Numerical Truncation Error Research Articles

Dynamics and control of a 5-DOF manipulator based on an H-4 parallel mechanism

This paper presents a design of a unique hybrid five-degree-of-freedom manipulator based on an H-4 family parallel mechanism with three translational movements and one rotational movement (orientation angle) together with a single axis rotating table. Forward or direct kinematics, inverse kinematics, and Jacobian are derived in details as well as the dynamic model. The dynamic model is derived from the Lagrangian formulation and is shown to be suitable in the real-time feedback control. The numerical results of the analysis of kinematics, inverse kinematics, and the dynamic model are compared with the results from a popular commercial software using a virtual modeling data, the ADAMS solver. Friction models obtained from the experiment are used to compensate for the actual friction of the system in the resolve acceleration control strategy. The inverse dynamics is implemented for the first four axis of the chosen configuration to perform feedback linearization. From the experimental results, the tracking performance is satisfied and can be improved by increasing the rigidity of the railway structure and reducing the numerical truncation error.

Implicit large-eddy simulation applied to turbulent channel flow with periodic constrictions

The subgrid-scale (SGS) model in a large-eddy simulation (LES) operates on a range of scales which is marginally resolved by discretization schemes. Accordingly, the discretization scheme and the subgrid-scale model are linked. One can exploit this link by developing discretization methods from subgrid-scale models, or the converse. Approaches where SGS models and numerical discretizations are fully merged are called implicit LES (ILES). Recently, we have proposed a systematic framework for the design, analysis, and optimization of nonlinear discretization schemes for implicit LES. In this framework parameters inherent to the discretization scheme are determined in such a way that the numerical truncation error acts as a physically motivated SGS model. The resulting so-called adaptive local deconvolution method (ALDM) for implicit LES allows for reliable predictions of isotropic forced and decaying turbulence and of unbounded transitional flows for a wide range of Reynolds numbers. In the present paper, ALDM is evaluated for the separated flow through a channel with streamwise-periodic constrictions at two Reynolds numbers Re = 2,808 and Re = 10,595. We demonstrate that, although model parameters of ALDM have been determined for isotropic turbulence at infinite Reynolds number, it successfully predicts mean flow and turbulence statistics in the considered physically complex, anisotropic, and inhomogeneous flow regime. It is shown that the implicit model performs at least as well as an established explicit model.

Towards implicit subgrid-scale modeling by particle methods

The numerical truncation error of vortex-in-cell methods is analyzed a-posteriori through the effective spectral numerical viscosity for simulations of three-dimensional isotropic turbulence. The interpolation kernels used for velocity-smoothing and re-meshing are identified as the most relevant components affecting the shape of the spectral numerical viscosity as a function of wave number. A linear combination of well-known standard kernels leads to new kernels assigned to the specific use for implicit large-eddy simulation of turbulent flows, i.e. their truncation errors acts as subgrid- scale model. Numerical results are provided to show the potential and drawbacks of the approach.

A proposed simplification of the adaptive local deconvolution method

T he adaptive local deconvolution method (ALDM) (9) provides as ystematic framework for the implicit large-eddy simulation (ILES) of turbulent flows. Exploiting numerical truncation errors, the subgrid scale (SGS) model of ALDM is implicitly contained within the discretization. An explicit computation of model terms therefore becomes unnecessary. Subject of the present paper is a modification of the numerical algorithm that allows for reducing the amount of computational operations without a!ecting the quality of the results. Computational results for isotropic turbulence and plane channel flow show that the simplified adaptive local deconvolution (SALD) method performs similarly to the original method ALDM and at least as well as established explicit models.

Mitigating the integration error in numerical simulations of Newtonian systems

AbstractWe introduce a method for mitigating the numerical integration errors of linear, second‐order initial value problems. We propose a methodology for constructing an optimal state‐space representation that gives minimum numerical truncation error, and in this sense, is the optimal state‐space representation for modelling given phase‐space dynamics. To that end, we utilize a simple transformation of the state‐space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in electromagnetism. We then utilize the gauge function to reduce the numerical integration error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time‐step. Moreover, we derive general expressions yielding the optimal gauge functions given a Newtonian one degree‐of‐freedom ODE. For the n degrees‐of‐freedom case we describe MATLAB® code capable of finding the optimal gauge functions and integrating the given system using the gauge‐optimized integration algorithm. In all of our illustrating examples, the gauge‐optimized integration outperforms the integration using standard state variables by a few orders of magnitude. Copyright © 2006 John Wiley & Sons, Ltd.

