₂/Tᵢ)
u = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
u = dt*μ*u + ν*uᵢ₋₁ + κ*uᵢ₋₂
tᵢ = μ*dt + ν*tᵢ₋₁ + κ*tᵢ₋₂
if i < mdeg
uᵢ₋₂ = uᵢ₋₁
uᵢ₋₁ = u
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
Tᵢ₋₂ = Tᵢ₋₁
Tᵢ₋₁ = Tᵢ
end
end
if integrator.alg.post_processing && (t+dt >= integrator.sol.prob.tspan[2])
Gₛ = integrator.g(u,p,tᵢ)
if (typeof(W.dW) <: Number) || is_diagonal_noise(integrator.sol.prob)
uᵢ₋₁ = Gₛ
else
WikRange = 1 .* (1:length(W.dW) .== 1)
uᵢ₋₁ = Gₛ*WikRange
end
winc = rand()*6
if winc < 1
u -= (sqrt(3*dt)*ccache.mc[mdeg-1])*uᵢ₋₁
elseif winc < 2
u += (sqrt(3*dt)*ccache.mc[mdeg-1])*uᵢ₋₁
end
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::SKSROCKCache)
@unpack uᵢ₋₁,uᵢ₋₂,k, Gₛ, WikRange = cache
@unpack t,dt,uprev,u,W,p,f = integrator
ccache = cache.constantcache
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
η = oftype(t,0.05)
mdeg = Int(floor(sqrt((abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t) + 1.5)/(2-η*4/3))+1))
mdeg = max(3,min(mdeg,200))
ω₀ = 1 + (η/(mdeg^2))
ωSq = (ω₀^2) - 1
Sqrt_ω = sqrt(ωSq)
cosh_inv = log(ω₀ + Sqrt_ω) # arcosh(ω₀)
ω₁ = (Sqrt_ω*cosh(mdeg*cosh_inv))/(mdeg*sinh(mdeg*cosh_inv))
μ, ν , κ = ω₁/ω₀, mdeg*ω₁/2, mdeg*ω₁/ω₀
Tᵢ₋₂ = oneunit(t)
Tᵢ₋₁ = convert(eltype(u),ω₀)
Tᵢ = Tᵢ₋₁
tᵢ₋₁ = t+dt*(ω₁/ω₀)
tᵢ = tᵢ₋₁
tᵢ₋₂ = t
#stage 1
integrator.g(Gₛ,uprev,p,t)
if (typeof(W.dW) <: Number) || is_diagonal_noise(integrator.sol.prob)
@.. u = Gₛ*W.dW
else
mul!(u,Gₛ,W.dW)
end
if integrator.alg.post_processing
@.. uᵢ₋₂ = uprev + ν*u
integrator.f(k,uᵢ₋₂,p,t)
@.. uᵢ₋₁ = uprev + (μ*dt)*k + κ*u + (ccache.mα[mdeg-1]*dt)*k
integrator.f(k,uprev,p,t)
@.. uᵢ₋₁ -= (ccache.mα[mdeg-1]*dt*2)*k
@.. uᵢ₋₂ = uprev - ν*u
integrator.f(k,uᵢ₋₂,p,t)
@.. uᵢ₋₁ += (ccache.mα[mdeg-1]*dt)*k
else
@.. uᵢ₋₁ = uprev + ν*u
integrator.f(k,uᵢ₋₁,p,t)
@.. uᵢ₋₁ = uprev + (μ*dt)*k + κ*u
end
@.. uᵢ₋₂ = uprev
for i in 2:mdeg
Tᵢ = 2*ω₀*Tᵢ₋₁ - Tᵢ₋₂
μ = 2*ω₁*(Tᵢ₋₁/Tᵢ)
ν = 2*ω₀*(Tᵢ₋₁/Tᵢ)
κ = (-Tᵢ₋₂/Tᵢ)
integrator.f(k,uᵢ₋₁,p,tᵢ₋₁)
@.. u = dt*μ*k + ν*uᵢ₋₁ + κ*uᵢ₋₂
tᵢ = μ*dt + ν*tᵢ₋₁ + κ*tᵢ₋₂
if i < mdeg
@.. uᵢ₋₂ = uᵢ₋₁
@.. uᵢ₋₁ = u
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
Tᵢ₋₂ = Tᵢ₋₁
Tᵢ₋₁ = Tᵢ
end
end
if integrator.alg.post_processing && (t+dt >= integrator.sol.prob.tspan[2])
integrator.g(Gₛ,u,p,tᵢ)
# println(Gₛ/length(W.dW))
if (typeof(W.dW) <: Number) || is_diagonal_noise(integrator.sol.prob)
@.. uᵢ₋₁ = Gₛ
else
WikRange .= 1 .* (1:length(W.dW) .== 1)
