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The Ladyzhenskaya-Prodi-Serrin type L s , r criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global L s , r norm is usually large and hence hard to control. Replacing the global L s , r norm with some kind of local norm is interesting. In this article, we introduce a local L s , r space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local L s , r type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local L s , r type norms is necessary and sufficient to affirmatively answer the millennium problem. [ABSTRACT FROM AUTHOR]

Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods. [ABSTRACT FROM AUTHOR]

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains. [ABSTRACT FROM AUTHOR]

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton’s method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form. [ABSTRACT FROM AUTHOR]

We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element. [ABSTRACT FROM AUTHOR]

Summary: We present a numerical method for modelling and simulation of transport and dispersion of phosphogypsum in the Jorf Lasfar coastal zone located on the Atlantic Ocean at Morocco. The governing equations consist of the well‐established barotropic ocean model including the barometric pressure, friction terms, Coriolis and wind stresses. To model transport and dispersion of phosphogypsum we consider an advection‐diffusion equation with an anisotropic dispersion tensor and source terms. As a numerical solver, we propose a novel multilevel adaptive semi‐Lagrangian finite element method. The proposed method combines the modified method of characteristics to deal with convection terms, the finite element discretization to handle complex geometries, a projection‐based algorithm to solve the Stokes problem, and an adaptive L2$$ {\mathrm{L}}^2 $$‐projection using quadrature rules to improve the efficiency and accuracy of the method. Numerical results are presented to demonstrate the high resolution of the proposed method and to confirm its capability to provide accurate and efficient simulations for transport and dispersion of phosphogypsum in the Jorf Lasfar coastal zone. [ABSTRACT FROM AUTHOR]

Keywords: Kinetic polymer models; FENE chain; incompressible Navier–Stokes equations; Fokker–Planck equations; nonhomogeneous dilute polymeric fluids; variable density.GraphBy Chuhui He and Endre SüliReported by Author; Author [Extracted from the article]

We present a numerical method specifically designed for simulating three-dimensional fluid-structure interaction (FSI) problems based on the reference map technique (RMT). The RMT is a fully Eulerian FSI numerical method that allows fluids and large-deformation elastic solids to be represented on a single fixed computational grid. This eliminates the need for meshing complex geometries typical in other FSI approaches and greatly simplifies the coupling between fluid and solids. We develop a three-dimensional implementation of the RMT, parallelized using the distributed memory paradigm, to simulate incompressible FSI with neo-Hookean solids. As part of our method, we develop a field extrapolation scheme that works efficiently in parallel. Through representative examples, we demonstrate the method's suitability in investigating many-body and active systems, as well as its accuracy and convergence. The examples include settling of a mixture of heavy and buoyant soft ellipsoids, lid-driven cavity flow containing a soft sphere, and swimmers actuated via active stress.

In a recent work (Castanon Quiroz & Di Pietro (2020) A hybrid high-order method for the incompressible Navier–Stokes problem robust for large irrotational body forces. Comput. Math. Appl. , 79 , 2655–2677), we have introduced a pressure-robust hybrid high-order method for the numerical solution of the incompressible Navier–Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error estimates for the velocity that are fully independent of the pressure. A crucial question was left open in that work, namely whether the proposed construction could be extended to general polytopal meshes. In this paper, we provide a positive answer to this question. Specifically, we introduce a novel divergence-preserving velocity reconstruction that hinges on the solution inside each element of a mixed problem on a subtriangulation, then use it to design discretizations of the body force and convective terms that lead to pressure robustness. An in-depth theoretical study of the properties of this velocity reconstruction, and their reverberation on the scheme, is carried out for arbitrary polynomial degrees |$k\geq 0$| and meshes composed of general polytopes. The theoretical convergence estimates and the pressure robustness of the method are confirmed by an extensive panel of numerical examples. [ABSTRACT FROM AUTHOR]

In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier–Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor–Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition-of-unity based error localization. We substantiate our methodology on 2D benchmark problems from computational fluid mechanics. [ABSTRACT FROM AUTHOR]

The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by ε, goes to zero, they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L∞(L2) ∩ L2(H1) sense on a uniform time interval independent of the small parameter ε. The proof consists of two parts: One is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces. Besides the careful energy estimates, a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions. The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak. [ABSTRACT FROM AUTHOR]

