This work presents an adaptive parameter scheme which is the finite
difference method with adaptive parameters for enhanced accuracy in
solving one-dimensional advection- diffusion equation, including variable
coefficient of Peclet number and source term. Generalized finite difference formulas containing adaptive parameters for the first-order and
second-order derivatives are introduced into the governing equations to
obtain a finite difference equation and its truncation error. The proposed
method employs a governing equation to differentiate and obtain a new
relative formula to increase the order of truncation error. The truncation
error can thus be derived from any order term and the optimal adaptive
parameter obtained from homogeneous governing equation in finite analysis
method. Finally, three examples are given to demonstrate the validity of this adaptive parameter scheme. The proposed method is more accurate than the weighting function scheme and achieves numerical solutions very close to exact solutions.