# 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.

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#### Timur

Criado: 2019-09-24 10:36:07, Ultima atualização: 2021-02-26 13:26:40

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 $y \prime = f(x,y)$ and enter the right side of the equation - $f(x,y)$ in the $y \prime$ field below.
You also need initial value as $y(x_0)=y_0$ and the point $x$ for which you want to approximate the $y$ 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

Digits after the decimal point: 6
Differential equation

Exact solution

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### Explicit Runge–Kutta methods

The general form of explicit Runge-Kutta method is
$y_{n+1}=y_n+h \sum_{i=1}^s b_i k_i$
where
$k_i=f(x_n+c_i h, y_n+h\sum_{j=1}^{i-1}a_{ij}k_j)$

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

$\begin{array}{c|cccc}c_1 & a_{11} & a_{12}& \dots & a_{1s}\\c_2 & a_{21} & a_{22}& \dots & a_{2s}\\\vdots & \vdots & \vdots& \ddots& \vdots\\c_s & a_{s1} & a_{s2}& \dots & a_{ss} \\\hline& b_1 & b_2 & \dots & b_s\\\end{array}$

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

#### Forward Euler method

$\begin{array}{c|c}0 & 0 \\\hline& 1 \\\end{array}$

#### Explicit midpoint method

$\begin{array}{c|cc}0 & 0 & 0 \\1/2 & 1/2 & 0 \\\hline& 0 & 1 \\\end{array}$

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

${\displaystyle {\begin{array}{c|ccc}0&0&0&0\\1&1&0&0\\1/2&1/4&1/4&0\\\hline &1/6&1/6&2/3\\\end{array}}}$

#### RK4 method

$\begin{array}{c|cccc}0 & 0 & 0 & 0 & 0\\1/2 & 1/2 & 0 & 0 & 0\\1/2 & 0 & 1/2 & 0 & 0\\1 & 0 & 0 & 1 & 0\\\hline& 1/6 & 1/3 & 1/3 & 1/6\\\end{array}$
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#### Calculadoras similares

PLANETCALC, Explicit Runge–Kutta methods