Calculator Inputs
Example Data Table
Example setup: y' = -2y, t₀ = 0, y₀ = 1, h = 0.1. The implicit update is yₙ₊₁ = yₙ / 1.2.
| Step | t | Approx y | Increment |
|---|---|---|---|
| 0 | 0.000000 | 1.000000 | 0.000000 |
| 1 | 0.100000 | 0.833333 | -0.166667 |
| 2 | 0.200000 | 0.694444 | -0.138889 |
| 3 | 0.300000 | 0.578704 | -0.115741 |
| 4 | 0.400000 | 0.482253 | -0.096451 |
Formula Used
yn+1 = yn + h f(tn+1, yn+1)
For
y' = a y, the step becomes yn+1 = yn / (1 - h a).
For
y' = a y + b t + c, use yn+1 = (yn + h(b tn+1 + c)) / (1 - h a).
The implicit step yields
(h r / K) yn+12 + (1 - h r) yn+1 - yn = 0. The calculator selects the practical root.
For
y' = a y² + b y + c, define g(z) = z - yn - h(a z² + b z + c). Newton's update is zk+1 = zk - g(zk) / g'(zk) with g'(z) = 1 - h(2 a z + b).
How to Use This Calculator
- Select the differential equation model that matches your problem.
- Enter the initial time, initial value, step size, and number of steps.
- Provide model parameters such as growth rate, carrying capacity, or forcing constants.
- Set the Newton tolerance and iteration limit for nonlinear problems.
- Press Submit to display the summary, plot, and step table above the form.
- Download the generated table as CSV or PDF when you need a report.
FAQs
1) What does the implicit Euler method do?
It advances an ordinary differential equation by evaluating the slope at the unknown next point. That makes each step more stable than explicit Euler for many stiff problems.
2) Why is it useful for stiff equations?
Implicit Euler is A-stable for the linear test equation. Because of that, it often tolerates larger step sizes without exploding when rapidly decaying modes are present.
3) Why are some models solved without iteration?
Certain equations turn the implicit step into an algebraic formula or quadratic equation. In those cases, the calculator uses the direct relation because it is faster and more reliable.
4) What does the residual column mean?
The residual measures how closely a computed value satisfies the implicit step equation. Smaller residuals indicate a more accurate algebraic solve at each time level.
5) When should I reduce the step size?
Reduce the step when you need better accuracy, when Newton iterations struggle, or when the graph loses important dynamics. Stability alone does not guarantee precise answers.
6) Does the plot always show an exact comparison curve?
No. Exact curves are only added for the models with closed-form analytical solutions. For other models, the graph focuses on the implicit approximation alone.
7) What if my denominator becomes zero?
A near-zero denominator means the chosen step and parameters create a singular update. The calculator stops and asks you to change the step size or coefficients.
8) Can I use this as a symbolic solver?
No. This page is a numerical stepper. It approximates solution values on a grid, summarizes errors where possible, and exports the computed table.