# Explicit Runge–Kutta methods

This online calculator implements several explicit Runge-Kutta methods so you can compare how they solve first degree differential equation with a given initial value.

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 forward to find 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 next approximation of a function curve. If you know the exact solution, you can enter it as well, and the calculator calculates an absolute error of each method.

Some theory can be found below the calculator.

### 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

*local_offer*#differentiation #Runge-Kutta 3/8-rule fourth-order method Classic fourth-order method differentiation Explicit midpoint method Forward Euler Heun's method Heun's third-order method Kutta's third-order method Math Numerical differential equations Ralston's fourth-order method Ralston's method Ralston's third-order method Third-order Strong Stability Preserving Runge-Kutta (SSPRK3)

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