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A model for the motion of slender vortex filaments is extended to include the effect of gravity. The model, initially introduced by Callegari and Ting ["Motion of a curved vortex filament with decaying vortical core and axial velocity," SIAM J. Appl. Math. 35, 148–175 (1978)], is based on a matched asymptotic expansion in which the outer solution, given by the Biot–Savart law, is matched with the inner solution derived from the Navier–Stokes equations. Building on recent work by Harikrishnan et al. ["On the motion of hairpin filaments in the atmospheric boundary layer," Phys. Fluids 35, 076603 (2023)], the Boussinesq approximation is applied such that the density variations only enter in the gravity term. However, unlike Harikrishnan et al. ["On the motion of hairpin filaments in the atmospheric boundary layer," Phys. Fluids 35, 076603 (2023)], the density variation enters at a lower order in the asymptotic expansion and, thus, has a more significant impact on the self-induced velocity of the vortex filament. In this regime, which corresponds to the regime studied by Chang and Smith ["The motion of a buoyant vortex filament," J. Fluid Mech. 857, R1 (2018)], the effect of gravity is given by an alteration of the core constant, which couples the motion of the filament to the motion within the vortical core, in addition to a change in the compatibility conditions (evolution equations), which determine the leading order azimuthal and tangential velocity fields in the vortex core. The results are used to explain certain properties of buoyant vortex rings, as well as qualitatively explore the impact of gravity on tornado-type atmospheric vortices. [ABSTRACT FROM AUTHOR]

This-36pt paper considers a discontinuous Galerkin finite element method for the 2D transient and incompressible Navier-Stokes model. Following the analysis of Heywood and Rannacher (SIAM J. Numer. Anal. 19:275–311, 1982), we derive optimal velocity and pressure error estimates in L ∞ (L 2) and L ∞ (L 2) -norms, respectively, for the discontinuous Galerkin case. We use standard L 2 -projection and modified Stokes operator but on appropriate broken Sobolev spaces, and then standard duality arguments to achieve these results. For sufficiently small data, uniform in time estimates are proved. Based on the backward Euler method, time discretization is carried out and fully discrete error estimates are derived. Finally, numerical experiments are conducted to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier–Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal.59 (2021) 2926–2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier–Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53(1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a divergence-free nonconforming virtual element discrete scheme for (Navier-) Stokes problem based on Raviart-Thomas-like virtual element space. By choosing different (compared to Zhao et al. (SIAM J. Numer. Anal. 57, 2730–2759, 2019)) degrees of freedom and global virtual element space, we realize H1- and L2-error estimates of the velocity that are independent of the continuous pressure. Moreover, the presented scheme can also deal with polygonal meshes (including non-convex and degenerate elements), arbitrary approximation orders k, and large Reynolds numbers. Finally, we investigate several classical numerical experiments of the incompressible flow to present the performance of this numerical scheme. [ABSTRACT FROM AUTHOR]

Space–time parallel methods can leverage modern parallel computing architectures to further accelerate numerical simulations where parallelizing only in space limits concurrency. In this paper, we develop a high-order space–time parallel computing method for solving the Navier–Stokes equations. The method is based on revisionist integral deferred correction (RIDC) algorithm proposed by Christlieb, Macdonald and Ong [SIAM J. Sci. Comput., 32 (2010), pp. 818-835]. This parallel-in-time algorithm yields high-order accurate results within a wall-clock time comparable to that of a forward or backward Euler simulation. We extend the RIDC algorithm to a space–time parallel method and apply the method to a compressible fluid solver using the message passing interface (MPI) framework. The global communicator is split into space and time communicators to manage parallel computing in spatial and temporal domains, respectively. Accuracy tests show that the method is high-order accurate. Numerical results encompassing both smooth flows and flows with discontinuities demonstrate that the proposed method can accelerate the computation robustly. Additionally, a three-dimensional space–time parallel computation of flow around a sphere is performed. The results indicate that RIDC has the potential to be used in the field of computational fluid dynamics. • A space–time parallel method for solving the N-S equations is developed. • The method is high-order accurate based on the RIDC method. • The method can break through the space-parallel bottleneck. • A 3D case indicates the potential application of the method in engineering. [ABSTRACT FROM AUTHOR]