This paper studies fully discrete finite element approximations to the Navier–Stokes equations using inf-sup stable elements and grad-div stabilization. For the time integration, two implicit–explicit second-order backward differentiation formulae (BDF2) schemes are applied. In both, the Laplacian is implicit while the nonlinear term is explicit, in the first one, and semiimplicit, in the second one. The grad-div stabilization allows us to prove error bounds in which the constants are independent of inverse powers of the viscosity. Error bounds of order |$r$| in space are obtained for the |$L^2$| error of the velocity using piecewise polynomials of degree |$r$| to approximate the velocity together with second-order bounds in time, both for fixed time-step methods and for methods with variable time steps. A Courant Friedrichs Lewy (CFL)-type condition is needed for the method in which the nonlinear term is explicit relating time-step and spatial mesh-size parameters. [ABSTRACT FROM AUTHOR]

The present paper addresses the convergence of a first-order in time incremental projection scheme for the time-dependent incompressible Navier–Stokes equations to a weak solution. We prove the convergence of the approximate solutions obtained by a semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield their existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes. [ABSTRACT FROM AUTHOR]

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton's method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases. [ABSTRACT FROM AUTHOR]

The present paper addresses the convergence of the implicit Marker-and-Cell scheme for time-dependent Navier–Stokes equations with variable density and density-dependent viscosity and forcing term. A priori estimates on the unknowns are obtained, and thanks to a topological degree argument, they lead to the existence of an approximate solution at each time step. Then, by compactness arguments relying on these same estimates, we obtain the convergence (up to the extraction of a subsequence), when the space and time steps tend to zero, of the numerical solutions to a limit; this latter is shown to be a weak solution to the continuous problem by passing to the limit in the scheme. [ABSTRACT FROM AUTHOR]

In this article we prove the global existence of weak solutions for a diffuse interface model in a bounded domain (both in 2D and 3D) involving incompressible magnetic fluids with unmatched densities. The model couples the incompressible Navier–Stokes equations, gradient flow of the magnetization vector and the Cahn–Hilliard dynamics describing the partial mixing of two fluids. The density of the mixture depends on an order parameter and the modelling (specifically the density dependence) is inspired from Abels et al. (Models Methods Appl Sci 22(3):1150013, 2011). [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the instantaneous radius of space analyticity for the solutions of 3D Navier–Stokes system with initial data in the Besov spaces B ˙ p , q s (R 3) for p ∈ ] 1 , ∞ [ , q ∈ [ 1 , ∞ ] and s ∈ [ - 1 + 3 p , 3 p [. Then for initial data u 0 ∈ L p (R 3) with p in ]3, 6[, we prove that 3D Navier–Stokes system has a unique solution u = u L + v with and v ∈ L ~ T ∞ ( B ˙ p , p 2 1 - 3 p ) ∩ L ~ T 1 ( B ˙ p , p 2 3 - 3 p ) for some positive time T. Furthermore, we derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in Hu and Zhang (Chin Ann Math Ser B 43:749–772, 2022) with initial data in L p (R 3) for p ∈ [ 3 , 18 / 5 [. [ABSTRACT FROM AUTHOR]

In this paper, I shall construct a family of global smooth solutions with large initial data for the incompressible Navier–Stokes equations, which are allowed to have a large three-dimensional part and a small three-dimensional part. In addition, one also acquires a global well-posedness with low regular data to the incompressible Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

A novel high-order projection method is presented in this study for the simulation of incompressible turbulent flows, utilising weighted essentially non-oscillatory (WENO) schemes. Spatial discretization involves using WENO schemes for nonlinear convection and standard central differences (CD) for the viscous term. The approach combines various orders of WENO and central difference schemes (WENO3/CD2, WENO5/CD4, and WENO7/CD6) to achieve third, fifth, and seventh-order spatial accuracy for velocity fields, respectively. Validating this method involved a thorough examination using the 2D Taylor Green vortex problem in both space and time, along with simulations of 3D Taylor-Green vortex problems at increasing Reynolds numbers (Re) of 1, 600, 16, 000, and 160, 000. The outcomes encompassed analyses of kinetic energy, enstrophy, energy spectra, and vortex fields, indicating the method's effectiveness in accurately simulating high Reynolds number turbulent flows without additional sub-grid scale models. [ABSTRACT FROM AUTHOR]

In this article, we develop and analyze a novel numerical scheme for the steady incompressible Navier–Stokes equations by the weak Galerkin methods. By employing the divergence‐preserving velocity reconstruction operator, our algorithm can achieve pressure‐robustness, which means, the velocity error is independent of the pressure and the irrotational body force. Error analysis has been established to show the rate of convergence. Numerical experiments are presented to validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an H (div) -conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the nonlinear convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method naturally converges to an H (div) -based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method. • Compatible finite element exterior calculus (FEEC) with a posteriori shock capturing feature. • Asymptotic-preserving (AP) semi-implicit scheme in the low Mach number limit. • Exactly divergence-free semi-implicit DG/FEEC scheme for incompressible Navier-Stokes. • Stable also in absence of physical viscosity. • Only two symmetric positive definite linear systems per time-step to solve. [ABSTRACT FROM AUTHOR]

In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes. • A new formulation involves the kinetic energy, a Lagrange multiplier, and a regularization parameter. • The regularization parameter allows using large time step sizes while ensuring the uniqueness of the Lagrange multiplier. • The proposed schemes require solving two linear Stokes systems and a scalar quadratic equation at each time step. • Numerical results in two and three dimensions confirm convergence, energy dissipation and robustness of the DRLM schemes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to justify the rigorous derivation of the incompressible Navier–Stokes equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, we first establish the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian, obtaining some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system. [ABSTRACT FROM AUTHOR]

We consider the fractional chemotaxis Navier-Stokes equations which are the fractional Keller-Segel model coupled with the Navier-Stokes fluid in the whole space, and prove the existence of global mild solutions with the small critical initial data in Besov-Morrey spaces. Our results enable us to obtain the self-similar solutions provided the initial data are homogeneous functions with small norms and considering the case of chemical attractant without degradation rate. Moreover, we show the asymptotic stability of solutions as the time goes to infinity and obtain a class of asymptotically self-similar ones. [ABSTRACT FROM AUTHOR]

Based on finite element discretization and a recent variational multiscale-stabilized method, we propose a three-step defect-correction algorithm for solving the stationary incompressible Navier-Stokes equations with large Reynolds numbers, where nonlinear slip boundary conditions of friction type are considered. This proposed algorithm consists of solving one nonlinear Navier-Stokes type variational inequality problem on a coarse grid in a defect step, and solving two stabilized and linearized Navier-Stokes type variational inequality problems which have the same stiffness matrices with only different right-hand sides on a fine grid in correction steps. In the defect step, an artificial viscosity term is used as a stabilizing factor, making the nonlinear system easier to solve. Error bounds of the approximate solutions in L2 norms for the velocity gradient and pressure are estimated. Scalings of the algorithmic parameters are derived. Some numerical results are given to support the theoretical predictions and test the validity of the present algorithm. [ABSTRACT FROM AUTHOR]

For N odd, a plane non-parallel Navier-Stokes flow in a square fluid domain exhibits N × N alternative vortices. The spectral problem linearized around the flow shows the occurrence of critical real spectral solutions, which lead to the existence of secondary steady-state flows bifurcating from the non-parallel flow. [ABSTRACT FROM AUTHOR]

Given initial data u0 ∈ Lp (ℝ3) for some p in , the auhtors first prove that 3D incompressible Navier-Stokes system has a unique solution u = uL+v with and for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631–1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in Ḣs(ℝ3) for . [ABSTRACT FROM AUTHOR]

Given initial data u0 ∈ Lp (ℝ3) for some p in , the auhtors first prove that 3D incompressible Navier-Stokes system has a unique solution u = uL+v with and for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631–1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in Ḣs(ℝ3) for . [ABSTRACT FROM AUTHOR]

We study the long-time behaviour of axisymmetric solutions without swirl for the three-dimensional Navier-Stokes equations in the whole space. Assuming that the initial vorticity is sufficiently localised, we compute explicitly the leading terms in the asymptotic expansion of the solution, both for the vorticity and the velocity field. In particular, we identify optimal temporal decay rates depending on the spatial localisation of the initial data. Our approach relies on accurate L p - L q estimates for the linearised evolution equation and its Taylor expansion in self-similar variables. [ABSTRACT FROM AUTHOR]

A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier–Stokes equations is formulated and analyzed. We prove unconditional stability of the proposed scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates. [ABSTRACT FROM AUTHOR]

It is essential for a Navier-Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present numerical solutions of the 2D Navier-Stokes equations using the fourth-order generalized harmonic polynomial cell (GHPC) method as the Poisson equation solver. Particular focus is on the local and global accuracy of the GHPC method on non-uniform grids. Our study reveals that the GHPC method enables the use of more stretched grids than the original HPC method. Compared with a second-order central finite difference method (FDM), global accuracy analysis also demonstrates the advantage of applying the GHPC method on stretched non-uniform grids. An immersed-boundary method is used to deal with general geometries involving the fluid-structure interaction problems. The Taylor-Green vortex and flow around a smooth circular cylinder and square are studied for the purpose of verification and validation. Good agreement with reference results in the literature confirms the accuracy and efficiency of the new 2D Navier-Stokes equation solver based on the present immersed-boundary GHPC method utilizing non-uniform grids. The present Navier-Stokes equations solver uses second-order central FDM and Quadratic Upstream Interpolation for Convective Kinematics scheme for the discretization of the diffusion term and advection term, respectively, which may be replaced by other higher-order schemes to further improve the accuracy. [ABSTRACT FROM AUTHOR]

