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Second-order Accuracy (second-order + accuracy)
Selected AbstractsSimple efficient algorithm (SEA) for shallow flows with shock wave on dry and irregular bedsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2008Alireza Zia Abstract An explicit Godunov-type solution algorithm called SEA (simple efficient algorithm) has been introduced for the shallow water equations. The algorithm is based on finite volume conservative discretisation method. It can deal with wet/dry and irregular beds. Second-order accuracy, in both time and space, is achieved using prediction and correction steps. A very simple and efficient flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purposes. In order to make sure about the balance between the flux gradient and the bed slope, treatment of the source term has been done using a new procedure inspired mainly by the physical rather than mathematical consideration. SEA has been applied to one-dimensional problems, although it can equally be applied to multi-dimensional problems. In order to assess the capability of proposed algorithm in dealing with practical applications, several test cases have been examined. Copyright © 2007 John Wiley & Sons, Ltd. [source] Volume fraction based miscible and immiscible fluid animationCOMPUTER ANIMATION AND VIRTUAL WORLDS (PREV: JNL OF VISUALISATION & COMPUTER ANIMATION), Issue 3-4 2010Kai Bao Abstract We propose a volume fraction based approach to effectively simulate the miscible and immiscible flows simultaneously. In this method, a volume fraction is introduced for each fluid component and the mutual interactions between different fluids are simulated by tracking the evolution of the volume fractions. Different techniques are employed to handle the miscible and immiscible interactions and special treatments are introduced to handle flows involving multiple fluids and different kinds of interactions at the same time. With this method, second-order accuracy is preserved in both space and time. The experiment results show that the proposed method can well handle both immiscible and miscible interactions between fluids and much richer mixing detail can be generated. Also, the method shows good controllability. Different mixing effects can be obtained by adjusting the dynamic viscosities and diffusion coefficients. Copyright © 2010 John Wiley & Sons, Ltd. [source] Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2004X. Zhou Abstract The primary objectives of the present exposition are to: (i) provide a generalized unified mathematical framework and setting leading to the unique design of computational algorithms for structural dynamic problems encompassing the broad scope of linear multi-step (LMS) methods and within the limitation of the Dahlquist barrier theorem (Reference [3], G. Dahlquist, BIT 1963; 3: 27), and also leading to new designs of numerically dissipative methods with optimal algorithmic attributes that cannot be obtained employing existing frameworks in the literature, (ii) provide a meaningful characterization of various numerical dissipative/non-dissipative time integration algorithms both new and existing in the literature based on the overshoot behavior of algorithms leading to the notion of algorithms by design, (iii) provide design guidelines on selection of algorithms for structural dynamic analysis within the scope of LMS methods. For structural dynamics problems, first the so-called linear multi-step methods (LMS) are proven to be spectrally identical to a newly developed family of generalized single step single solve (GSSSS) algorithms. The design, synthesis and analysis of the unified framework of computational algorithms based on the overshooting behavior, and additional algorithmic properties such as second-order accuracy, and unconditional stability with numerical dissipative features yields three sub-classes of practical computational algorithms: (i) zero-order displacement and velocity overshoot (U0-V0) algorithms; (ii) zero-order displacement and first-order velocity overshoot (U0-V1) algorithms; and (iii) first-order displacement and zero-order velocity overshoot (U1-V0) algorithms (the remainder involving high-orders of overshooting behavior are not considered to be competitive from practical considerations). Within each sub-class of algorithms, further distinction is made between the design leading to optimal numerical dissipative and dispersive algorithms, the continuous acceleration algorithms and the discontinuous acceleration algorithms that are subsets, and correspond to the designed placement of the spurious root at the low-frequency limit or the high-frequency limit, respectively. The conclusion and design guidelines demonstrating that the U0-V1 algorithms are only suitable for given initial velocity problems, the U1-V0 algorithms are only suitable for given initial displacement problems, and the U0-V0 algorithms are ideal for either or both cases of given initial displacement and initial velocity problems are finally drawn. For the first time, the design leading to optimal algorithms in the context of a generalized single step single solve framework and within the limitation of the Dahlquist barrier that maintains second-order accuracy and unconditional stability with/without