mul!(uᵢ₋₁,Gₛ,WikRange)
end
winc = rand()*6
if winc < 1
@.. u -= (sqrt(3*dt)*ccache.mc[mdeg-1])*uᵢ₋₁
elseif winc < 2
@.. u += (sqrt(3*dt)*ccache.mc[mdeg-1])*uᵢ₋₁
end
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::TangXiaoSROCK2ConstantCache)
@unpack t,dt,uprev,u,W,p,f = integrator
@unpack recf, recf2, mα, mσ, mτ, mn̂, c1, c2 = cache
n̂ = mn̂[integrator.alg.version_num]
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
(integrator.alg.version_num <= 2) && (cache.mdeg = Int(floor(sqrt((abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)+1.5)/0.811)+1)))
(integrator.alg.version_num > 2) && (cache.mdeg = Int(floor(sqrt((abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)+1.5)/0.611)+1)))
cache.mdeg = max(4,min(cache.mdeg,200))-2
choose_deg!(integrator,cache)
mdeg = cache.mdeg
start = cache.start
deg_index = cache.deg_index
α = mα[integrator.alg.version_num]
σ = (1-α)*1//2 + α*mσ[deg_index]
τ = 1//2*((1-α)^2) + 2*α*(1-α)*mσ[deg_index] + (α^2)*(mσ[deg_index]*(mσ[deg_index]+mτ[deg_index]))
η₁ = (rand() < 1//2) ? -1 : 1
η₂ = (rand() < 1//2) ? -1 : 1
sqrt_dt = sqrt(abs(dt))
Û₁ = zero(u)
Û₂ = zero(u)
t̂₁ = t̂₂ = zero(t)
tᵢ = tᵢ₋₁ = tᵢ₋₂ = tₓ = t
uᵢ₋₂ = uprev
for i in 0:mdeg+1
if i == 1
μ = recf[start]
tᵢ = tᵢ₋₁ = t + α*dt*μ
uᵢ = integrator.f(uprev,p,t)
uᵢ₋₁ = uprev + α*dt*μ*uᵢ
elseif i > 1 && i <= mdeg
μ, ν, κ = recf[start + 2*(i-2) + 1], 1 + recf[start + 2*(i-2) + 2], recf[start + 2*(i-2) + 2]
uᵢ = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
uᵢ = α*dt*μ*uᵢ + ν*uᵢ₋₁ - κ*uᵢ₋₂
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
elseif i == mdeg + 1
μ, ν, κ = recf2[(deg_index-1)*4 + 1], 1 + recf2[(deg_index-1)*4 + 2], recf2[(deg_index-1)*4 + 2]
uᵢ = integrator.f(uᵢ₋₁,p,tᵢ₋₁)
tₓ = tᵢ₋₁ + 2*τ*dt
uₓ = uᵢ₋₁ + (2*τ*dt)*uᵢ
u = uᵢ₋₁ + (2*σ - 1//2)*dt*uᵢ
uᵢ = α*dt*μ*uᵢ + ν*uᵢ₋₁ - κ*uᵢ₋₂
tᵢ = α*dt*μ + ν*tᵢ₋₁ - κ*tᵢ₋₂
end
j = i - mdeg - 1 + n̂
if j > 0
j += cache.start_mcs - 1
if i == 0
Û₁ += c1[j]*uprev
t̂₁ += c1[j]*t
Û₂ += c2[j]*uprev
t̂₂ += c2[j]*t
elseif i == 1
Û₁ += c1[j]*uᵢ₋₁
t̂₁ += c1[j]*tᵢ₋₁
Û₂ += c2[j]*uᵢ₋₁
t̂₂ += c2[j]*tᵢ₋₁
else
Û₁ += c1[j]*uᵢ
t̂₁ += c1[j]*tᵢ
Û₂ += c2[j]*uᵢ
t̂₂ += c2[j]*tᵢ
end
end
if i > 1 && i < mdeg + 1
uᵢ₋₂ = uᵢ₋₁
uᵢ₋₁ = uᵢ
tᵢ₋₂ = tᵢ₋₁
tᵢ₋₁ = tᵢ
end
end
if (typeof(W.dW) <: Number) || (length(W.dW) == 1)
Gₛ = integrator.g(Û₁,p,t̂₁)
uₓ += Gₛ*W.dW
uₓ = integrator.f(uₓ,p,tₓ)
u += (1//2*dt)*uₓ + Gₛ*((W.dW^2 - abs(dt))/(η₁*sqrt_dt) - W.dW)
Û₁ -= (η₁*sqrt_dt/2)*Gₛ
Û₂ += (η₁*sqrt_dt/2)*Gₛ
Gₛ = integrator.g(Û₂,p,t̂₂)
u += Gₛ*W.dW
Gₛ = integrator.g(Û₁,p,t̂₁)
u += Gₛ*(W.dW - (W.dW^2 - abs(dt))/(η₁*sqrt_dt))
elseif is_diagonal_noise(integrator.sol.prob)
Gₛ = integrator.g(Û₁,p,t̂₁)
uᵢ₋₁ = Gₛ .* W.dW
Û₁ -= (1//2*η₁*sqrt_dt)*Gₛ
Û₂ += (1//2*η₁*sqrt_dt)*Gₛ
uₓ += uᵢ₋₁
uₓ = integrator.f(uₓ,p,tₓ)
u += (1//2)*dt*uₓ
u .+= Gₛ .* ((W.dW .^ 2 .- abs(dt)) ./ (η₁*sqrt_dt) .- W.dW)
Gₛ = integrator.g(Û₂,p,t̂₂)
u .+= Gₛ .* W.dW
Gₛ = integrator.g(Û₁,p,t̂₁)
u .-= Gₛ .* ((W.dW .^ 2 .- abs(dt)) ./ (η₁*sqrt_dt) .- W.dW)
else
Gₛ = integrator.g(Û₁,p,t̂₁)
for i in 1:length(W.dW)
(i == 1) && (uᵢ₋₁ = @view(Gₛ[:,i])*W.dW[i])
(i != 1) && (uᵢ₋₁ += @view(Gₛ[:,i])*W.dW[i])
end
uₓ += uᵢ₋₁
uₓ = integrator.f(uₓ,p,tₓ)
u += (1//2*dt)*uₓ - uᵢ₋₁
for i in 1:length(W.dW)
uᵢ₋₁ = Û₁ - (1//2*η₁*sqrt_dt)*@view(Gₛ[:,i])
Gₛ₁ = integrator.g(uᵢ₋₁,p,t̂₁)
u += @view(Gₛ₁[:,i])*W.dW[i] + (@view(Gₛ[:,i]) - @view(Gₛ₁[:,i]))*((W.dW[i]^2 - abs(dt))/(η₁*sqrt_dt))
end
for i in 1:length(W.dW)
for j in 1:length(W.dW)
(i > j) && (WikJ = (1//2)*(1+η₂)*W.dW[j])
(i < j) && (WikJ = (1//2)*(1-η₂)*W.dW[j])
(i == j) && (WikJ = (1//2)*(η₁*sqrt_dt))
uᵢ₋₁ += @view(Gₛ[:,j])*WikJ
end
Gₛ₁ = integrator.g(uᵢ₋₁,p,t̂₂)
u += @view(Gₛ₁[:,i])*W.dW[i]
end
end
integrator.u = u
end
@muladd function perform_step!(integrator,cache::TangXiaoSROCK2Cache)
@unpack uᵢ, uₓ, uᵢ₋₁, uᵢ₋₂, Û₁, Û₂, k, Gₛ, Gₛ₁ = cache
@unpack t,dt,uprev,u,W,p,f = integrator
@unpack recf, recf2, mα, mσ, mτ, mn̂, c1, c2 = cache.constantcache
n̂ = mn̂[integrator.alg.version_num]
ccache = cache.constantcache
alg = unwrap_alg(integrator, true)
alg.eigen_est === nothing ? maxeig!(integrator, cache) : alg.eigen_est(integrator)
(integrator.alg.version_num <= 2) && (ccache.mdeg = Int(floor(sqrt((abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)+1.5)/0.811)+1)))
(integrator.alg.version_num > 2) && (ccache.mdeg = Int(floor(sqrt((abs(dt)*integrator.opts.internalnorm(integrator.eigen_est,t)+1.5)/0.611)+1)))
ccache.mdeg = max(4,min(ccache.mdeg,200))-2
choose_deg!(integrator,cache)
mdeg = ccache.mdeg
start = ccache.start
deg_index = ccache.deg_index
α = oftype(t,1.33)
σ = (1-α)*1//2 + α*mσ[deg_index]
τ = 1//2*((1-α)^2) + 2*α*(1-α)*mσ[deg_index] + (α^2)*(mσ[deg_index]*(mσ[deg_index]+mτ[deg_index]))
η₁ = (rand() < 1//2) ? -1 : 1
η₂ = (rand() < 1//2) ? -1 : 1
sqrt_dt = sqrt(abs(dt))
@.. Û₁ = zero(eltype(u))
@.. Û₂ = zero(eltype(u))