Boundary layer analysis of the Navier-Stokes equations with Generalized Navier boundary conditions

We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $\epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $\epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $\epsilon^{3/8 - \delta}$ near the boundary and $\epsilon^{3/4 - \delta'}$ in the interior, where $\delta, \delta'$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.Comment: Additional references and several typos fixe

Analysis of mixed elliptic and parabolic boundary layers with corners

We study the asymptotic behavior at small diffusivity of the solutions, u??, to a convection-diffusion equation in a rectangular domain. The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on ???, between u?? and the corresponding limit solution, u0, we propose asymptotic expansions of u?? at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners of , the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference of u?? and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces.open

Stability of Vortex Solutions to an Extended Navier-Stokes System

We study the long-time behavior an extended Navier-Stokes system in $\R^2$ where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego '07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu '04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne '05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study.Comment: 24 pages, 1 figure, updated to add authors' contact information and to address referee's comment

INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS WITH IMPLICIT TIME-INTEGRATION TECHNIQUES FOR NONLINEAR PARABOLIC EQUATIONS

Abstract. We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin (IPDG) methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l 2 (H 1 ) and l ∞ (L 2 ) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin (SIPG) scheme with the implicit θ-schemes in time, which include backward Euler and Crank-Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments

Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition

We study the asymptotic behavior, at small viscosity ??, of the Navier-Stokes equations in a 2D curved domain. The Navier-Stokes equations are supplemented with the slip boundary condition, which is a special case of the Navier friction boundary condition where the friction coefficient is equal to two times the curvature on the boundary. We construct an artificial function, which is called a corrector, to balance the discrepancy on the boundary of the Navier-Stokes and Euler vorticities. Then, performing the error analysis on the corrected difference of the Navier-Stokes and Euler vorticity equations, we prove convergence results in the L2 norm in space uniformly in time, and in the norm of H1 in space and L2 in time with rates of order ??3/4 and ??1/4, respectively. In addition, using the smallness of the corrector, we obtain the convergence of the Navier-Stokes solution to the Euler solution in the H1 norm uniformly in time with rate of order ??1/4.close0

Semi-analytic shooting methods for Burgers equation

We implement new semi-analytic shooting methods for the stationary viscous Burgers' equation by modifying the classical time differencing methods. When the viscosity is small, a very stiff boundary layer appears and this boundary layer causes significant difficulties to approximate the solution for Burgers' equation. To overcome this issue and improve the numerical quality of the shooting methods with the classical Integrating Factor (IF) methods and Exponential Time Differencing (ETD) methods, we first employ the singular perturbation analysis for Burgers' equation, and derived the so-called correctors that approximate the stiff part of the solution. Then, we build our new semianalytic shooting methods for the stationary viscous Burgers' equation by embedding these correctors into the IF and ETD methods. By performing numerical simulations, we verify that our new schemes, enriched with the correctors, give much better approximations, compared with the classical schemes.(c) 2022 Elsevier B.V. All rights reserved

Semi-analytic time differencing methods for singularly perturbed initial value problems

We implement our new semi-analytic time differencing methods, applied to singularly perturbed non-linear initial value problems. It is well-known that, concerning the singularly perturbed initial problem, a very stiff layer, called initial layer, appears when the perturbation parameter is small, and this stiff initial layer causes significant difficulties to approximate the solution. To improve numerical quality of the classical integrating factor (IF) methods and exponential time differencing (ETD) methods for stiff problems, we first derived the so-called correctors, which are analytic approximations of the stiff part of the solution. Then, by embedding these correctors into the IF and ETD methods, we build our new enriched schemes to improve the IF Runge-Kutta and ETD Runge-Kutta schemes. By performing numerical simulations, we verify that our new enriched schemes give much better approximations of solutions to the stiff problems, compared with the classical schemes without using the correctors

Validation of a 2D cell-centered Finite Volume method for elliptic equations

Following the approach in Gie and Temam(2010) and Gie and Temam(2015), we construct the Finite Volume (FV) approximations of a class of elliptic equations and perform numerical computations where a 2D domain is discretized by convex quadrilateral meshes. The FV method with Taylor Series Expansion Scheme (TSES), which is properly adjusted from a version widely used in engineering, is tested in a box, annulus, and in a domain which includes a topography at the bottom boundary. By comparing with other related convergent FV schemes in Sheng and Yuan(2008), Aavatsmark(2002), Hermeline(2000) and Faureet al. (2016), we numerically verify that our FV method is a convergent 2nd order scheme that manages well the complex geometry. The advantage of our scheme is on its simple structure which do not require any special reconstruction of dual type mesh for computing the nodal approximations or discrete gradients. (C) 2019 International Association for Mathematics and Computers in Simulation (IMACS)