• The optimal convergence rate of L 2 -error for the velocity is completely proven. • The corresponding algorithm of our method is simple to be implemented by codes. • The remarkable advantage of our method is demonstrated via adequate numerical experiments. In this work, we develop a high-order pressure-robust method for the rotation form of the incompressible Navier–Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H (div) -conforming velocity reconstruction operator. In order to match the rotation form and ease error analysis, a skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proved to achieve the pressure-independent velocity errors. Optimal convergence rates of H 1 , L 2 -error for the velocity and L 2 -error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and remarkable performance of the proposed method. [ABSTRACT FROM AUTHOR]

We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument. [ABSTRACT FROM AUTHOR]

In this paper we present the comparison of experiments and numerical simulations for bubble cutting by a wire. The air bubble is surrounded by water. In the experimental setup an air bubble is injected on the bottom of a water column. When the bubble rises and contacts the wire, it is separated into two daughter bubbles. The flow is modeled by the incompressible Navier–Stokes equations. A meshfree method is used to simulate the bubble cutting. We have observed that the experimental and numerical results are in very good agreement. Moreover, we have further presented simulation results for liquid with higher viscosity. In this case the numerical results are close to previously published results. [ABSTRACT FROM AUTHOR]

In this paper, linearly implicit backward differentiation formulas with variable step-sizes are proposed to solve numerically the two-dimensional incompressible Navier-Stokes equations (formulated in vorticity-stream function). With Fourier pseudospectral methods for spatial discretization, the diffusion term is discretized implicitly and the nonlinear convection term is treated by a combination of implicit and explicit discretizations. As a result, only linear solvers are needed at each time step to achieve the desired temporal accuracy. With the help of a priori assumption and aliasing error control techniques, the error estimates for one-step and two-step backward differentiation formulas are established in several norms under appropriate step-size constraints. Compared with the numerical results of implicit-explicit (the nonlinear convection term is treated explicitly) BDF2 method and fully implicit Crank-Nicolson method, it demonstrates that the proposed linearly implicit variable step-size BDF2 method is effective and robust. [ABSTRACT FROM AUTHOR]

We present a numerical method specifically designed for simulating three-dimensional fluid-structure interaction (FSI) problems based on the reference map technique (RMT). The RMT is a fully Eulerian FSI numerical method that allows fluids and large-deformation elastic solids to be represented on a single fixed computational grid. This eliminates the need for meshing complex geometries typical in other FSI approaches and greatly simplifies the coupling between fluid and solids. We develop a three-dimensional implementation of the RMT, parallelized using the distributed memory paradigm, to simulate incompressible FSI with neo-Hookean solids. As part of our method, we develop a field extrapolation scheme that works efficiently in parallel. Through representative examples, we demonstrate the method's suitability in investigating many-body and active systems, as well as its accuracy and convergence. The examples include settling of a mixture of heavy and buoyant soft ellipsoids, lid-driven cavity flow containing a soft sphere, and swimmers actuated via active stress. [ABSTRACT FROM AUTHOR]

We introduce a robust and efficient preconditioner for a hybrid Newton–GMRES method for solving the nonlinear systems arising from incompressible Navier–Stokes equations. When the Reynolds number is relatively high, these systems often involve millions of degrees of freedom (DOFs), and the nonlinear systems are difficult to converge, partially due to the strong asymmetry of the system and the saddle‐point structure. In this work, we propose to alleviate these issues by leveraging a multilevel ILU preconditioner called HILUCSI, which is particularly effective for saddle‐point problems and can enable robust and rapid convergence of the inner iterations in Newton–GMRES. We further use Picard iterations with the Oseen systems to hot‐start Newton–GMRES to achieve global convergence, also preconditioned using HILUCSI. To further improve efficiency and robustness, we use the Oseen operators as physics‐based sparsifiers when building preconditioners for Newton iterations and introduce adaptive refactorization and iterative refinement in HILUCSI. We refer to the resulting preconditioned hybrid Newton–GMRES as HILUNG. We demonstrate the effectiveness of HILUNG by solving the standard 2D driven‐cavity problem with Re 5000 and a 3D flow‐over‐cylinder problem with low viscosity. We compare HILUNG with some state‐of‐the‐art customized preconditioners for INS, including two variants of augmented Lagrangian preconditioners and two physics‐based preconditioners, as well as some general‐purpose approximate‐factorization techniques. Our comparison shows that HILUNG is much more robust for solving high‐Re problems and it is also more efficient in both memory and runtime for moderate‐Re problems. [ABSTRACT FROM AUTHOR]