Solver inputs
Formula used
Stiff problems require implicit timestepping for stability.
How to use this calculator
- Select a model. Use presets to test stiff behavior quickly.
- Choose an implicit method. Start with Backward Euler for stability.
- Set t₀, tₑₙd, h, and the initial state y₀.
- Tune Newton tolerance and max iterations if convergence is hard.
- Press Solve ODE. Results appear below the header.
- Use Download CSV or Download PDF for reporting.
Example data table
Sample output structure for Van der Pol with μ = 1000, using Backward Euler.
| t | y1 | y2 | Comment |
|---|---|---|---|
| 0.0000 | 2.0000 | 0.0000 | Initial condition |
| 0.0010 | 1.9999 | -0.0020 | Fast transient begins |
| 0.0100 | 1.9978 | -0.1940 | Strong damping from stiffness |
| 0.1000 | 1.7800 | -1.9500 | Slow manifold phase |
Professional article: stiff ODE solving for physics workflows
1) Stiffness in real physics models
Stiff ordinary differential equations appear when fast decay and slow evolution coexist. They show up in reaction kinetics, thermal relaxation, and strongly damped motion. A practical warning sign is a time-scale ratio above 103, where explicit schemes need extremely small h for stability.
2) Why implicit methods are preferred
Implicit time integration keeps the scheme stable at larger steps by evaluating the dynamics at the unknown future state. This calculator solves each step with Newton iterations, enabling step sizes that may be orders of magnitude larger than explicit Euler when stiffness dominates.
3) Backward Euler as a robust baseline
Backward Euler is A-stable and strongly damping, which suppresses spurious high-frequency modes. It is the safest first choice when explicit runs blow up or ring. Accuracy is first order, but stability remains strong even for very negative stiffness parameters such as λ ≈ −1000.
4) Trapezoidal rule for higher accuracy
The implicit trapezoidal rule is second order and A-stable, improving accuracy on smooth solutions. It is not strongly damping, so very stiff modes can cause mild ringing. The calculator can compare Trapezoidal against Backward Euler to form an error indicator and suggest smaller steps.
5) BDF2 for smooth stiff dynamics
BDF2 is a two-step method that often performs well once the solution has settled onto a slow manifold. It is widely used in diffusion-like and relaxation-dominated problems. Because it needs a startup step, the solver automatically bootstraps with Backward Euler before switching to BDF2 updates.
6) Newton iterations and Jacobian data
Each implicit step solves a residual equation F(y)=0. Newton’s method often converges in 3–8 iterations when the step is reasonable. The Jacobian is approximated by finite differences, and the resulting linear system is solved with Gaussian elimination to update the state efficiently.
7) Choosing step size and tolerances
For severe stiffness, start with h = 10−3 to 10−2 (model units) and a Newton tolerance near 10−8. If Newton fails, reduce h first. If results look overly damped, increase h gradually or switch from Backward Euler to Trapezoidal or BDF2.
8) Reproducible reporting with exports
The output table records time, state components, and basic diagnostics such as Newton convergence. CSV export supports plotting and post-processing, while the PDF report summarizes method settings, initial conditions, and a compact trajectory table for lab notes and technical reporting.
FAQs
1) What makes an ODE “stiff”?
An ODE is stiff when stable integration requires a much smaller step than accuracy would suggest, usually because fast decaying modes coexist with slow dynamics. This often happens when eigenvalues span several orders of magnitude.
2) Which method should I choose first?
Start with Backward Euler if you want maximum stability. If the solution is smooth and you need better accuracy, try Implicit Trapezoidal. For long runs on smooth stiff behavior, BDF2 can be efficient.
3) Why does Newton fail, and what should I change?
Newton can fail when the step is too large, the initial guess is poor, or the problem is highly nonlinear during a transient. Reduce h, increase max Newton iterations, or relax tolerance slightly, then try again.
4) How do I pick a good time step h?
Begin with a small h during fast transients, then increase gradually once the solution becomes smooth. If your output shows repeated Newton failures or large error estimates, reduce h. If the solution is stable but too damped, increase h.
5) Can I solve vector systems with a custom expression?
Custom expressions are intended for scalar dy/dt = f(t,y). For vector presets, choose Van der Pol or Robertson, which are built-in multi-variable models. For custom multi-variable systems, extend the preset section in the code.
6) What do the CSV and PDF exports include?
CSV includes the computed time series and state components for each accepted step. The PDF report summarizes chosen method, parameters, integration interval, and a compact table of results. Use them for plotting, archiving, and sharing.
7) Are the example table numbers exact?
No. The example table is illustrative and shows the expected structure and qualitative behavior for a stiff run. Use the Solve button to generate the exact trajectory for your chosen model, step size, and tolerances.