Discrete approximations of \(BV\) solutions to doubly nonlinear degenerate parabolic equations. (English) Zbl 0963.65094
The authors consider the differential equation
\[
{\partial u\over\partial t}+{\partial\over\partial x} \{f(u)\}= {\partial\over\partial x} \Biggl\{A \Biggl(b(u){\partial u\over\partial x}\Biggr)\Biggr\}, A(s)= \int^s_0 a(\zeta) d\zeta, a(s),b(s)\geq 0, u(x,0)= u_0(x).
\]
They discuss the possibility of shocks arising and the associated entropy condition. Discrete approximations involving three-point implicit difference schemes are introduced and regularity estimates are obtained. It is shown that the schemes suggested give approximate solutions which converge to a weak solution.
Reviewer: Ll.G.Chambers (Bangor)
MSC:
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
35K65 | Degenerate parabolic equations |
35L65 | Hyperbolic conservation laws |
76L05 | Shock waves and blast waves in fluid mechanics |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |