Estimate stiff and nonlinear initial value solutions with confidence. Use backward Euler, trapezoidal, and BDF2. Review iterations, errors, charts, and downloadable solution summaries easily.
The page uses one vertical layout. Only the input grid expands across columns.
This calculator solves the initial value problem y' = f(t, y) with y(t₀) = y₀ by forming an implicit equation at each time step and solving it with Newton iteration.
yₙ₊₁ = yₙ + h·f(tₙ₊₁, yₙ₊₁)
The unknown future state appears inside the derivative term. This makes the method very stable for stiff problems, though usually first-order accurate.
yₙ₊₁ = yₙ + (h/2)·[f(tₙ, yₙ) + f(tₙ₊₁, yₙ₊₁)]
This method is second-order accurate and averages the old slope with the unknown future slope. It is often more accurate than Backward Euler.
yₙ₊₁ - (4/3)yₙ + (1/3)yₙ₋₁ = (2h/3)·f(tₙ₊₁, yₙ₊₁)
BDF2 is a second-order multistep implicit method. The first step is automatically started with Backward Euler.
Yₘ₊₁ = Yₘ - G(Yₘ) / G′(Yₘ)
Each implicit step becomes a nonlinear equation G(Y) = 0. The calculator tracks Newton iterations, residual size, and step-to-step change.
Example setup: Linear Decay, Backward Euler, y' = -2y, y(0) = 1, step size = 0.1.
| Step | t | Backward Euler y | Exact y | Abs. Error |
|---|---|---|---|---|
| 0 | 0.000000 | 1.000000 | 1.000000 | 0.000000 |
| 1 | 0.100000 | 0.833333 | 0.818731 | 0.014603 |
| 2 | 0.200000 | 0.694444 | 0.670320 | 0.024124 |
| 3 | 0.300000 | 0.578704 | 0.548812 | 0.029892 |
| 4 | 0.400000 | 0.482253 | 0.449329 | 0.032924 |
It computes the next solution value from an equation that already contains that unknown future value. Implicit methods are usually preferred for stiff or hard-to-stabilize differential equations.
Use Backward Euler when stability matters most. It is robust for stiff decay, chemical kinetics, and rapidly damped systems, even though it is only first-order accurate.
Each implicit step becomes a nonlinear algebraic equation. Newton iteration solves that equation efficiently by using both the residual and its derivative with respect to the unknown state.
The residual shows how well the computed value satisfies the selected implicit formula. Smaller residuals mean the step equation is being solved more accurately.
BDF2 is a two-step method, so it needs two past values. The calculator generates the first new point with Backward Euler before switching to full BDF2 updates.
This version focuses on curated models for reliability and clean derivatives. That design keeps Newton steps stable and avoids unsafe or broken expression parsing.
Start with a moderate step size and a strict tolerance like 1e-8 or 1e-10. Reduce the step size if errors grow or Newton convergence becomes inconsistent.
Exact errors appear only when the selected test problem has a closed-form solution implemented in the page. For models without one, the calculator still reports residuals and iteration data.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.