Finite difference schemes differ from the other mentioned schemes, because it does not rely on a functional discretization, but only represents the functions used for the solution of the governing equations by a mapping of points and values. Even though this does not lead to a solution function, it can be shown that for many cases that such a method is consistent and convergent and produces adequate solutions. Especially for time-stepping, finite difference schemes are often used in combination with other discretization schemes so that the results obtained by the simulation are defined as continuous functions on single time slices.
3. 4 Finite Difference Schemes
The backward Euler scheme is a time discretization scheme using finite differences for the time discretization and, for instance, finite elements for spatial discretization.
Finite difference schemes is very flexibly employable and a common basis for all operations cannot be defined by introducing a function space and treating the results of the finite difference scheme in the same manner as, e.g., the result of a finite element solution function. Finite difference schemes treat functions as defined by argument-value pairs and form interpolation functions ad-hoc for the special field of application.
The aim of this section is to find a formulation based on the systematics according to Section 2.5 and investigate the differences of the topological base operations.