For any partial differential equation, initial and/or boundary conditions requirements depend on the direction of information propagation and the domain of dependence. Second-order partial differential equations are classified as elliptic, parabolic or hyperbolic. In elliptic equations, information propagates in all directions simultaneously. The solution domain is, therefore, a closed domain and the resulting solution is always smooth. In contrast, in hyperbolic equations, information propagates along characteristic directions with finite speed. These equations can, therefore, be solved using marching techniques. The solution may contain discontinuities (like shocks) because of the non-dissipative nature of these equations. For parabolic equations, the solution domain is open, but always yields a smooth solution due to its dissipative nature. A general, second-order partial differential equation in N independent variables (x1, x2,..xN) can be written:

This equation can be classified on the basis of eigenvalues, X, of a matrix with entries Ajk (Fletcher, 1991). The eigenvalues (X) are roots of the following equation:

where I is a unit vector. If any eigenvalue is zero, the equation is parabolic. If all eigenvalues are non-zero and are of the same sign, the equation is elliptic. If all eigenvalues are non-zero and all but one are of the same sign, the equation is hyperbolic. It should be noted that the coefficients Ajk might be functions of the dependent variable <p or independent variables. In such a case, the same equation may be locally parabolic, elliptic and hyperbolic depending upon the local conditions. In many cases, equations governing complex flow processes exhibit mixed properties and are difficult to formally classify under any particular type. The ideas discussed above should be kept in mind when formulating appropriate boundary conditions and when selecting appropriate numerical methods to solve the model equations. Commonly used boundary conditions are discussed below.

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