Implicit Euler and @btime report Warning: Instability detected. Aborting

[malinjlu]

using DifferentialEquations
using Plots
using BenchmarkTools

function f1(du,u,p,t)
du[1] = u[2]
du[2] = -u[1]-0.15u[2]
du[3] = u[2]+2u[3]
du[4] = -u[3]+0.8u[4]
du[5] = 0.3u[4]+u[5]
du[6] = u[5]+0.8u[6]
du[7] = -u[6]+u[7]
du[8] = u[7]-u[8]
du[9] = 3u[8]-u[9]
du[10] = u[9]-u[10]
end

function j!(J, x)
for i=1:10
for j=1:10
J[i,j]=0
end
end
J[1,2] = 1
J[2,1] = -1
J[2,2] =-0.15
J[3,2] = 1
J[3,3] = 2
J[4,3] =-1
J[4,4] =0.8
J[5,4] =0.3
J[5,5] =1
J[6,5]=1
J[6,6]=0.8
J[7,6]=-1
J[7,7]=1
J[8,7]=1
J[8,8]=-1
J[9,8]=3
J[9,9]=-1
J[10,9]=1
J[10,10]=-1
end

u0 = [1.0;0.0;0.0;1.0;0.5;-0.3;1.0;0.5;-0.3;-0.1]
tspan = (0.0,10000.0)

f=ODEFunction(f1;jac_prototype=j!)
prob=ODEProblem(f,u0,tspan)

sol= solve(prob,ImplicitEuler(autodiff=false,diff_type=Val{:central}))

print("end solve!!!!")

---

I use Implicit Euler to solve the ODEs, but it shows that

Warning: Instability detected. Aborting
└ @ SciMLBase C:\Users\PC.julia\packages\SciMLBase\QqtZA\src\integrator_interface.jl:525

Interpolation: 3rd order Hermite

when I use @Btime, not the implicit euler.

[ChrisRackauckas]

Yes, the solution blows up to infinity in finite time. It has an analytical solution:

A = zeros(10,10)
j!(A, nothing)
t = 50
exp(A*t)*u0

so it's clearly just a property of the solution (can also just be checked by the unstable eigenvalues).