In this section, we introduce some central concepts from object-oriented programming: abstract classes, subclasses, and inheritance. To guide you through these concepts, we consider another mathematical example: one-step methods for solving a differential equation.
The generic form of an ordinary initial value problem is as follows:
The data is the right-hand side function 
, the initial value 
, and the interval of interest 
.
The solution to this problem is a function 
. A numerical algorithm gives this solution as a vector 
of discrete values 
being approximations to 
. Here, 
and 
are discretized values of the independent variable 
, which in physical models often represents time.
A one-step method constructs the solution values 
by the recursion steps:
Here, 
is a step function that characterizes the individual methods, see also [28]:
-
Explicit Euler:
_ -
Midpoint rule:
-
Runge–Kutta 4:
with 
What we did here is the typical way of describing...
