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Second-order Convergence (second-order + convergence)
Selected AbstractsNumerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010Roberto Croce Abstract In this paper we present a three-dimensional Navier,Stokes solver for incompressible two-phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third-order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second-order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first-order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three-dimensional results with those of quasi-two-dimensional and two-dimensional simulations. This comparison clearly shows the need for full three-dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd. [source] Flow simulation on moving boundary-fitted grids and application to fluid,structure interaction problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2006Martin Engel Abstract We present a method for the parallel numerical simulation of transient three-dimensional fluid,structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non-overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time-dependent domains. To this end, we present a technique to solve the incompressible Navier,Stokes equation in three-dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time-dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian,Eulerian formulation of the Navier,Stokes equations. Here the grid velocity is treated in such a way that the so-called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well-known MAC-method to a staggered mesh in moving boundary-fitted coordinates which uses grid-dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second-order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid,structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid,structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd. [source] A quasi-planar incident wave excitation for time-domain scattering analysis of periodic structuresINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 5 2006David Degerfeldt Abstract We present a quasi-planar incident wave excitation for time-domain scattering analysis of periodic structures. It uses a particular superposition of plane waves that yields an incident wave with the same periodicity as the periodic structure itself. The duration of the incident wave is controlled by means of its frequency spectrum or, equivalently, the angular spread in its constituting plane waves. Accuracy and convergence properties of the method are demonstrated by scattering computations for a planar dielectric half-space. Equipped with the proposed source, a time-domain solver based on linear elements yields an error of roughly 1% for a resolution of 20 points per wavelength and second-order convergence is achieved for smooth scatterers. Computations of the scattering characteristics for a sinusoidal surface and a random rough surface show similar performance. Copyright © 2006 John Wiley & Sons, Ltd. [source] A uniformly stable conformal FDTD-method in Cartesian gridsINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 2 2003I.A. Zagorodnov Abstract A conformal finite-difference time-domain algorithm for the solution of electrodynamic problems in general perfectly conducting 3D geometries is presented. Unlike previous conformal approaches it has the second-order convergence without the need to reduce the maximal stable time step of conventional staircase approach. A novel proof for the local error rate for general geometries is given, and the method is verified and compared to other approaches by means of several numerical 2D examples. Copyright © 2003 John Wiley & Sons, Ltd. [source] A Petrov-Galerkin method with quadrature for semi-linear second-order hyperbolic problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2006B. Bialecki Abstract We propose and analyze a Crank,Nicolson quadrature Petrov,Galerkin (CNQPG) -spline method for solving semi-linear second-order hyperbolic initial-boundary value problems. We prove second-order convergence in time and optimal order H2 norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order Hk, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source] Lp error estimates and superconvergence for covolume or finite volume element methodsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2003So-Hsiang Chou Abstract We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the Lp norm, 2 , p , ,, are derived. We also show second-order convergence in the Lp norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the "supercloseness" results in Chou and Li [Math Comp 69(229) (2000), 103,120] to the Lp based spaces, duality arguments, and the discrete Green's function method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 463,486, 2003 [source] | ||||||||||