numerically dissipative features is described for structural dynamics computations; thereby, providing closure to the class of LMS methods. Copyright © 2003 John Wiley & Sons, Ltd. [source] Numerical approximation of a thermally driven interface using finite elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2003P. Zhao Abstract A two-dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh. The model tracks the position of the sharp solid,liquid interface using a set of marker points placed on the interface. The simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it; at the interface the heat flux is discontinuous due to the release of latent heat during the solidification (melting) process. Two ways to calculate the temperature gradients at the interface, evaluating their interpolants at Gauss points, were proposed. Using known one- and two-dimensional solutions to stable solidification problems (the Stefan problem), it was shown that the method converges with second-order accuracy. When applied to the unstable solidification of a crystal into an undercooled liquid, it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface, i.e. different approximations and different meshes can yield different solutions. The cause of these difficulties is examined, the effect of different types of interpolation on the simulations is investigated, and the necessary criteria to ensure converged solutions are established. Copyright © 2003 John Wiley & Sons, Ltd. [source] Flow-induced vibrations of non-linear cables.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2002Part 1: Models, algorithms Abstract In this paper, we develop governing equations for non-linear cables as well as a formulation for the coupled flow-structure problem. The structure is discretized with second-order accuracy while the flow is discretized using spectral/hp elements in the context of the arbitrary Lagrangian,Eulerian formulation (ALE). Several benchmark problems are considered and the computational implementation is detailed. In the second part of this work large-scale simulation examples are presented. Copyright © 2002 John Wiley & Sons, Ltd. [source] Some results on the accuracy of an edge-based finite volume formulation for the solution of elliptic problems in non-homogeneous and non-isotropic mediaINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2009Darlan Karlo Elisiário de Carvalho Abstract The numerical simulation of elliptic type problems in strongly heterogeneous and anisotropic media represents a great challenge from mathematical and numerical point of views. The simulation of flows in non-homogeneous and non-isotropic porous media with full tensor diffusion coefficients, which is a common situation associated with the miscible displacement of contaminants in aquifers and the immiscible and incompressible two-phase flow of oil and water in petroleum reservoirs, involves the numerical solution of an elliptic type equation in which the diffusion coefficient can be discontinuous, varying orders of magnitude within short distances. In the present work, we present a vertex-centered edge-based finite volume method (EBFV) with median dual control volumes built over a primal mesh. This formulation is capable of handling the heterogeneous and anisotropic media using structured or unstructured, triangular or quadrilateral meshes. In the EBFV method, the discretization of the diffusion term is performed using a node-centered discretization implemented in two loops over the edges of the primary mesh. This formulation guarantees local conservation for problems with discontinuous coefficients, keeping second-order accuracy for smooth solutions on general triangular and orthogonal quadrilateral meshes. In order to show the convergence behavior of the proposed EBFV procedure, we solve three benchmark problems including full tensor, material heterogeneity and distributed source terms. For these three examples, numerical results compare favorably with others found in literature. A fourth problem, with highly non-smooth solution, has been included showing that the EBFV needs further improvement to formally guarantee monotonic solutions in such cases. Copyright © 2008 John Wiley & Sons, Ltd. [source] A gradient smoothing method (GSM) for fluid dynamics problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2008G. R. Liu Abstract A novel gradient smoothing method (GSM) based on irregular cells and strong form of governing equations is presented for fluid dynamics problems with arbitrary geometries. Upon the analyses about the compactness and the positivity of coefficients of influence of their stencils for approximating a derivative, four favorable schemes (II, VI, VII and VIII) with second-order accuracy are selected among the total eight proposed discretization schemes. These four schemes are successively verified and carefully examined in solving Poisson's equations, subjected to changes in the number of nodes, the shapes of cells and the irregularity of triangular cells, respectively. Numerical results imply us that all the four schemes give very good results: Schemes VI and VIII produce a slightly better accuracy than the other two schemes on irregular cells, but at a higher cost in computation. Schemes VII and VIII that consistently rely on gradient smoothing