A new class of fractional step techniques for the incompressible Navier–Stokes equations using direction splitting

A new direction-splitting-based fractional time stepping is introduced for solving the incompressible Navier–Stokes equations. The main originality of the method is that the pressure correction is computed by solving a sequence of one-dimensional elliptic problems in each spatial direction. The method is very simple to program in parallel, very fast, and has exactly the same stability and convergence properties as the Poisson-based pressure-correction technique, either in standard or rotational form.     

A mixed problem for the steady Navier–Stokes equations

We consider a mixed boundary problem for the Navier–Stokes equations in a bounded Lipschitz two-dimensional domain: we assign a Dirichlet condition on the curve portion of the boundary and a slip zero condition on its straight portion. We prove that the problem has a solution provided the boundary datum and the body force belong to a Lebesgue’s space and to the Hardy space respectively.     

Parallel spectral-element direction splitting method for incompressible Navier–Stokes equations

An efficient parallel algorithm for the time dependent incompressible Navier–Stokes equations is developed in this paper. The time discretization is based on a direction splitting method which only requires solving a sequence of one-dimensional Poisson type equations at each time step. Then, a spectral-element method is used to approximate these one-dimensional problems. A Schur-complement approach is used to decouple the computation of interface nodes from that of interior nodes, allowing an efficient parallel implementation. The unconditional stability of the full discretized scheme is rigorously proved for the two-dimensional case. Numerical results are presented to show that this algorithm retains the same order of accuracy as a usual spectral-element projection type schemes but it is much more efficient, particularly on massively parallel computers.     

High order time discretisation methods for incompressible Navier–Stokes equations

In this paper we solve incompressible Navier–Stokes equations with higher order time integration schemes such as Radau-IIA and Lobatto-IIIC methods. We compare the results with 4th order DIRK and ROW methods to show the advantages of higher order time integration methods. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A convergent FEM-DG method for the compressible Navier–Stokes equations

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.     

A logarithmically improved regularity criterion for the Navier–Stokes equations

In this note we prove a logarithmically improved regularity criterion in terms of the Besov space norm for the Navier–Stokes equations. The result shows that if a mild solution u satisfies ${\int_{0}^{T}\frac{\|u (t,\cdot)\|_{{\dot{B}}_{\infty,\infty}^{-r}}^{\frac{2}{1-r}}}{1+\ln(e+\| u(t,\cdot)\|_{H^{s}})}\text{d}t < \infty}$ for some 0?≤ r?<?1 and ${s\geq\frac{n}{2}-1}$ , then u is regular at t?=?T.     

A class of hemivariational inequalities for nonstationary Navier–Stokes equations

This paper is devoted to the study of a class of hemivariational inequalities for the time-dependent Navier–Stokes equations, including both boundary hemivariational inequalities and domain hemivariational inequalities. The hemivariational inequalities are analyzed in the framework of an abstract hemivariational inequality. Solution existence for the abstract hemivariational inequality is explored through a limiting procedure for a temporally semi-discrete scheme based on the backward Euler difference of the time derivative, known as the Rothe method. It is shown that solutions of the Rothe scheme exist, they contain a weakly convergent subsequence as the time step-size approaches zero, and any weak limit of the solution sequence is a solution of the abstract hemivariational inequality. It is further shown that under certain conditions, a solution of the abstract hemivariational inequality is unique and the solution of the abstract hemivariational inequality depends continuously on the problem data. The results on the abstract hemivariational inequality are applied to hemivariational inequalities associated with the time-dependent Navier–Stokes equations.     

Analysis of the Yosida method for the incompressible Navier–Stokes equations

The Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier–Stokes equations. From the algebraic viewpoint, it can be regarded as an inexact factorization of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic stabilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)). The motivation of this method as well as an extensive numerical validation were given in (Quarteroni et al., to appear).In this paper we carry out the analysis of this scheme. In particular, we consider a first-order time advancing unsplit method. In the case of the Stokes problem, we prove unconditional stability and moreover that the splitting error introduced by the Yosida scheme does not affect the overall accuracy of the solution, which remains linear with respect to the time step. Some numerical experiments, for both the Stokes and Navier–Stokes equations, are presented in order to substantiate our theoretical results.     

A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

In this note we provide a criterion for the existence of globally defined solutions for any regular initial data for the 3D Navier–Stokes system in Serrin’s classes.     

Unique solvability for the density-dependent Navier–Stokes equations

In this paper we consider the incompressible Navier–Stokes equations with a density-dependent viscosity in a bounded domain $\Omega$ of ${\mathbf{R}}^n(n=2,3)$. We prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition. This condition is also necessary for a very general initial data. Moreover, we provide a blow-up criterion for the regularity of the strong solution. For these results, the initial density need not be strictly positive. It may vanish in an open subset of $\Omega$.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.     

A regularity criterion for the Navier–Stokes equations with mass diffusion

In this work, a regularity criterion is proved for local strong solutions of the Navier–Stokes equations in the presence of mass diffusion.