Convergence of Finite Volume Scheme for Degenerate Parabolic Problem with Zero Flux Boundary Condition

Abstract

This note is devoted to the study of the finite volume methods used in the discretization of degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The notion of an entropy-process solution, successfully used for the Dirichlet problem, is insufficient to obtain a uniqueness and convergence result because of a lack of regularity of solutions on the boundary. We infer the uniqueness of an entropy-process solution using the tool of the nonlinear semigroup theory by passing to the new abstract notion of integral-process solution. Then, we prove that numerical solution converges to the unique entropy solution as the mesh size tends to 0.

Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions

“…In [14] Gazibo proposes and analyzes an implicit finite volume scheme for (6.1). Andreianov and Gazibo prove convergence of that scheme to an entropy solution in [2,14].…”

“…These authors consider conservation laws with a general dissipative boundary condition, which includes as particular cases the Dirichlet, Neumann (flux), Robin, and obstacle boundary conditions. Well-posedness results for degenerate parabolic problems have been provided by Andreianov and Gazibo [1,2].…”

“…To our knowledge, there are no previous published results on the subject of convergence of finite difference schemes for the zero-flux boundary problem (1.1), even for the important onedimensional case. However, there is a recent convergence result for an implicit finite volume method for degenerate parabolic equations (with zero flux condition), due to Andreianov and Gazibo [2]. The convergence proof in [2] is rather involved, relying on sophisticated energy (weak BV) estimates and nonlinear weak convergence techniques.…”

“…However, there is a recent convergence result for an implicit finite volume method for degenerate parabolic equations (with zero flux condition), due to Andreianov and Gazibo [2]. The convergence proof in [2] is rather involved, relying on sophisticated energy (weak BV) estimates and nonlinear weak convergence techniques. Our proof, on the other hand, is short and elementary, relying on BV estimates obtained through a slight modification of the total variation functional (the only intricate trick involved).…”

Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

“…Further, when (2) fails the question of what is the correct definition of solutions to the zero-flux problem remains open (cf. [7,8] for the purely hyperbolic case); it is demonstrated numerically in [22,6] that the formulation of [13,5] is not appropriate in absence of (2).…”