operations are more accurate than Schemes II and VI in which directional correction is imposed. It is interestingly found that GSM is insensitive to the irregularity of meshes, indicating the robustness of the presented GSM. Among the four schemes of GSM, Scheme VII outperforms the other three schemes, for its outstanding overall performance in terms of numerical accuracy, stability and efficiency. Finally, GSM solutions with Scheme VII to some benchmarked compressible flows including inviscid flow over NACA0012 airfoil, laminar flow over flat plate and turbulent flow over an RAE2822 airfoil are presented, respectively. Copyright © 2008 John Wiley & Sons, Ltd. [source] Temporal accuracy analysis of phase change convection simulations using the JFNK-SIMPLE algorithmINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2007Katherine J. Evans Abstract The incompressible Navier,Stokes and energy conservation equations with phase change effects are applied to two benchmark problems: (1) non-dimensional freezing with convection; and (2) pure gallium melting. Using a Jacobian-free Newton,Krylov (JFNK) fully implicit solution method preconditioned with the SIMPLE (Numerical Heat Transfer and Fluid Flow. Hemisphere: New York, 1980) algorithm using centred discretization in space and three-level discretization in time converges with second-order accuracy for these problems. In the case of non-dimensional freezing, the temporal accuracy is sensitive to the choice of velocity attenuation parameter. By comparing to solutions with first-order backward Euler discretization in time, it is shown that the second-order accuracy in time is required to resolve the fine-scale convection structure during early gallium melting. Qualitative discrepancies develop over time for both the first-order temporal discretized simulation using the JFNK-SIMPLE algorithm that converges the nonlinearities and a SIMPLE-based algorithm that converges to a more common mass balance condition. The discrepancies in the JFNK-SIMPLE simulations using only first-order rather than second-order accurate temporal discretization for a given time step size appear to be offset in time. Copyright © 2007 John Wiley & Sons, Ltd. [source] An eigenvector-based linear reconstruction scheme for the shallow-water equations on two-dimensional unstructured meshesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2007Sandra Soares Frazão Abstract This paper presents a new approach to MUSCL reconstruction for solving the shallow-water equations on two-dimensional unstructured meshes. The approach takes advantage of the particular structure of the shallow-water equations. Indeed, their hyperbolic nature allows the flow variables to be expressed as a linear combination of the eigenvectors of the system. The particularity of the shallow-water equations is that the coefficients of this combination only depend upon the water depth. Reconstructing only the water depth with second-order accuracy and using only a first-order reconstruction for the flow velocity proves to be as accurate as the classical MUSCL approach. The method also appears to be more robust in cases with very strong depth gradients such as the propagation of a wave on a dry bed. Since only one reconstruction is needed (against three reconstructions in the MUSCL approach) the EVR method is shown to be 1.4,5 times as fast as the classical MUSCL scheme, depending on the computational application. Copyright © 2006 John Wiley & Sons, Ltd. [source] Theoretical analysis for achieving high-order spatial accuracy in Lagrangian/Eulerian source termsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2006David P. Schmidt Abstract In a fully coupled Lagrangian/Eulerian two-phase calculation, the source terms from computational particles must be agglomerated to nearby gas-phase nodes. Existing methods are capable of accomplishing this particle-to-gas coupling with second-order accuracy. However, higher-order methods would be useful for applications such as two-phase direct numerical simulation and large eddy simulation. A theoretical basis is provided for producing high spatial accuracy in particle-to-gas source terms with low computational cost. The present work derives fourth- and sixth-order accurate methods, and the procedure for even higher accuracy is discussed. The theory is also expanded to include two- and three-dimensional calculations. One- and two-dimensional tests are used to demonstrate the convergence of this method and to highlight problems with statistical noise. Finally, the potential for application in computational fluid dynamics codes is discussed. It is concluded that high-order kernels have practical benefits only under limited ranges of statistical and spatial resolution. Additionally, convergence demonstrations with full CFD codes will be extremely difficult due to the worsening of statistical errors with increasing mesh resolution. Copyright © 2006 John Wiley & Sons, Ltd. [source] Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfacesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2005Anton Smolianski Abstract The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two-fluid interfacial flows, having in mind possible interface topology changes (like merger or break-up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator-splitting