# Warning dt\<dtmin in ODE Solver not solved with FAQ

**URL:** https://discourse.julialang.org/t/warning-dt-dtmin-in-ode-solver-not-solved-with-faq/76999
**Category:** General Usage
**Tags:** differentialequation
**Created:** [February 23, 2022, 11:45pm UTC](https://discourse.julialang.org/t/warning-dt-dtmin-in-ode-solver-not-solved-with-faq/76999 "2022-02-23T23:45:57Z")
**Posts on this page:** 4
**Page:** 1

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### Author: ![gnicolosi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gnicolosi/32/48031_2.png) [@gnicolosi](https://discourse.julialang.org/u/gnicolosi)
#### Post date: [February 23, 2022, 11:45pm UTC](https://discourse.julialang.org/t/warning-dt-dtmin-in-ode-solver-not-solved-with-faq/76999/1 "2022-02-23T23:45:57Z")

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Dear all, I have run into the following issue, which is described in details also here: [Frequently Asked Questions · DifferentialEquations.jl (sciml.ai)](https://diffeq.sciml.ai/stable/basics/faq/).

I have tried all suggestions on the link and made a great amount of small modifications to test alternatives. Also, the same exact implementation presents no instability in MATHEMATICA, so I strongly believe the model is correct.

```julia
Warning: dt <= dtmin. Aborting. There is either an error in your model specification or the true solution is unstable.
└ @ SciMLBase C:\Users\18143\.julia\packages\SciMLBase\L7Nun\src\integrator_interface.jl:345

```

Here is the main part of my code relevant to this:

```julia
## 
##Define the parameters of the dynamics
dt_step=0.10
time=collect(dt_step:dt_step:4.0)
initialcond=collect(0.1:0.2:0.9)
N=4; M=4; T=4; L=1; Nsamples=(N+1)*(M+1)

##Problem-related parameters 
β=0.5; ξ=0.5; α=-2; C=2;

### --- Functions --- ###
function uₜₑᵣₘₛ(i,j,t,x₀)
    uₜₑᵣₘₛ = aₘₙ[i,j]*cos((i-1)*π*x₀/L)*cos((j-1)*π*t/T)
end

function uₐₚₚ(t,x₀)
    uₐₚₚ = sum(uₜₑᵣₘₛ(i,j,t,x₀) for i=1:(M+1), j=1:(N+1))
end

### --- More Functions --- ###
using DifferentialEquations

function ϕᵣₑₐₗ(x₀)
    f(ϕ,p,t) = β*ϕ*(1-ϕ)*(uₐₚₚ(t,x₀)-ξ)
    ϕ₀=x₀
    tspan=(0.0,4.0)
    prob=ODEProblem(f,ϕ₀,tspan)
    sol=solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8) #Tested many of these already
    return sol
end

#Numerical Integration of Functional - Trapezoidal Rule
function quad_trap(f,a,b,NN,x₀) 
    h = (b-a)/NN
    int = h * ( f(a,x₀) + f(b,x₀) ) / 2
    for k=1:NN-1
        xk = (b-a) * k/NN + a
        int = int + h*f(xk,x₀)
    end
    return int
end

function RealFunctional(t,x₀)
    α*ϕᵣₑₐₗ(x₀)(t)*uₐₚₚ(t,x₀) + C*uₐₚₚ(t,x₀)^2
end

function RealFunctionalCost(x₀)
    Int = quad_trap(RealFunctional,0,T,size(time)[1],x₀)
    return Int
end

function RealJₐ()
       return sum(RealFunctionalCost(x₀) for x₀=initialcond)
end

#Gradient Descent 
aₘₙ=rand(Uniform(-1,1),(M+1),(N+1)); ϵ=10e-2;
ApproxRealGrad=Matrix{Float64}(undef,(M+1),(N+1))

#Gradient Computation -- I know this is lousy but it needs to be like this
    FunctionalI=RealJₐ()
    println(FunctionalI)
    a_δ=aₘₙ
        for i=1:(M+1), j=1:(N+1)
            aₘₙ=a_δ
            aₘₙ[i,j] = a_δ[i,j] + ϵ
            FunctionalF=RealJₐ()
            ApproxRealGrad[i,j] = (FunctionalF-FunctionalI)/ϵ
        end    
    aₘₙ=a_δ

τ=0.1
    #Descent 
    aₘₙ -= τ*ApproxRealGrad

```

There is a loop in the above omitted here, because I am running iteration by iteration “by hand” in Jupyter.

Also, I know there would be many alternatives to the computation of this derivative, but for our application this is a necessary step. The same step in the descent step worked fine in MATHEMATICA as well.

Please help me, mainly with the ODE `ϕᵣₑₐₗ(x₀)` , which is the source of this.

I spent a lot of time already trying to get out of this warning. Thank you a lot for your time and help.

Gabriel

---

<div class="post-metadata">

### Author: ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [February 24, 2022, 1:18pm UTC](https://discourse.julialang.org/t/warning-dt-dtmin-in-ode-solver-not-solved-with-faq/76999/2 "2022-02-24T13:18:20Z")

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Following:

> [@PSA: How to help yourself debug differential equation solving issues](https://discourse.julialang.org/t/psa-how-to-help-yourself-debug-differential-equation-solving-issues/62489):
>
> Debugging differential equation solver issues almost always boils down to doing the same thing, so this is a summary to help you out. For more information on specific issues, check out the FAQ section of the DifferentialEquations.jl documentation which highlights common issues and questions: [https://diffeq.sciml.ai/dev/basics/faq/](https://diffeq.sciml.ai/dev/basics/faq/)How do I debug why the differential equation solver is diverging? dt \<= dtmin. Aborting. There is either an error in your model specification or the true solution i…

Was the instability with all solvers? Including CVODE and the Fortran methods? Those are the same solvers you can use from Mathematica, so you should get the same results if you choose them.

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<div class="post-metadata">

### Author: ![gnicolosi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gnicolosi/32/48031_2.png) [@gnicolosi](https://discourse.julialang.org/u/gnicolosi)
#### Post date: [February 25, 2022, 11:12pm UTC](https://discourse.julialang.org/t/warning-dt-dtmin-in-ode-solver-not-solved-with-faq/76999/3 "2022-02-25T23:12:05Z")

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Thank you again for your support. I made it work with the Sundials.jl. Now I have exactly the same output as in Mathematica.

Gabriel

---

<div class="post-metadata">

### Author: ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [February 25, 2022, 11:39pm UTC](https://discourse.julialang.org/t/warning-dt-dtmin-in-ode-solver-not-solved-with-faq/76999/4 "2022-02-25T23:39:12Z")

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Could you post an example that can be explored? I’m curious why for example `FBDF()` wouldn’t work if Sundials does. That’s worth fixing if that’s the case.