Explicit Runge–Kutta methods

Runge–Kutta methods are the methods for the numerical solution of the ordinary differential equation (numerical differentiation). The methods start from an initial point and then take a short step toward finding the next solution point. Here you can find online implementation of 11 explicit Runge-Kutta methods listed here, including Forward Euler method, Midpoint method and classic RK4 method.

To use the calculator you should have differential equation in the form and enter the right side of the equation - in the field below.
You also need initial value as and the point for which you want to approximate the value.
The last parameter of a method - a step size- is literally a step to compute a function curve's next approximation. If you know the exact solution, you can enter it as well, and the calculator calculates an absolute error of each method.

You can find a theory below the calculator.

Explicit Runge–Kutta methods

y'

Initial x

Initial y

Point of approximation

Step size

Exact solution (optional)

Calculation precision

Digits after the decimal point: 6

Calculate

Differential equation

 

Exact solution

 

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Explicit Runge–Kutta methods

The general form of explicit Runge-Kutta method is

where

A particular method is specified by providing the integer s (the number of stages), and the coefficients (for 1 ≤ j < i ≤ s), called the Runge-Kutta matrix, (for i = 1, 2, ..., s), called weights, and (for i = 2, 3, ..., s), called nodes. Coefficients are usually arranged in a mnemonic form, known as a Butcher tableau (after John C. Butcher):

Here are some examples of a Butcher tableau with s equals to 1, 2, 3 and 4 respectively:

Forward Euler method

Explicit midpoint method

Third-order Strong Stability Preserving Runge-Kutta (SSPRK3)

RK4 method