4th-Order Runge-Kutta (RK4) vs. Euler Method ODE Step Solver Calculator

Compare 4th-order Runge-Kutta (RK4) and forward Euler numerical ODE integration against the exact analytical solution.

RK4 Solution y(2)
2.718 y(x)
Exact Analytical Solution y₀·e^(rx) 2.718 y
Forward Euler Approximation 2.594 y
Euler Global Truncation Error 0.125 |err|
Inputs
⚡ Scenarios:
y₀
Initial value y(0) at x = 0.
r
Coefficient r in dy/dx = r·y.
h
Integration step width h.
steps
Number of integration steps N.
Results & Graph
RK4 Solution y(2)
2.718 y(x)
Exact Analytical Solution y₀·e^(rx)
2.718 y
Forward Euler Approximation
2.594 y
Euler Global Truncation Error
0.125 |err|

RK4 Solution y(2)

Graph 1 • Runge-Kutta (RK4) vs. Euler • Domain Architecture & Live Telemetry Maps Initial Value Y0, Growth Rate Coeff R, Step Size H through Runge-Kutta (RK4) vs. Euler Formula.
RK4 Solution y(2): 2.718 y(x) Runge-Kutta (RK4) vs. Euler • Invariants & Metrics RK4 Solution y(2) 2.718 y(x) Exact Analytical Solution y₀ 2.718 y Forward Euler Approximation 2.594 y Euler Global Truncation Erro 0.125 |err|
Graph 2 • RK4 Solution y(2) Across Initial Value Y0 Sweep (40% → 160%) Evaluates y_{n+1} = y_n + (h / 6) × (k_1 + 2k_2 + 2k_3 + k_4); Euler: y_{n+1} = across 7 operating points (current input highlighted).
0 y(x) 2.39 y(x) 4.78 y(x) 1.09 0.400y₀ 1.63 0.600y₀ 2.17 0.800y₀ 2.72 1y₀ 3.26 1.20y₀ 3.81 1.40y₀ 4.35 1.60y₀
Graph 3 • Joint Sensitivity: Initial Value Y0 × Growth Rate Coeff R Compares RK4 Solution y(2) when Growth Rate Coeff R is shifted +25% (solid) vs. −25% (dashed).
0 y(x) 2.50 y(x) 4.99 y(x) --- Growth Rate Coeff R @ 75% 2.44 -30% 2.97 -15% 3.49 +0% 4.01 +15% 4.54 +30%
Summary: At x = 2 (10 steps of h=0.2), RK4 gives 2.71828 (error 0.000002) vs. Euler 2.593742 (error 0.1245).
Summary: At x = 2 (10 steps of h=0.2), RK4 gives 2.71828 (error 0.000002) vs. Euler 2.593742 (error 0.1245).
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Formula & Mathematical Methodology: Runge-Kutta (RK4) vs. Euler Formula

yn+1 = yn + (h / 6) × (k1 + 2k2 + 2k3 + k4)
Euler: yn+1 = yn + h × f(xn, yn)
y_{n+1} = y_n + (h / 6) × (k_1 + 2k_2 + 2k_3 + k_4); Euler: y_{n+1} = y_n + h × f(x_n, y_n)
  • initialValueY0: Initial Value Y0 (y₀)
  • growthRateCoeffR: Growth Rate Coeff R (r)
  • stepSizeH: Step Size H (h)
  • numberOfStepsN: Number Of Steps N (steps)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Runge-Kutta (RK4) vs. Euler

Compare 4th-order Runge-Kutta (RK4) and forward Euler numerical ODE integration against the exact analytical solution.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: y_{n+1} = y_n + (h / 6) × (k_1 + 2k_2 + 2k_3 + k_4); Euler: y_{n+1} = y_n + h × f(x_n, y_n).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Runge-Kutta (RK4) vs. Euler via y_{n+1} = y_n + (h / 6) × (k_1 + 2k_2 + 2k_3 + k_4); Euler: y_{n+1} = y_n + h × f(x_n, y_n).

Frequently Asked Questions

How does the 4th-Order Runge-Kutta (RK4) vs. Euler Method ODE Step Solver Calculator compute the solution?

It evaluates y_{n+1} = y_n + (h / 6) × (k_1 + 2k_2 + 2k_3 + k_4); Euler: y_{n+1} = y_n + h × f(x_n, y_n) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Numerical Analysis & Calculus

4th-Order Runge-Kutta (RK4) vs. Euler Method ODE Step Solver Calculator: inputs, method, and how to read the result

Compare 4th-order Runge-Kutta (RK4) and forward Euler numerical ODE integration against the exact analytical solution.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Initial Value Y0: Initial value y(0) at x = 0.
  • Growth Rate Coeff R: Coefficient r in dy/dx = r·y.
  • Step Size H: Integration step width h.
  • Number Of Steps N: Number of integration steps N.

What the calculation includes

With the default values shown, RK4 Solution y(2) is 2.718 y(x). The calculation follows Runge-Kutta (RK4) vs. Euler Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Initial Value Y0 from 1 y₀ to 2 y₀ changes RK4 Solution y(2) from 2.718 y(x) to 5.437 y(x). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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