Extensively revised edition of Computational Methods in Partial Differential Equations. A more general approach has been adopted for the splitting of operators for parabolic and hyperbolic equations to include Richtmyer and Strang type splittings in addition to alternating direction implicit and locally one dimensional methods. A description of the now standard factorization and SOR/ADI iterative techniques for solving elliptic difference equations has been supplemented with an account or preconditioned conjugate gradient methods which are currently gaining in popularity. Prominence is also given to the Galerkin method using different test and trial functions as a means of constructing difference approximations to both elliptic and time dependent problems. The applications of finite difference methods have been revised and contain examples involving the treatment of singularities in elliptic equations, free and moving boundary problems, as well as modern developments in computational fluid dynamics. Emphasis throughout is on clear exposition of the construction and solution of difference equations. Material is reinforced with theoretical results when appropriate.

Aqueous foam concentrate (AFC) 380 foam was developed by Sandia National Laboratory as a blast mitigation foam for unexploded ordnance (UXO) and its ''engineered foam structure'' is reported to be able to ''envelop chemical or biological aerosols'' [1]. It is similar to commercial fire-fighting foams, consisting mostly of water with small amounts of two alcohols, an ether and surfactant. It also contains xanthan gum, probably, to strengthen the foam film and delay drainage. The concentrate is normally diluted in a 6:94 ratio with water for foaming applications. The diluted solution is normally foamed with air to an expansion factor of about 100 (density 0.01 g/cc), which is called ''dry'' foam. Higher density foam (0.18> {rho}> 0.03 g/cc) was discovered which had quite different characteristics from ''dry'' foam and was called ''wet'' foam. Some characterization of these foams has also been carried out, but the major effort described in this document is the evaluation, at the small and medium scale, of chemical, mechanical and thermal approaches to defoaming AFC 380 foam. Several chemical approaches to defoaming were evaluated including oxidation and precipitation of the xanthan, use of commercial oil-emulsion or suspension defoamers, pH modification, and cation exchange with the surfactant. Of these the commercial defoamers were most effective. Two mechanical approaches to defoaming were evaluated: pressure and foam rupture with very fine particles. Pressure and vacuum techniques were considered too difficult for field applications but high surface area silica particles worked very well on dry foam. Finally simple thermal techniques were evaluated. An order-disorder transition occurs in xanthan solutions at about 60 C, which may be responsible for the effectiveness of hot air as a defoamer. During defoaming of 55 gallons of foam with hot air, after about 70% of the AFC 380 foam had been defoamed, the effectiveness of hot air was dramatically reduced. Approximately 15 gal of residual foam containing mostly small bubbles was resistant to further defoaming by methods that had been effective on the original, dry foam. In this paper the residual foam is referred to as ''wet'' and the original foam is referred to as ''dry''. Methods for generating ''wet'' foam in small to moderate quantities for defoaming experiments have been developed. Methods for defoaming wet foam are currently under study.