High-order Difference Scheme Research Articles

High-Throughput TVD-Based Simulation of Tracer Flow

Abstract High-throughput, total-variation-diminishing based, numerical techniques for simulating tracer flow are developed. High-throughput (HT) timestepping enables explicit differencing techniques to use timesteps that are larger than allowed by normal stability constraints. Total-variation-diminishing (TVD) based techniques are a means of controlling the solution oscillations associated with high-order differencing schemes. The concept of ‘upstream stability’ is introduced to help combine HT timestepping with TVD-based differencing. Additionally, an improved HT timestepping algorithm for solving physically dispersive flow problems is developed. Numerical experiments indicate that HT timestepping is most beneficial for solving highly convective flows that have both a physical Peclet number and a diffusion-only numerical Peclet number greater than one. Experiments comparing standard and alternating direction TVD-based techniques with both the Superbee (SB) and third-order based (L3) limiters indicate that the TVD-SB scheme is the most accurate for purely convective problems. For physically dispersive problems, the TVD-SB scheme may be too compressive and the TVD-L3 is recommended. The alternating direction TVD scheme is the least sensitive to timestepping and the most sensitive to grid-orientation.

A convection scheme sensitized to the convection direction of a scalar quantity

It is generally believed that higher order differencing schemes for the convection transport term, e.g., the QUICK scheme and its variants, are superior to the first order simple upwind differencing scheme in the sense that the former produces less numerical diffusion than the latter. In this paper it is shown that this conclusion is no more correct when the flow changes its direction quite rapidly and the grid density is not sufficient. In this situation the simple upwind differencing returns much steeper change of convective variables than higher order schemes. The failure of usual higher order schemes for this flow condition is attributed to the ignorance of convection direction of variables, and a new convection scheme sensitized to the direction of convective transport of a scalar quantity is devised and applied to typical benchmark flows. Results show that the proposed scheme is sufficiently accurate for computation of scalar fields, and also show the optimum behavior for tested problems.

High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients

In this paper, we develop implicit difference schemes of O( k 4 + k 2 h 2 + h 4), where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions. The proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable ADI method of O( k 4 + k 2 h 2 + h 4). Some physical nonlinear problems are provided to demonstrate the accuracy of the implementation.

High-order difference schemes by modification of the right-hand side of three-dimensional Poisson equation

High-order difference schemes by modification of the right-hand side of three-dimensional Poisson equation

High Order Difference Schemes for Unsteady One-Dimensional Diffusion-Convection Problems

For unsteady 1D diffusion-convection problems, this paper develops an extensive analysis of two-level three-point finite difference schemes of order 2 in time and 4 in space. This general class of FDS includes several schemes independently proposed by different authors. One main objective is the identification of difference schemes yielding satisfactory numerical results for strongly convective problems (i.e., when the cell Reynolds number bα = λ h/2 is greater than unity). The stability and the oscillatory behaviour of the schemes are carefully studied and the analyses are completed by some numerical experiments. We outline some key points: (i) the great difficulty to obtain accurate numerical results for large values of α (ii) the possibility of virtually optimum schemes is essentially theoretical and requires, in practice, careful experiments; (iii) for strongly convective problems, some second-order explicit schemes are almost as efficient (and less costly) than implicit fourth-order schemes.

Investigation of turbulence modelling in thermal stratification analysis

In-vessel thermal stratification analysis was carried out using a multidimensional thermohydraulic analysis code, in which a higher-order finite difference scheme was applied to the convection terms. Discussions centred on the buoyancy modelling in the vicinity of the stratification interface through comparisons between experiment and calculation. Computational results were obtained from the following three turbulence models: (i) the k−ϵ model with a constant turbulent Prandtl number Pr t, (ii) the k−ϵ model with the turbulent Prandtl number being dependent on the local Richardson number Ri, and (iii) the algebraic stress model. Numerical analysis of the stratification phenomena using the higher-order scheme showed that, in general, the modelling of the buoyancy terms appearing in the turbulence transport equations was the most important key to successful results. When the k−ϵ model was used, it was pointed out that a dependence on the local Richardson number must be carefully included in the turbulent Prandtl number. In this case, however, the range of applicability was limited to the phenomena observed in the water system in general because the model was constructed and calibrated for water experiments. Overall it was found that the calculated stratification interface rise agreed well with experimental results in water and sodium insofar as the algebraic stress model was utilized. As a conclusion, in predicting the behaviour of the thermal stratification phenomena in liquid metal cooled reactors, the coupled use of the higher-order difference scheme and the algebraic stress model was most appropriate and recommended.

