Showing papers on "Temporal discretization published in 1998"

DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research

Abstract: ▪ Abstract We review the direct numerical simulation (DNS) of turbulent flows. We stress that DNS is a research tool, and not a brute-force solution to the Navier-Stokes equations for engineering problems. The wide range of scales in turbulent flows requires that care be taken in their numerical solution. We discuss related numerical issues such as boundary conditions and spatial and temporal discretization. Significant insight into turbulence physics has been gained from DNS of certain idealized flows that cannot be easily attained in the laboratory. We discuss some examples. Further, we illustrate the complementary nature of experiments and computations in turbulence research. Examples are provided where DNS data has been used to evaluate measurement accuracy. Finally, we consider how DNS has impacted turbulence modeling and provided further insight into the structure of turbulent boundary layers.

Finite Elements for Shallow-Water Equation Ocean Models

Abstract: The finite-element spatial discretization of the linear shallow-water equations on unstructured triangular meshes is examined in the context of a semi-implicit temporal discretization. Triangular finite elements are attractive for ocean modeling because of their flexibility for representing irregular boundaries and for local mesh refinement. The semi-implicit scheme is beneficial because it slows the propagation of the high-frequency small-amplitude surface gravity waves, thereby circumventing a severe time step restriction. High-order computationally expensive finite elements are, however, of little benefit for the discretization of the terms responsible for rapidly propagating gravity waves in a semi-implicit formulation. Low-order velocity/surface-elevation finite-element combinations are therefore examined here. Ideally, the finite-element basis-function pair should adequately represent approximate geostrophic balance, avoid generating spurious computational modes, and give a consistent discretization of the governing equations. Existing finite-element combinations fail to simultaneously satisfy all of these requirements and consequently suffer to a greater or lesser extent from noise problems. An unconventional and largely unknown finite-element pair, based on a modified combination of linear and constant basis functions, is shown to be a good compromise and to give good results for gravity-wave propagation.

A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation

Abstract: A second-order splitting method is applied to a KdV-like Rosenau equation in one space variable. Then an orthogonal cubic spline collocation procedure is employed to approximate the resulting system. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index 1. Error estimates in L2 and L∞ norms have been obtained for the semidiscrete approximations. For the temporal discretization, the time integrator RADAU5 is used for the resulting system. Some numerical experiments have been conducted to validate the theoretical results and to confirm the qualitative behaviors of the Rosenau equation. Finally, orthogonal cubic spline collocation method is directly applied to BBM (Benjamin-Bona-Mahony) and BBMB (Benjamin-Bona-Mahony-Burgers) equations and the well-known decay estimates are demonstrated for the computed solution.

Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows

Abstract: A higher order accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for 2D or 3D fluid flow problems It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth and sixth order compact finite-difference schemes for spatial discretization The particular difficulty of satisfying the divergence-free velocity field required in incompressible fluid flow is resolved by solving a Poisson equation for pressure It is demonstrated that for consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation Special care is also required to achieve the formal temporal accuracy of the Runge-Kutta schemes The accuracy of the present procedure is demonstrated by application to several pertinent benchmark problems

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Abstract: Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-directior is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton-Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm.

Optimal Control in Navier-Stokes Equations

Abstract: This paper presents a formulation for optimal control of a forced convection flow. The state equation that governs the forced convection flow can be expressed as the incompressible Navier-Stokes equations and energy equation. The optimal control can be formulated as finding a control force to minimize a performance function that is defined to evaluate a control object. The stabilized finite element method is used for the spatial discretization, while the Crank-Nicolson scheme is used for the temporal discretization. The Sakawa-Shindo method, which is an iterative procedure, is applied for minimizing the performance function.

Mathematical and Numerical Analyses of Dynamical Structure of Numerical Solutions of Two-Dimensional Fluid Equations

Abstract: In the present paper the structure of dynamical systems which were produced by discretizing the two-dimensional Burgers' equation is analyzed. An analytical approach and several numerical nonlinear dynamics approaches such as bifurcation diagram and so on are applied in order to discuss the structure of asymptotic numerical solutions. In particular, the dependence of the nonlinear structure of the numerical solutions on the temporal discretization parameter (Δ t ) are considered. Furtheremore, these numerical approaches are applied to the analyses of the results of more practical two-dimensional CFD (Computational Fluid Dynamics) calculations. The subsonic flow around a circular cylinder is selected as a model and the dependence of the structure of dynamical systems which are given from the computed time series of the drag coefficient on the temporal discretization parameter is discussed.

