An average Reynolds equation for predicting the effects of deterministic periodic roughness, taking Jakobsson, Floberg, and Olsson mass flow preserving cavitation model into account, is introduced based upon the double scale analysis approach. This average Reynolds equation can be used both for a microscopic interasperity cavitation and a macroscopic one. The validity of such a model is verified by numerical experiments both for one-dimensional and two-dimensional roughness patterns.

This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.

In this paper a higher order characteristics time discretization scheme is analyzed for a variable coefficient convection‐(possibly degenerate)diffusion‐reaction equation with mixed Dirichlet–Robin boundary conditions. First, the proposed second order time discretization scheme is rigorously introduced for exact and approximate characteristics. Next, under not very restrictive hypotheses on the data, the $l^{\infty}(L^2)$ stability is proved and $l^{\infty}(L^2)$ error estimates of order $O(\Delta t^2)$ are obtained. Lagrange–Galerkin schemes using different finite elements spaces will be analyzed in the second part of this work [to appear in SIAM J. Numer. Anal.], where quadrature formulas are proposed for practical implementation.