  Conventional finite-difference schemes for solving partial differential
equations are based on approximating derivatives by finite-differences. In this
work, an alternative theory is proposed which view finite-difference schemes as
systematic ways of matching up to the operator solution of the partial
differential equation. By completely abandon the idea of approximating
derivatives directly, the theory provides a unified description of explicit
finite-difference schemes for solving a general linear partial differential
equation with constant coefficients to any time-marching order. As a result,
the stability of the first-order algorithm for an entire class of linear
equations can be determined all at once. Because the method is based on
solution-matching, it can also be used to derive any order schemes for solving
the general nonlinear advection equation.
