In the linear PDE of second order in two variables,
if is formally replaced by , by , by , by and by , then associated with equation (7) is a polynomial of degree two in and
The mathematical nature of the solutions of equation (7) are largely determined by the algebraic properties of the polynomial . In turn, the computational strategy that one selects to numerically solve (7) is strongly influenced by the mathematical nature of the solution. Thus, before embarking on a quantitative analysis of a partial differential equation of the form (7), it is important that a computational scientist have an idea of the qualitative nature of the solution. Much of this qualitative understanding of the solution can be obtained via the following classification scheme. P(a,b) and along with it, the PDE (7) is classified as hyperbolic, parabolic, or elliptic according as its discriminant, , is positive, zero, or negative. Note that the type of equation (7) is determined solely by its principal part (the terms involving the highest-order derivatives of u) and that the type will generally change with position in the xy-plane unless a,b, and c are constants.