for temporal discretization and the level-set method for interface representation. We show that the finite element implementation of the level-set approach brings some additional benefits as compared to the standard, finite difference level-set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second-order accuracy of the interface normal, curvature and mass conservation. The operator-splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal-order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh,Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd. [source] An implicit three-dimensional fully non-hydrostatic model for free-surface flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2004Hengliang Yuan Abstract An implicit method is developed for solving the complete three-dimensional (3D) Navier,Stokes equations. The algorithm is based upon a staggered finite difference Crank-Nicholson scheme on a Cartesian grid. A new top-layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non-hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two-dimensional vertical plane problems, for each of which a block tri-diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free-surface flow problems using a very small number of vertical layers (e.g. two,four layers). The developed model is second-order accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd. [source] The design of improved smoothing operators for finite volume flow solvers on unstructured meshesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2001Benjamin de Foy Abstract Spatial operators used in unstructured finite volume flow solvers are analysed for accuracy using Taylor series expansion and Fourier analysis. While approaching second-order accuracy on very regular grids, operators in common use are shown to have errors resulting in accuracy of only first-, zeroth- or even negative-order on three-dimensional tetrahedral meshes. A technique using least-squares optimization is developed to design improved operators on arbitrary meshes. This is applied to the fourth-order edge sum smoothing operator. The improved numerical dissipation leads to a much more accurate prediction of the Strouhal number for two-dimensional flow around a cylinder and a reduction of a factor of three in the loss coefficient for inviscid flow over a three-dimensional hump. Copyright © 2001 John Wiley & Sons, Ltd. [source] Higher order explicit time integration schemes for Maxwell's equationsINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 5-6 2002Holger Spachmann Abstract The finite integration technique (FIT) is an efficient and universal method for solving a wide range of problems in computational electrodynamics. The conventional formulation in time-domain (FITD) has a second-order accuracy with respect to spatial and temporal discretization and is computationally equivalent with the well-known finite difference time-domain (FDTD) scheme. The dispersive character of the second-order spatial operators and temporal integration schemes limits the problem size to electrically small structures. In contrast higher-order approaches result not only in low-dispersive schemes with modified stability conditions but also higher computational costs. In this paper, a general framework of explicit Runge,Kutta and leap-frog integrators of arbitrary orders N is derived. The powerful root-locus method derived from general system theory forms the basis of the theoretical mainframe for analysing convergence, stability and dispersion characteristics of the proposed integrators. As it is clearly stated, the second- and fourth-order leap-frog scheme are highly preferable in comparison to any other higher order Runge,Kutta or leap-frog scheme concerning stability, efficiency and energy conservation. Copyright © 2002 John Wiley & Sons, Ltd. [source] Some numerical properties of approaches to physics,dynamics coupling for NWPTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 614 2006Mark Dubal Abstract At the present time there exist a number of different approaches to the problem of coupling parametrized physical processes to the dynamical core in operational numerical weather-prediction (NWP) and climate models. Motivated by the various strategies in use, some idealized representative coupling schemes are constructed and subsequently analysed using a methodology in which the physics and dynamics terms are represented in a simplified way. Particular numerical properties of the idealized schemes which are of interest are the ability to capture correct steady-state solutions and to be second-order accurate in time. In general, the schemes require specific choices for the time-differencing of certain coupled processes if correct steady-state solutions are to be obtained. This has implications for the overall numerical stability of a coupling strategy. An alternative physics,dynamics coupling approach is then described and analysed. A multiple-sweep predictor,corrector coupling scheme is shown to capture the correct steady-state solution and to allow for second-order accuracy, provided that the convective process is coupled explicitly. This approach has a number of advantages over those currently used in operational NWP models. Copyright © 2006 Crown copyright [source] | ||||||||||