# What is a numerical solution

At**ordinary differential equations** the function you are looking for depends only on a variable. Ordinary derivatives of the function can therefore occur in this one variable. The order of the differential equation corresponds to the highest occurring derivative.

A distinction is made between explicit and implicit differential equations, depending on whether or not the equation can be solved for the highest derivative that occurs.

In practice, the variable is often time.

The differential equation describes the change in behavior of the quantities sought in relation to one another.

Explicit algorithms for the solution exist only for a few differential equations. For many solutions, even an explicit solution representation is not possible, so that a lot of numerical approximation is used here.

**Explicit Euler method**

This is the simplest form of an explicit one-step procedure. In each step, the change given by the differential equation is determined and the next step is determined with its help. In clear terms, the change rule is integrated in every step with the help of the rectangle rule on the left. The explicit Euler method can also be viewed as a Runge-Kutta method of order 1.

**Heun procedure**

Heun's method is a simple one from the class of the Runge-Kutta methods. The differential equation is now evaluated several times in each step, namely at the current point and at the next point provided by the explicit Euler method. Both pieces of information are averaged and are used in the next step. In clear terms, the change specification is integrated in every step with the help of the trapezoid rule. It is a 2-stage explicit Runge-Kutta method of the 2nd order.

**2nd order Runge-Kutta method**

Another 2nd order Runge-Kutta method. The differential equation is now evaluated several times in each step, namely at the current point and at the next point provided by the explicit Euler method. In the next step, however, only the last evaluation is included (in contrast to Heun's method). It is a 2-stage explicit Runge-Kutta method of the 2nd order.

**3rd order Runge-Kutta method**

The differential equation is now evaluated several times in each step, namely at the current point, an intermediate step and at the next point. All three pieces of information are weighted and are used in the next step. In clear terms, the change specification is integrated in every step with the help of the Simpson rule. It is a 3-stage explicit Runge-Kutta method of the 3rd order.

**Classical 4th order Runge-Kutta method**

The differential equation is now evaluated several times in each step, namely at the current point, twice at an intermediate step and at the next point. All of the four pieces of information are weighted and are used in the next step. It is a 4-stage explicit Runge-Kutta method of the 4th order.

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