What is the Runge-Kutta method?

The Runge-Kutta method

The Runge-Kutta (R-K) technique is an efficient and commonly used approach for solving initial-value problems of differential equations.

It's used to generate high-order accurate numerical methods without the necessity for high-order derivatives of functions. The Runge-Kutta method addresses Euler's method challenge in selecting a sufficiently short step size to provide satisfactory accuracy in problem resolution.

Note: Want to read more about ODEs? Refer here.

Formula

Consider an ordinary differential equation $dy/dx = f(x, y)$ with the initial condition $y(x_0) = y_0$.

The formulae for Runge-Kutta methods are defined as follows.

$1^{st}$ order R-K method

The formula is defined as follows:

This equation is equivalent to Euler's method.

$2^{nd}$ order R-K method

The following series of formulae are involved in calculating $2^{nd}$ order R-K method.

Here,

$3^{rd}$ order R-K method

This level of R-K method comprises the following:

Here,

$4^{th}$ order R-K method

The R-K4 method is the most frequently used R-K method for solving differential equations.

The Runge-Kutta method estimates $y$ for a given position $x$. The local truncation errorIt is the error introduced by a single iteration., in this case, is of the order of $O(h^5)$, whereas the accumulated errorIt is defined as the error introduced by a many iterations. is on the order of $O(h^4)$.

Equations for calculating $4^{th}$ order are stated below:

Here,

• $h$ is the interval size

• $k_1$ is the slope at the beginning using $y$

• $k_2$ is the midpoint slop using $y$and $k_1$

• $k_3$ is again the midpoint slope using $y$ and $k_2$

• $k_4$ is the slope at the end of the interval using $y$ and $k_3$

Example

Find $y(0.1)$ for $y′=\frac{x-y}{4}$ , $y(0) = 1$, with step length $0.1$ using R-K4 method.

Solution

Given the function $y'$, $y(0)=1,h=0.1,y(0.2)= ?$

Step 1: Find $k_1$, $k_2$, $k_3$, and $k_4$using the above-aforementioned formulae for R-K4.

Step 2: Now put the values in $y$ to get $y_{1}$ that leads to the result.

Continue the iterations till the optimized desired outcome is obtained.

Applications

The R-K methods are widely employed in numerous disciplines, primarily in fluid dynamics and mechanics, to optimize fluid solutions. Simulation and games are two further real-world applications of this method.