S. Evje et Kh. Karlsen, Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations, NUMER MATH, 86(3), 2000, pp. 377-417
In this paper we present and analyse certain discrete approximations of sol
utions to scalar, doubly nonlinear degenerate, parabolic problems of the fo
rm
partial derivative(t)u + partial derivative(x)f(u) = partial derivative(x)A
(b(u)partial derivative(x)u), u(x, 0) = u(0)(x),
(P)
A(s) = integral(0)(s) a(xi) d xi, a(s) greater than or equal to 0, b(s) gre
ater than or equal to 0,
under the very general structural condition A(+/-infinity) = +/-infinity. T
o mention only a few examples: the heat equation, the porous medium equatio
n, the two-phase flow equation, hyperbolic conservation laws and equations
arising from the theory of non-Newtonian fluids are all special cases of (P
). Since the diffusion terms a(s) and b(s) are allowed to degenerate on int
ervals, shock waves will in general appear in the solutions of (P). Further
more, weak solutions are not uniquely determined by their data. For these r
easons we work within the framework of weak solutions that are of bounded v
ariation (in space and time) and, in addition, satisfy an entropy condition
. The well-posedness of the Cauchy problem (P) in this class of so-called B
V entropy weak solutions follows from a work of Yin [18]. The discrete appr
oximations are shown to converge to the unique BV entropy weak solution of
(P). Mathematics Subject Classification (1991): 65M12, 35K65, 35L65.