Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes

We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, which we choose of the Brezzi-Pitk\"aranta type. The scheme features two essential properties: the discrete gradient is the transposed of the divergence terms and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem, the steady and the transient Navier-Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the $L^2$-convergence of the components of the velocity, and, in the steady case, the weak $L^2$-convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov's theorem. The limit of this subsequence is then shown to be a weak solution of the Navier-Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and the nonlinear case.Comment: submitted September 0

Finite volume schemes and Lax-Wendroff consistency

We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wendroff theorems. We then state a Lax-Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a recent generalization of the flux consistency notion designed to cope with general discrete functions

Finite volume approximation of a diffusion-dissolution model and application to nuclear waste storage

International audienceThe study of two phase flow in porous media under high capillary pressures, in the case where one phase is incompressible and the other phase is gaseous, shows complex phenomena. We present in this paper a numerical approximation method, based on a two pressures formulation in the case where both phases are miscible, which is shown to also handle the limit case of immiscible phases. The space discretization is performed using a finite volume method, which can handle general grids. The efficiency of the formulation is shown on three numerical examples related tounderground waste disposal situations

Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in space; the latter is in fact a class of methods that includes conforming and nonconforming finite elements, discontinuous Galerkin methods and several others. The main result is an error estimate which holds without supplementary regularity hypothesis on the solution. This result states that the approximation error has the same order as the sum of the interpolation error and the conformity error. The proof of this result relies on an inf-sup inequality in Hilbert spaces which can be used both in the continuous and the discrete frameworks. The error estimate result is illustrated by numerical examples with low regularity of the solution

POD‐identification reduced order model of linear transport equations for control purposes

Intrusive reduced order modeling techniques require access to the solver's discretization and solution algorithm, which are not available for most computational fluid dynamics codes. Therefore, a nonintrusive reduction method that identifies the system matrix of linear fluid dynamical problems with a least-squares technique is presented. The methodology is applied to the linear scalar transport convection-diffusion equation for a 2D square cavity problem with a heated lid. The (time-dependent) boundary conditions are enforced in the obtained reduced order model (ROM) with a penalty method. The results are compared and the accuracy of the ROMs is assessed against the full order solutions and it is shown that the ROM can be used for sensitivity analysis by controlling the nonhomogeneous Dirichlet boundary conditions

Baseline neutrophil-to-lymphocyte ratio as a predictive and prognostic biomarker in patients with metastatic castration-resistant prostate cancer treated with cabazitaxel versus abiraterone or enzalutamide in the CARD study

Abiraterona; Cabazitaxel; Factor pronósticoAbiraterona; Cabazitaxel; Factor pronòsticAbiraterone; Cabazitaxel; Prognostic factorBackground There is growing evidence that a high neutrophil-to-lymphocyte ratio (NLR) is associated with poor overall survival (OS) for patients with metastatic castration-resistant prostate cancer (mCRPC). In the CARD study (NCT02485691), cabazitaxel significantly improved radiographic progression-free survival (rPFS) and OS versus abiraterone or enzalutamide in patients with mCRPC previously treated with docetaxel and the alternative androgen-receptor-targeted agent (ARTA). Here, we investigated NLR as a biomarker. Patients and methods CARD was a multicenter, open-label study that randomized patients with mCRPC to receive cabazitaxel (25 mg/m2 every 3 weeks) versus abiraterone (1000 mg/day) or enzalutamide (160 mg/day). The relationships between baseline NLR [< versus ≥ median (3.38)] and rPFS, OS, time to prostate-specific antigen progression, and prostate-specific antigen response to cabazitaxel versus ARTA were evaluated using Kaplan–Meier estimates. Multivariable Cox regression with stepwise selection of covariates was used to investigate the prognostic association between baseline NLR and OS. Results The rPFS benefit with cabazitaxel versus ARTA was particularly marked in patients with high NLR {8.5 versus 2.8 months, respectively; hazard ratio (HR) 0.43 [95% confidence interval (CI) 0.27-0.67]; P < 0.0001}, compared with low NLR [7.5 versus 5.1 months, respectively; HR 0.69 (95% CI 0.45-1.06); P = 0.0860]. Higher NLR (continuous covariate, per 1 unit increase) independently associated with poor OS [HR 1.05 (95% CI 1.02-1.08); P = 0.0003]. For cabazitaxel, there was no OS difference between patients with high versus low NLR (15.3 versus 12.9 months, respectively; P = 0.7465). Patients receiving an ARTA with high NLR, however, had a worse OS versus those with low NLR (9.5 versus 13.3 months, respectively; P = 0.0608). Conclusions High baseline NLR predicts poor outcomes with an ARTA in patients with mCRPC previously treated with docetaxel and the alternative ARTA. Conversely, the activity of cabazitaxel is retained irrespective of NLR.This work was supported by Sanofi Genzyme (no grant number). The authors were responsible for all content and editorial decisions and received no honoraria for development of this manuscript

Probabilistic analysis of the upwind scheme for transport

We provide a probabilistic analysis of the upwind scheme for multi-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we prove that the scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all a>0, for a Lipschitz continuous initial datum. Our analysis provides a new interpretation of the numerical diffusion phenomenon