Second-Order Implicit Schemes that Satisfy the GCL for Flow Computations on Dynamic Grids

Abstract: We consider the solution of three-dimensi onal flow problems with moving boundaries using the Arbitrary Lagrangian Eulerian formulation or dynamic meshes. We focus on the case where spatial discretization is performed by unstructured finite volumes or finite elements. We formulate the consequence of the Geometric Conservation Law on the second-order implicit temporal discretization of the semi-discrete equations governing such problems, and use it as a guideline to construct a new family of second-order time-accurate and geometrically conservative implicit numerical schemes for flow computations on moving grids. We apply these new algorithms to the solution of three-dimensi onal flow problems with moving and deforming boundaries, demonstrate their superior accuracy and computational efficiency, and highlight their impact on the simulation of fluid/structure interaction problems.

Low-Storage Semi-Implicit Runge-Kutta Methods for Reactive Flow Computations

Abstract: The numerical methods developed in this paper are motivated by our current research project on direct numerical simulation of laminar-turbulent transition of reacting hypersonic boundary layers. In such simulations, the full Navier-Stokes equations with reactive source terms are simulated by resolving all flow time and length scales using high-order spatial and temporal discretization methods. Such reactive flow equations are stiff because the source terms modeling finite-rate thermochemical processes contain a wide range of time scales. Implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations. In addition, highorder accuracy is required for the overall methods in direct numerical simulation. This paper presents two families of low-storage semi-implicit Runge-Kutta methods and extends previous derivations (JCP 96) of general semi-implicit Runge-Kutta methods for split differential equations in the form of u = f ( t , u} +g(t, u), where / is treated explicitly and g is simultaneously treated implicitly. Two versions of such schemes, namely autonomous and non-autonomous methods have been derived and tested such that they are both high-order accurate and strongly stable for efficient computations.

A discrete-time approach to the steady state analysis of distributed nonlinear autonomous circuits

Abstract: We present a new method for the steady state analysis of autonomous circuits with transmission lines and generic nonlinear elements. With the temporal discretization of the equations that describe the circuit, we obtain a nonlinear algebraic formulation where the unknowns to be determined are the samples of the variables directly in the steady state, along with the oscillation period, the main unknown in autonomous circuits. An efficient scheme to build the Jacobian matrix with exact partial derivatives with respect to the oscillation period and with respect to the samples of the unknowns is described. To illustrate the proposed technique, the time-delayed Chua's circuit is analyzed in its periodic zones.

Efficient incompressible flow calculations using PQ2Q1 element

Abstract: Implementation of an equal-order-interpolation velocity–pressure element pair is presented for the finite element solution of incompressible viscous flows. A fractional-step method is employed for temporal discretization. The element pair, also called a pseudo-biquadratic velocity/bilinear pressure element (pQ2Q1), consists of a bilinear pressure element and bilinear velocity elements defined on subdivisions of the pressure element. This pair satisfies the so-called ‘Ladyzhenskaya–Babuska–Brezzi’ condition. Considerable savings in computational cost are achieved due to the reduced number of elements for pressure. A modification of the element is realized for a better representation of curved surfaces. Two test cases, namely the lid-driven cavity flow and impulsively started circular cylinder in cross-flow, are used to assess the accuracy and efficiency of the element compared to a regular bilinear velocity–pressure (Q1Q1) element pair. Computational results presented show that the pQ2Q1 element solutions require less memory and CPU time compared to Q1Q1 element solutions, for at least the same accuracy. © 1998 John Wiley & Sons, Ltd.

Computation of Steady and Unsteady Laminar Flames: Theory

Abstract: In this paper we describe the numerical analysis underlying our efforts to develop an accurate and reliable code for simulating flame propagation using complex physical and chemical models. We discuss our spatial and temporal discretization schemes, which in our current implementations range in order from two to six. In space we use staggered meshes to define discrete divergence and gradient operators, allowing us to approximate complex diffusion operators while maintaining ellipticity. Our temporal discretization is based on the use of preconditioning to produce a highly efficient linearly implicit method with good stability properties. High order for time accurate simulations is obtained through the use of extrapolation or deferred correction procedures. We also discuss our techniques for computing stationary flames. The primary issue here is the automatic generation of initial approximations for the application of Newton's method. We use a novel time-stepping procedure, which allows the dynamic updating of the flame speed and forces the flame front towards a specified location. Numerical experiments are presented, primarily for the stationary flame problem. These illustrate the reliability of our techniques, and the dependence of the results on various code parameters.

On the origin of model errors. Part I. Effects of the temporal discretization for Hamiltonian systems

Abstract: There are two main sources of errors in numerical modeling of the atmosphere: the errors of initial conditions and that of the models themselves. The algebraic structure of equations is strongly related to their integrability, thus the problem of sensitivity to initial data cannot be handled separately from that of model formulation (i.e. of model errors). In a nonintegrable system the utilization of a numerical solution algorithm is a must. The present review paper (in two parts) deals with model errors originating from the inevitable discretization of the continuous equations in space and time. The effects of these errors are investigated in case of Hamiltonian systems in terms of phase space structure. It is pointed out in Part I that (1) even if the discretized ordinary differential equations were perfect the time integration schemes unavoidably introduce errors, (2) the time discretization errors have a strong influence on the time evolution of the probability distribution function in the phase space, thus an ensemble of numerical runs is always distorted by model errors, (3) the initial spin-up process is unavoidable, even if the true state of the system can be observed. In a Hamiltonian system the accuracy of a numerical solution can always be improved by increasing the order of the integration scheme and decreasing the length of the time step. The concepts introduced are illustrated with simple examples and with numerical experiments carried out with the 2D vorticity equation and the two-layer quasi-geostrophic model.

A Fully—Implicit 2‐D Navier-Stokes Algorithm for Unstructured Grids

Abstract: A fully-implicit algorithm is developed for the two-dimensional, compressible, Favre-averaged Navier-Stokes equations. It incorporates the standard k-∊ turbulence model of Launder and Spalding and the low Reynolds number correction of Chien. The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. A generalization of wall functions including pressure gradient effects is implemented to solve the near-wall region for turbulent flows using a separate algorithm and a hybrid grid. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, Trapezoidal or 3-Point Backward dif...

Numerical methods for problems in computational aeroacoustics

Abstract: A goal of computational aeroacoustics is the accurate calculation of noise from a jet in the far field. This work concerns the numerical aspects of accurately calculating acoustic waves over large distances and long time. More specifically, the stability, efficiency, accuracy, dispersion and dissipation in spatial discretizations, time stepping schemes, and absorbing boundaries for the direct solution of wave propagation problems are determined. Efficient finite difference methods developed by Tam and Webb, which minimize dispersion and dissipation, are commonly used for the spatial and temporal discretization. Alternatively, high order pseudospectral methods can be made more efficient by using the grid transformation introduced by Kosloff and Tal-Ezer. Work in this dissertation confirms that the grid transformation introduced by Kosloff and Tal-Ezer is not spectrally accurate because, in the limit, the grid transformation forces zero derivatives at the boundaries. If a small number of grid points are used, it is shown that approximations with the Chebyshev pseudospectral method with the Kosloff and Tal-Ezer grid transformation are as accurate as with the Chebyshev pseudospectral method. This result is based on the analysis of the phase and amplitude errors of these methods, and their use for the solution of a benchmark problem in computational aeroacoustics. For the grid transformed Chebyshev method with a small number of grid points it is, however, more appropriate to compare its accuracy with that of high-order finite difference methods. This comparison, for an order of accuracy 10$\sp{-3}$ for a benchmark problem in computational aeroacoustics, is performed for the grid transformed Chebyshev method and the fourth order finite difference method of Tam. Solutions with the finite difference method are as accurate. and the finite difference method is more efficient than, the Chebyshev pseudospectral method with the grid transformation. The efficiency of the Chebyshev pseudospectral method is further improved by developing Runge-Kutta methods for the temporal discretization which maximize imaginary stability intervals. Two new Runge-Kutta methods, which allow time steps almost twice as large as the maximal order schemes, while holding dissipation and dispersion fixed, are developed. In the process of studying dispersion and dissipation, it is determined that maximizing dispersion minimizes dissipation, and vice versa. In order to determine accurate and efficient absorbing boundary conditions, absorbing layers are studied and compared with one way wave equations. The matched layer technique for Maxwell equations is equivalent to the absorbing layer technique for the acoustic wave equation introduced by Kosloff and Kosloff. The numerical implementation of the perfectly matched layer for the acoustic wave equation with a large damping parameter results in a small portion of the wave transmitting into the absorbing layer. A large damping parameter also results in a large portion of the wave reflecting back into the domain. The perfectly matched layer is implemented on a single domain for the solution of the second order wave equation, and when implemented in this manner shows no advantage over the matched layer. Solutions of the second order wave equation, with the absorbing boundary condition imposed either by the matched layer or by the one way wave equations, are compared. The comparison shows no advantage of the matched layer over the one way wave equation for the absorbing boundary condition. Hence there is no benefit to be gained by using the matched layer, which necessarily increases the size of the computational domain.

Selection of Spatial and Temporal Discretization in Wetland Modelling

Abstract: Spatialandtemporaldiscretizationsarekey factorsdecidingtheoptimaluseof computerresourcesin a wetlandmodelingapplication.Thediscretizationshouldbesufficientlyfineto describethesolutionwith areasonableresolution,andpreventexcessi ve numericalerrors.It shouldnotbetoofineto preventtheruntimesbecomingexcessi ve. Thepaperdescribesa studyaimedat understandingthevariationof numericalerrors andtherun timesof 1-D and2-D overlandandgroundwaterflow models,in termsof non-dimensional spaceandtime discretizations. Fourieranalysisof thelinearizedgoverningequationis usedin thestudyto obtain analyticalexpressionsfor numericalerrorsandrun times.Numericalexperimentsare carriedout with explicit modelto obtainresultsfor comparison.Thepaperdescribes theuseof theresultsin selectingspaceandtime discretizationsfor wetlandmodeling applications. INTRODUCTION Overlandandgroundwatercomponentsof flow in wetlandmodelsare governedby nonlinearandlinearparabolicpartialdifferentialequations.TheNaturalSystemModel (NSM) andtheSouthFloridaWatermanagement Model (SFWMM), (Fennema, etal., 1994, Lal, 1998) are two modelsbasedon diffusion flow that are usedto simulate flow in theEverglades.Other2-D overlandmodelsusingthediffusionflow assumption include the WETFLOW model by Fengand Molz (1997), and the modelsby HromadkaandLai (1985). Two dimensionalapplicationof the MODFLOW model (McDonaldandHarbough,1984)is basedonasimilargoverningequation.Numerical methodsavailableto solve parabolicequationsusingrectangularspatialgrids include the explicit method,the alternatingdirectionexplicit methods(ADE) in the caseof theNSM/SFWMM models,the implicit methodsasin WETFLOW andMODFLOW models,andtheADI method.

A discrete-time app oach to the steady state analysis of distri uted nonlinear autq~q~qus circuits

Abstract: We present a new method for the steady state analysis of autonomous circuits with transmission lines and generic nonlinear elements. With the temporal discretization of the equations that describe the circuit, we obtain a nonlinear algebraic formulation where the unknowns to be determined are the samples of the variables directly in the steady state, along with the oscillation period, the main unknown in autonomous circuits. An efficient scheme to build the Jacobian matrix with exact partial derivatives with respect to the oscillation period and with respect to the samples of the unknowns is described. To illustrate the proposed technique, the time-delayed Chua's circuit is analized in its periodic zones.

Time-domain analysis of flexible foundations using lanczos vectors

Abstract: An analytical procedure to obtain the response of embedded flexible foundation in time domain is described. The procedure makes use of large domain for discretization along with coordinate transformation using Lanczos vectors. The responses are obtained in time domain using an adaptive direct integration method. The scheme has the ability of error estimation due to temporal discretization and coordinate transformation. The procedure has been applied to three-dimensional flexible foundations. The compliance functions of the foundations have been obtained for all modes of vibration. The computational scheme has also been used to analyze the response of a machine foundation transmitting nonharmonic but periodic forces. The present method has all the advantages of time domain scheme, which is local in space and time with small computational effort.

Designing high-order, time-domain numerical solvers for Maxwell's equations

Abstract: Considerable time and energy has been devoted to the design and development of second order, time-domain schemes for Maxwell's equations. Schemes, for example, such as the finite-difference time-domain (FDTD) method have been applied to many diverse problems ranging from radar cross-section analysis to ionospheric radio wave propagation studies. Yet, the second-order nature of the schemes prevents one from obtaining accurate data for geometries or propagation distances that span tens of wavelengths. This property is clearly manifested by frequency domain analyses, which demonstrate the high accumulation of phase and dissipation errors over many time steps. To mitigate the effect of phase and dissipation errors, two key options exist: (1) decrease the size of the discretization cell and the time step or (2) increase the order of accuracy. We consider only option two. In essence, the design of a time domain solver is subdivided into three essential tasks: (1) spatial discretization, (2) temporal discretization and (3) evaluation of the fully discrete system.