Showing papers by "Stanley Osher published in 1976"

Hyperbolic equations in regions with characteristic boundaries or with corners

Abstract: Publisher Summary This chapter discusses hyperbolic equations in regions with characteristic boundaries or with corners. In the past decade, a general theory of hyperbolic mixed problems for first order systems has been developed. A parallel theory has been partially worked out for the more complicated finite difference analogue. Unfortunately, the hypotheses required by this theory have been restrictive on two counts: (1) the boundary must be smooth and (2) the boundary must be noncharacteristic. The chapter reviews the theory in the smooth boundary noncharacteristic case, describes the results for noncharacteristic corner problems, and presents several examples and counter­examples. It also discusses the difficulties in the nonuniformly characteristic smooth boundary case and also describes some work in progress with A. Majda on the shallow water equations. These results are contrasted with those of the analogous elliptic and parabolic boundary value problems and their numerical analogues.