Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure

In this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem; this yields the convergence of a subsequence to a weak solution of the parabolic system.     

A Partial Differential Inequality in Geological Models

Sedimentation and erosion processes in sedimentary basins can be modeled by a parabolic equation with a limiter on the fluxes and a constraint on the time variation.This limiter happens to satisfy a stationary scalar hyperbolic inequality,within a constraint,for which the authors prove the existence and the uniqueness of the solution.Actually,this solution is shown to be the maximal element of a convenient convex set of functions.The existence proof is obtained thanks to the use of a numerical scheme.     

Nucleotide Analogues Bearing a C2′ or C3′-Stereogenic All-Carbon Quaternary Center as SARS-CoV-2 RdRp Inhibitors

The design of novel nucleoside triphosphate (NTP) analogues bearing an all-carbon quaternary center at C2′ or C3′ is described. The construction of this all-carbon stereogenic center involves the use of an intramoleculer photoredox-catalyzed reaction. The nucleoside analogues (NA) hydroxyl functional group at C2′ was generated by diastereoselective epoxidation. In addition, highly enantioselective and diastereoselective Mukaiyama aldol reactions, diastereoselective N-glycosylations and regioselective triphosphorylation reactions were employed to synthesize the novel NTPs. Two of these compounds are inhibitors of the RNA-dependent RNA polymerase (RdRp) of SARS-CoV-2, the causal virus of COVID-19.     

Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data

We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time step and the ratio of the space step to the time step tend to zero. We then consider a second finite volume scheme, obtained from the first one by addition of some uniform numerical viscosity. We prove that this scheme converges to the continuous solution, as the space step tends to zero, whereas the ratio of the space step to the time step remains bounded by below and by above, and under assumption of uniform regularity of the mesh. This is obtained via an improved discrete Sobolev inequality and a sharp weak BV estimate, under some additional assumptions on the transport flow. Examples show the optimality of these assumptions.     

A vanishing diffusion process in unsaturated soils

A model for simulating water flow and air flow in unsaturated soils is presented herein. Drainage from a one-dimensional soil column is specifically investigated. A mathematical analysis of the equations reveals that, due to a vanishing diffusion process, the gradient of the water degree of saturation is infinite at the water table. Moreover, this discontinuity propagates at a velocity that satisfies a local non-linear equation involving only the properties of the material. A numerical simulation of this problem serves to confirm the results obtained.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems

In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation of the solution of the nonlinear partial differential equation u_t+div(qf(u))−ΔΦ(u)=0 in a 1D, 2D or 3D domain. The function Φ is supposed to be strictly increasing, but some values s such that Φ′(s)=0 can exist. The method is based on the solution, at each interface between two control volumes, of the nonlinear elliptic two point boundary value problem (qf(υ)+(Φ(υ))′)′=0 with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show the increase of accuracy due to the use of this scheme, compared to some other schemes.     

A mixed finite volume scheme for anisotropic diffusion problems on any grid

We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. The approximate solution is shown to converge to the continuous one as the size of the mesh tends to 0, and an error estimate is given. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids and for various diffusion matrices.     

Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

Foundations of Computational Mathematics - This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible...