Numerical solution of ordinary differential equations
In ordinary differential equations the unknown function depends only on one variable. Therefore only ordinary derivatives of the function in this one variable can occur. The order of the differential equation is the highest occurring derivative. One differentiates between explicit and implicit differential equations, depending on whether you can solve the equation for the highest occurring derivative or not.
In application, often the time is the variable. Thus, the differential equation describing the changing behavior of the required quantities.
Only for a few differential equations exist explicit algorithms to solve. For many solutions is not even an explicit solution representation possible so that here much resorting to numerical approximation.
This is the simplest form of an explicit step process. In each step, the change predetermined by the differential equation is determined and with their help the next step is calculated. Figuratively speaking, the change rule is integrated in each step using the left-hand rectangle rule. The explicit Euler method can also be considered as Runge-Kutta methods of 1st order.
The method of Heun is a simple from the class of Runge-Kutta methods. The differential equation is now repeatedly evaluated in each step, namely, at the current location as well as foreseen by the forward Euler method next point. Both pieces of information are averaged and are included in the next step. Figuratively speaking, the change rule is integrated in each step using the trapezoidal rule. It is a 2-stage explicit Runge-Kutta method of 2nd order.
Another Runge-Kutta method of 2nd order. The differential equation is now repeatedly evaluated in each step, namely, at the current location as well as foreseen by the forward Euler method next point. In the next step, however, only the last evaluation is used (in contrast to the method of Heun). It is a 2-stage explicit Runge-Kutta methods of 2nd order.
The differential equation will now be evaluated several times in each step, namely, at the current position, an intermediate step and at the next location. All three pieces of information are weighted and are included in the next step. Figuratively speaking, the change rule is integrated in each step using Simpson's rule. It is a 3-stage explicit Runge-Kutta methods of 3rd order.
The differential equation will now be evaluated several times in each step, namely, at the current position, twice at an intermediate step and at the next location. All the four information are weighted and are included in the next step. It is a 4-stage explicit Runge-Kutta methods of 4th order.