## First Order Runge - Kutte Methods

The Runge - Kutta method uses a modification of the Euler method to find a solution to the initial value problem

The local truncation error can be reduced by using a more accurate method for estimating

the integral than Euler's e.g., the trapezium rule:

gives a local truncation error of order 3, but it cannot be used directly as we do not knowWe can estimateusing Euler's formulathen

leading to the new scheme

We can estimate the local truncation error of this scheme, using the Taylor series expansion

Substitutingand

Hence(1)

This is only one example of the Runge – Kutta method. The general 2nd order Runge -Kutta scheme takes the form

Repeating the earlier analysis, we see that

So

Comparing this expression with (1) we get

Since we have 3 equations and 4 unknowns, there are infinitely many solutions. The most popular are:

Modied Euler:

Midpoint method:

Heun's method: