  A systematic digital-discrete method for obtaining continuous functions with
smoothness to a certain order (C^(n)) from sample data is designed. This method
is based on gradually varied functions and the classical finite difference
method. This new method has been applied to real groundwater data and the
results have validated the method. This method is independent from existing
popular methods such as the cubic spline method and the finite element method.
The new digital-discrete method has considerable advantages for a large number
of real data applications. This digital method also differs from other
classical discrete methods that usually use triangulations. This method can
potentially be used to obtain smooth functions such as polynomials through its
derivatives f^(k) and the solution for partial differential equations such as
harmonic and other important equations.