Numerical Hydrodynamics from Gas-Kinetic Theory

We present a new hydrodynamics code, based on the solution of the Bhatnagar-Gross-Krook model of the Boltzmann equation. The basic idea is to construct an approximate, locally valid solution of a set of non-linear integral equations for the equilibrium Maxwell-Boltzmann distribution, and use this solution to solve the BGK equation for the velocity distribution function at cell walls. Once this is known it is possible to compute time-dependent fluxes of mass, momentum, and energy between cells. All steps are carried out explicitly for the case of a perfect gas; the details of the distribution functions disappear in the final code. In common with other "Boltzmann-type" codes employing distribution functions, the Riemann problem does not arise. The novelty of the present code consists in solving for the distribution function, taking account of collisions, rather than assuming a convenient form for it. Results are presented for a number of standard test cases, with features ranging from cavitation (Sjögreen test) to the collision of strong shocks (Woodward-Colella test). All cases are run with precisely the same code, and all cells are treated in the same way. Our code appears to behave as well as current high-order difference schemes at shocks and to give better results for rarefaction waves. It is competitive with current codes that do not employ special means of diagnosing and treating contact discontinuities—the inclusion of such devices in our scheme remains an option.

A synthetically accelerated scheme for radiative transfer calculations

Recently, we have developed a one-dimensional radiation-hydrodynamics code called SARA ( Synthetically Accelerated Radiation-Hydrodynamics Algorithm) which solves the multigroup radiation transport equations implicitly. A high order differencing scheme as well as multifrequency grey synthetic acceleration techniques are used to obtain a robust algorithm that can be used for a large variety of energy transport problems. The advantages of such an algorithm are, first, that it solves the true ( S n ) transport equations, secondly, a positive differencing scheme which verifies the asymptotic diffusion limit is used, and, finally, the S n equations are accelerated by the S 2 synthetic acceleration method, which allows the acceleration of high order schemes in plane and curvilinear geometries. The code has been validated with some Marshak wave propagation problems found in the bibliography, and successfully used to simulate the conversion experiments of laser and ion beams into X-rays.

Improving the numerical modeling of river water quality by using high order difference schemes

A one-dimensional numerical model is presented for the description of water quality in rivers. The model is an improvement over existing models, since it involves two differential schemes of third-order truncation error (QUICK and QUICKEST), which avoid the stability problems of central difference, while remaining relatively free of the inaccuracies of numerical dispersion associated with the upwind difference scheme. The model is successfully applied to four simple, steady-state and transient water quality problems. Results show that the commonly used upwind scheme performs very poorly. The central difference scheme performs satisfactorily in steady-state implicit calculations, but not in transient problems, where it shows numerical dispersion and significant oscillations. Calculations with QUICK and QUICKEST are stable and accurate involving the minimum error and minimum level of numerical dispersion in both steady-state and transient problems.

特性多項式解析法に基づく高次差分ＬＥＣＵＳＳＯスキームの数値振動解析

High-order difference schemes for thermohydraulic equations tend to show solutions suffering from numerical oscillations. Basic considerations on the occurrence and behaviors of numerical oscillations are presented, and two aspects of dynamic oscillations and static oscillations in the numerical oscillations are pointed out. A practically useful method for static numerical oscillations analysis based on a characteristic polynomial technique is proposed.The LECUSSO scheme, which determines the upwinding weight such that the difference scheme satisfies an analytical solution of the convection-diffusion equation, is extended so as it can be used in nonuniform mesh size grids. The effects of computational grids on static numerical oscillations are discussed with the characteristic polynomial analysis on the extended LECUSSO scheme. It is found that the extended LECUSSO scheme shows solutions free from numerical oscillations only in uniform mesh size grids. Limitations on the application of the static oscil a ions analysis based on the characteristic polynomial technique are discussed, and its applicability for transient problems, multidimensional problems and nonlinear problems are examined through numerical experiments.

Numerical diffusion in single-phase multi-dimensional thermal-hydraulic analysis

Higher-order difference schemes, including QUICK and its related schemes, are discussed through several numerical experiments with the focus on their performance to reduce numerical diffusion and to suppress local unphysical oscillations. Finally, these schemes are applied to in-vessel thermal-hydraulic analysis and their effectiveness on the analysis is demonstrated.

Min-max truncation: An accurate and stable filtering method for difference calculation of convection

Higher-order difference schemes always cause oscillation in convection calculation though they require much effort in the calculation. Overshoot and undershoot of the oscillation is troublesome particularly in the calculations of temperature, energy, and turbulence parameters. Filtering techniques improve both accuracy and stability, and some of them can completely suppress the overshoot or undershoot, when they are combined with the higher-order difference schemes. In the present study, a new min-max truncation (MMT) method is proposed. It is evaluated in two test problems of pure convection, and compared with other filtered and not-filtered difference schemes.

Studies in numerical computations of recirculating flows

AbstractThis paper considers the use of various finite differencing schemes for the computation of flows involving regions of recirculation. Standard first‐order hybrid schemes, vector (or skew) schemes and second‐order schemes are used to predict laminar flows in a channel containing a constriction and over a normal flat plate with a downstream splitter plate. In the former case the results are compared with those of other workers and with the implications of analytic theories for the viscous dominated flow around the sharp corner.Attention is concentrated on the effects of errors arising from the use of non‐uniform grids and it is shown that higher‐order differencing schemes are generally much less susceptible to these than the simpler schemes. The major conclusion is that for flows containing regions where pressure gradients largely balance the convective terms in the momentum equations, in addition to other regions where convection and diffusion balance, higher order differencing schemes are likely to be essential if accurate predictions are required on grids without excessive numbers of nodes. It is argued that similar conclusions must hold for high Reynolds number turbulent flows.

Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem

On construit une famille de schemas aux differences finies uniformement precis pour le probleme du type −eu''+a(x)u'+b(x)u=f(x). On donne une analyse complete de l'erreur de discretisation basee sur les resultats de stabilite. On presente des essais numeriques

Supplement to Uniform High-Order Difference Schemes for a Singularly Perturbed Two-Point Boundary Value Problem

Supplement to Uniform High-Order Difference Schemes for a Singularly Perturbed Two-Point Boundary Value Problem

High order difference schemes with reduced dispersion for hyperbolic differential equations

We investigate difference schemes for systems of first order hyperbolic differential equations in two space dimensions, possessing the following characteristics: 1. The spatial discretizations are fourth order accurate. 2. The time discretization is of explicit Runge-Kutta type and is also fourth order accurate. 3. The scaled stability boundary is approximately 1 2 2 . 4. The weights in the space discretizations and the Runge-Kutta parameters can be adapted in order to reduce the dispersion of dominant Fourier components. This method is illustrated by applying it to the shallow water equations simulating the motion of water in a shallow sea due to tidal forces. Since in such problems the dominant frequencies in the solution are known in advance, the method can take fully advantage of the possibility to tune the various parameters to these dominant frequencies.

Direct evaluation of critical conditions for thermal explosion and catalytic reaction

A novel method of evaluating the critical conditions of a solid explosive is suggested. The procedure can be also used for calculation of ignition and extinction points of an exothermic catalytic reaction. The method is straightforward and can be used for any arbitrary geometry. The transport equations are replaced by a high-order difference scheme and the critical conditions are calculated numerically from an analytic condition for branching. The accuracy of the results is improved by an extrapolation procedure. Richardson's method and the ϵ-algorithm have been used for extrapolation. The method is very simple, inexpensive, and can be used also for multidimensional problems.

Stability and convergence of high-order difference schemes for parabolic partial differential equations

Stability and convergence of high-order difference schemes for parabolic partial differential equations

Construction of high-order accuracy difference schemes for quasilinear elliptic equations III. Two-dimensional case

Construction of high-order accuracy difference schemes for quasilinear elliptic equations III. Two-dimensional case

High order difference schemes for the wave equation

AbstractADI difference schemes for the solution of the wave equation with variable coefficients are given. Mckee1 has given a formula of order O(k2 + h4), while Ciment and Leventhal2 have given a formula of order O(k4 + h4). Both the formulas are conditionally stable with the condition λ√c λ √3−1. In this paper we have derived uncoditionally stable ADI schemes of orders O(k2 + h2) and O(k2 + h4). These schemes allow large time steps and the computational effort is thus reduced. Numerical experiments suggest the usefulness of our formulas.