An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale

In this paper we present a massively parallel open source solver for Richards equation, named the RichardsFOAM solver. This solver has been developed in the framework of the open source generalist computational fluid dynamics tool box OpenFOAM (R) and is capable to deal with large scale problems in both space and time. The source code for RichardsFOAM may be downloaded from the CPC program library website. It exhibits good parallel performances (up to similar to 90% parallel efficiency with 1024 processors both in strong and weak scaling), and the conditions required for obtaining such performances are analysed and discussed. These performances enable the mechanistic modelling of water fluxes at the scale of experimental watersheds (up to few square kilometres of surface area), and on time scales of decades to a century. Such a solver can be useful in various applications, such as environmental engineering for long term transport of pollutants in soils, water engineering for assessing the impact of land settlement on water resources, or in the study of weathering processes on the watersheds. (C) 2014 Elsevier B.V. All rights reserved.

L-p estimates for the wave equation associated to the Grushin operator

We prove that the solution of the wave equation associated to the Grushin operator G = -Delta -vertical bar x vertical bar(2)partial derivative(2)(t) is bounded on L-P (Rn+1), with 1 < p < infinity, when vertical bar 1/p - 1/2 vertical bar < 1/n+2.

Internal noise-driven generalized Langevin equation from a nonlocal continuum model

Starting with a micropolar formulation, known to account for nonlocal microstructural effects at the continuum level, a generalized Langevin equation (GLE) for a particle, describing the predominant motion of a localized region through a single displacement degree of freedom, is derived. The GLE features a memory-dependent multiplicative or internal noise, which appears upon recognizing that the microrotation variables possess randomness owing to an uncertainty principle. Unlike its classical version, the present GLE qualitatively reproduces the experimentally measured fluctuations in the steady-state mean square displacement of scattering centers in a polyvinyl alcohol slab. The origin of the fluctuations is traced to nonlocal spatial interactions within the continuum, a phenomenon that is ubiquitous across a broad class of response regimes in solids and fluids. This renders the proposed GLE a potentially useful model in such cases.

EXACT INTERNAL CONTROLLABILITY FOR THE WAVE EQUATION IN A DOMAIN WITH OSCILLATING BOUNDARY WITH NEUMANN BOUNDARY CONDITION

In this paper, we study the exact controllability of a second order linear evolution equation in a domain with highly oscillating boundary with homogeneous Neumann boundary condition on the oscillating part of boundary. Our aim is to obtain the exact controllability for the homogenized equation. The limit problem with Neumann condition on the oscillating boundary is different and hence we need to study the exact controllability of this new type of problem. In the process of homogenization, we also study the asymptotic analysis of evolution equation in two setups, namely solution by standard weak formulation and solution by transposition method.

HOMOGENIZATION OF A HYPERBOLIC EQUATION WITH HIGHLY CONTRASTING DIFFUSIVITY COEFFICIENTS

We study a hyperbolic problem in the framework of periodic homogenization assuming a high contrast between the diffusivity coefficients of the two components M-epsilon and B-epsilon of the heterogeneous medium. There are three regimes depending on the ratio between the size of the period and the amplitude a, of the diffusivity in B-epsilon. For the critical regime alpha(epsilon) similar or equal to epsilon, the limit problem is a strongly coupled system involving both the macroscopic and the microscopic variables. We also include the results in the non critical case.

HOMOGENIZATION OF A HYPERBOLIC EQUATION WITH HIGHLY CONTRASTING DIFFUSIVITY COEFFICIENTS

We study a hyperbolic problem in the framework of periodic homogenization assuming a high contrast between the diffusivity coefficients of the two components M-epsilon and B-epsilon of the heterogeneous medium. There are three regimes depending on the ratio between the size of the period and the amplitude a, of the diffusivity in B-epsilon. For the critical regime alpha(epsilon) similar or equal to epsilon, the limit problem is a strongly coupled system involving both the macroscopic and the microscopic variables. We also include the results in the non critical case.