On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations

The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for coarse spatial resolutions and small time step sizes. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly.Comment: 31 page

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com

A comparison of various quasi-newton schemes for partitioned fluid-structure interaction

During the last 5 years, quasi-Newton schemes have proven to be a robust and efficient way to couple partitioned fluid-structure interaction. We showed in previous work that they also allow to perform a parallel coupling. Bogaers et al. introduced a new variant based on a multi-vector update [14]. This variant renders a tuning of the reuse of old information unnecessary as all old iterations are implicitly covered in a Jacobian update. In this work, we compare this multi-vector variant in an inverse formulation to the classical IQN-ILS algorithm for serial as well as parallel coupling

On Fluid Dynamics of Freshwater and Seawater in Marine Systems

Interaction between freshwater and seawater can significantly influence marine life and create important productive and dynamic natural habitats, such as estuaries. We study such processes and for this purpose we develop a new mathematical model with the system of coupled nonlinear conservation equations that govern the interaction of the two incompressible fluids with different properties. The system is solved by using implicit operator splitting and mixed finite elements methods. Finally, the numerical scheme has been used to provide simulations of two real world scenarios

On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics

This paper describes the development and analysis of finite-volume methods for the Landau–Lifshitz Navier–Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatiotemporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order Runge–Kutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations.Together with a novel method for discretizing the stochastic stress tensor in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit

Institute for Computational Mechanics in Propulsion (ICOMP) fourth annual review, 1989

The Institute for Computational Mechanics in Propulsion (ICOMP) is operated jointly by Case Western Reserve University and the NASA Lewis Research Center. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1989 are described

Robust finite volume schemes for 2D shallow water models. Application to flood plain dynamics

This study proposes original combinations of higher order Godunov type finite volume schemes and time discretization schemes for the 2d shallow water equations, leading to fully second-order accuracy with well-balanced property. Also accuracy, positiveness and stability properties in presence of dynamic wet/dry fronts is demonstrated. The test cases are the classical ones plus extra new ones representing the geophysical flow features and difficulties