Computing a flow system a number of times with different samples of flow parameters is a common practice in many uncertainty quantification applications, which can be prohibitively expensive for complex nonlinear flow problems. This report presents two second order, stabilized, scalar auxiliary variable (SAV) ensemble algorithms for fast computation of the Navier--Stokes flow ensembles: Stab-SAV-CN and Stab-SAV-BDF2. The proposed ensemble algorithms are based on the ensemble timestepping idea which makes use of a quantity called the ensemble mean to construct a common coefficient matrix for all realizations at the same time step after spatial discretization, in which case efficient block solvers, e.g., block GMRES, can be used to significantly reduce both storage and computational time. The adoption of a recently developed SAV approach that treats the nonlinear term explicitly results in a constant shared coefficient matrix among all realizations at different time steps, which further cuts down the computational cost, yielding an extremely efficient ensemble algorithm for simulating nonlinear flow ensembles with provable long time stability without any time step conditions. The SAV approach for the Navier--Stokes equations for a single realization was proved to be unconditionally stable in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1-22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465-2491]. However we found the SAV approach has very low accuracy that compromises its stability in our initial numerical investigations for several commonly tested benchmark flow problems. In this report, we propose to use the stabilization in Stab-SAV-BDF2 to address this issue. We prove that both of our ensemble algorithms are long time stable under one parameter fluctuation condition, without any time step constraints. For a single realization, both algorithms are unconditionally stable and have better accuracy than the SAV methods studied in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1-22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465-2491] for our test problems. Extensive numerical experiments are performed to show the efficiency of the proposed ensemble algorithms and the effectiveness of the stabilization for increasing accuracy and stability. [ABSTRACT FROM AUTHOR]

This paper is the second part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we construct optimal hp-approximations for two classes of functions: (1) functions with general Sobolev regularity which lead to algebraic convergence rates using locally quasi-uniform meshes, and (2) singular functions belonging to countably normed (weighted) spaces which lead to exponential convergence using geometrically graded meshes. In turn, the finite element spaces we considered in part I [M. Ainsworth and C. Parker, SIAM J. Numer. Anal., 59 (2021), pp. 1218-1244] have optimal convergence properties, which is demonstrated by two numerical examples: Moffatt eddies in a wedge and flow in a T-shaped cavity. [ABSTRACT FROM AUTHOR]

The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a pointwise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming, and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressurerobustness by only changing the right-hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor-Hood and mini elements by Lederer et al. [SIAM J. Numer. Anal., 55 (2017), pp. 1291-1314]. We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods. [ABSTRACT FROM AUTHOR]

The framework of inner product norm preserving relaxation Runge--Kutta methods [D. I. Ketcheson, SIAM J. Numer. Anal., 57 (2019), pp. 2850--2870] is extended to general convex quantities. Conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addition of a relaxation parameter that multiplies the Runge--Kutta update at each step. Moreover, other desirable stability (such as strong stability preservation) and efficiency (such as low storage requirements) properties are preserved. The technique can be applied to both explicit and implicit Runge--Kutta methods and requires only a small modification to existing implementations. The computational cost at each step is the solution of one additional scalar algebraic equation for which a good initial guess is available. The effectiveness of this approach is proved analytically and demonstrated in several numerical examples, including applications to high order entropy-conservative and entropy-stable semidiscretizations on unstructured grids for the compressible Euler and Navier--Stokes equations. [ABSTRACT FROM AUTHOR]

In [M. Benzi and M. A. Olshanskii, SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113] a preconditioner of augmented Lagrangian type was presented for the two-dimensional stationary incompressible Navier-Stokes equations that exhibits convergence almost independent of Reynolds number. The algorithm relies on a highly specialized multigrid method involving a custom prolongation operator and for robustness requires the use of piecewise constant finite elements for the pressure. However, the prolongation operator and velocity element used do not directly extend to three dimensions: the local solves necessary in the prolongation operator do not satisfy the inf-sup condition. In this work we generalize the preconditioner to three dimensions, proposing alternative finite elements for the velocity and prolongation operators for which the preconditioner works robustly. The solver is effective at high Reynolds number: on a three-dimensional lid-driven cavity problem with approximately one billion degrees of freedom, the average number of Krylov iterations per Newton step varies from 4.5 at Re = 10 to 3 at Re = 1000 and 5 at Re = 5000. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the numerical treatment of a recent diffuse interface model for two-phase flow of electrolyte solutions (Campillo-Funollet et al., SIAM J. Appl. Math. 72(6), 1899-1925, 2012). This model consists of a Nernst-Planck-system describing the evolution of the ion densities and the electrostatic potential which is coupled to a Cahn-Hilliard-Navier-Stokes-system describing the evolution of phase-field, velocity field, and pressure. In the first part, we present a stable, fully discrete splitting scheme, which allows to split the governing equations into different blocks, which may be treated sequentially and thereby reduces the computational costs significantly. This scheme comprises different mechanisms to reduce the induced numerical dissipation. In the second part, we investigate the impact of these mechanisms on the scheme's sensitivity to the size of the time increment using the example of a falling droplet. Finally, we shall present simulations showing ion induced changes in the topology of charged droplets serving as a qualitative validation for our discretization and the underlying model. [ABSTRACT FROM AUTHOR]

Abstract: In [J.H. Bramble, I.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997) 1072–1092], a nonlinear Uzawa algorithm for solving generalized saddle point problems iteratively was considered. In [Z. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171], Cao gave another nonlinear Uzawa algorithm in order to accelerate convergence. In this paper, a new nonlinear Uzawa method, which is defined by two nonlinear approximate inverses, is proposed and its convergence result is deduced. At the same time, we analyze convergence results of three nonlinear Uzawa methods. The results of numerical experiments are presented when we apply them to Stokes equations discretized by mixed finite element. [Copyright &y& Elsevier]