Reducing reflections from mesh refinement interfaces in numerical relativity

Full interpretation of data from gravitational wave observations will require accurate numerical simulations of source systems, particularly binary black hole mergers. A leading approach to improving accuracy in numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a manifestation of numerical interface truncation error which appears as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh refinement implementations, potentially compromising the effectiveness of mesh refinement techniques for some numerical relativity applications if left untreated. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite differencing stencil modifications, termed mesh-adapted differencing (MAD), which eliminate this pathology in both our model problem and in numerical relativity examples.

Finescale Topography and the MC2 Dynamics Kernel

Abstract The Canadian Mesoscale Compressible Community (MC2) model provided daily forecasts across the Alps at 3-km resolution during the Mesoscale Alpine Programme (MAP) field phase of 1999. Among the results of this endeavor, some have had an immediate impact on MC2 itself as it increasingly became evident that the model was spuriously too sensitive to finescale orographic forcing. The model solves the Euler equations of motion using a semi-implicit semi-Lagrangian scheme in an oblique terrain-following coordinate. To improve model behavior, typical approaches were tried at first. These included a generalization of the coordinate transformation to make the terrain influence decay much more quickly with height as well as the introduction of nonisothermal basic states to diminish the amplitude of numerical truncation errors. The concept of piecewise-constant finite elements was invoked to reduce coding arbitrariness. But it was later pointed out that the problem was very specific and due to a numerical inconsistency. The true height of model grid points is fixed and known in height-based coordinates. Nevertheless, it was discovered that for this semi-Lagrangian scheme to be consistent, the departure height is an unknown that must be obtained in the same manner as the other unknowns.

Error analysis of finite difference methods for two-dimensional advection–dispersion–reaction equation

In this paper, the numerical errors associated with the finite difference solutions of two-dimensional advection–dispersion equation with linear sorption are obtained from a Taylor analysis and are removed from numerical solution. The error expressions are based on a general form of the corresponding difference equation. The variation of these numerical truncation errors is presented as a function of Peclet and Courant numbers in X and Y direction, a Sink/Source dimensionless number and new form of Peclet and Courant numbers in X–Y plane. It is shown that the Crank–Nicolson method is the most accurate scheme based on the truncation error analysis. The effects of these truncation errors on the numerical solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution for predicting contaminant plume distribution in uniform flow field. Considering computational efficiency, an alternating direction implicit method is used for the numerical solution of governing equation. The results show that removing these errors improves numerical result and reduces differences between numerical and analytical solution.

Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results

We present here both analytical and numerical results of hydrodynamic stability investigations of rotationally supported circumstellar flows using the shearing box formalism. Asymptotic scaling arguments justifying the shearing box approximation are systematically derived, showing that there exist two limits which we call small shearing box (SSB) and large shearing box (LSB). The physical meaning of these two limits and their relationship to model equations implemented by previous investigators are discussed briefly. Two dimensional (2D) dynamics of the SSB are explored and shown to contain transiently growing (TG) linear modes, whose nature is first discussed within the context of linear theory. The fully nonlinear regime in 2D is investigated numerically for very high Reynolds (Re) numbers. Solutions exhibiting long-term dynamical activity are found and manifest episodic but recurrent TG behavior and these are associated with the formation and long-term survival of coherent vortices. The life-time of this spatio-temporal complexity depends on the Re number and the strength and nature of the initial disturbance. The dynamical activity in finite Re solutions ultimately decays with a characteristic time increasing with Re. However, for large enough Re and appropriate initial perturbation, a large number of TG episodes recur before any viscous decay begins to clearly manifest itself. In cases where nominally (i.e. any dissipation resulting only from numerical truncation errors), the dynamical activity persists for the entire duration of the simulation (hundreds of box orbits). Because the SSB approximation used here is equivalent to a 2D incompressible flow, the dynamics can not depend on the Coriolis force. Therefore, three dimensional (3D) simulations are needed in order to decide if this force indeed suppresses nonlinear hydrodynamical instability in rotationally supported disks in the shearing box approximation, and if recurrent TG behavior can still persist in three dimensions as well – possibly giving rise to a subcritical transition to long-term spatio-temporal complexity.

Numerical errors of explicit finite difference approximation for two-dimensional solute transport equation with linear sorption

The numerical errors associated with explicit upstream finite difference solutions of two-dimensional advection—Dispersion equation with linear sorption are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation. The numerical truncation errors are defined using Peclet and Courant numbers in the X and Y direction, a sink/source dimensionless number and new Peclet and Courant numbers in the XY plane. The effects of these truncation errors on the explicit solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution in uniform flow field. The results show that these errors are not negligible and correcting the finite difference scheme for them results in a more accurate solution.

A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.

Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography

A model based on the finite-volume method is developed for unsteady, two-dimensional, shallow-water flow over arbitrary topography with moving lateral boundaries caused by flooding or recession. The model uses Roe's approximate Riemann solver to compute fluxes, while the monotone upstream scheme for conservation laws and predictor-corrector time stepping are used to provide a second-order accurate solution that is free from spurious oscillations. A robust, novel procedure is presented to efficiently and accurately simulate the movement of a wet/dry boundary without diffusing it. In addition, a new technique is introduced to prevent numerical truncation errors due to the pressure and bed slope terms from artificially accelerating quiescent water over an arbitrary bed. Model predictions compare favorably with analytical solutions, experimental data, and other numerical solutions for one- and two-dimensional problems.

Eulerian Time-Domain Filtering for Spatial Large-Eddy Simulation

An approach to large-eddy simulation is developed whose subgrid-scale model incorporates Eulerian time-domain filtering, in contrast to conventional approaches that exploit spatial filtering. For applications to large-eddy simulation, time-domain filters enjoy certain advantages relative to spatial filters. Among these, they more naturally commute with differentiation operators than do spatial filters, and there is typically wide separation between numerical truncation error and the dissipation of the subgrid-scale model. Eulerian time-domain filtering is most appropriate for flows whose large coherent structures convect approximately at a common characteristic velocity. Such flows include jets, mixing layers, and wakes. The method is demonstrated in the large-eddy simulation of a heated, subsonic, axisymmetric jet, and results are compared with those obtained from well-resolved direct numerical simulation

Microcomputer based Harmonic Estimation of Thyristor Converter Waveform using Recursive Discrete Fourier Transform

The paper presents an application of recursive discrete Fourier transform (RDFT) method involving real number computations for estimating the harmonic magnitudes in the waveforms of a 6 pulse, full wave thyristor bridge circuit. The signal is modelled in state variable form as the output of a dynamic system. A dead-beat spectral observer in discrete time, operating at the Nyquist sampling rate estimates the harmonic amplitude components recursively since it performs one step per-sample updating. The estimates rapidly converge to yield the transform coefficient after exactly the requisite number of samples decided by Nyquist rate, even in the presence of numerical truncation errors. A microcomputer based instrumentation scheme for on-line implementation of the RDFT is suggested.

Conformal hyperbolic formulation of the Einstein equations

We propose a reformulation of the Einstein evolution equations that cleanly separates the conformal degrees of freedom and the nonconformal degrees of freedom with the latter satisfying a first order strongly hyperbolic system. The conformal degrees of freedom are taken to be determined by the choice of slicing and the initial data, and are regarded as given functions (along with the lapse and the shift) in the hyperbolic part of the evolution. We find that there is a two parameter family of hyperbolic systems for the nonconformal degrees of freedom for a given set of trace free variables. The two parameters are uniquely fixed if we require the system to be ``consistently trace-free,'' i.e., the time derivatives of the trace free variables remain trace-free to the principal part, even in the presence of constraint violations due to numerical truncation error. We show that by forming linear combinations of the trace free variables a conformal hyperbolic system with only physical characteristic speeds can also be constructed.

Truncation errors in finite difference models for solute transport equation with first-order reaction

The truncation errors associated with finite difference solutions of the advection-dispersion equation with first-order reaction are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation and a temporally and spatially weighted parametric approach is used for differentiating among the various finite difference schemes. The numerical truncation errors are defined using Peclet and Courant numbers and a new Sink/Source dimensionless number. It is shown that all of the finite difference schemes suffer from truncation errors. In particular it is shown that the Crank–Nicolson approximation scheme does not have second order accuracy for this case. The effects of these truncation errors on the solution of an advection–dispersion equation with a first order reaction term are demonstrated by comparison with an analytical solution. The results show that these errors are not negligible and that correcting the finite difference scheme for them results in a more accurate solution.

Permeability of the stratospheric vortex edge: On the mean mass flux due to thermally dissipating, steady, non‐breaking Rossby waves

AbstractAs part of an assessment of the flowing‐processor hypothesis of Tuck et al. (1993) and references–see also Rosenlof et al. (1997)–this paper estimates possible contributions to flow through the edge of the stratospheric polar vortex due solely to distortion of the vortex by thermally dissipating Rossby waves forced from below. To isolate such contributions in a clear‐cut way, and to eliminate questions about numerical dissipation and truncation error, an idealized model is studied analytically. It assumes steady conditions and non‐breaking waves, the waves being stationary in some rotating frame such as that of the earth. The model is studied using two approaches: first via the generalized Lagrangian‐mean formalism of Andrews and McIntyre (1978), simplified by assuming small wave amplitude a; and second via a direct consideration of the three‐dimensional, finite‐amplitude undulations of the vortex edge, as defined by isentropic contours of potential vorticity, avoiding the use of any mean‐and‐deviation formalism. It is shown, in particular, that under quasi‐geostrophic scaling the Lagrangian‐mean meridional velocity V−L is given correct to O(a2) by equation image , where θB is the basic‐state potential temperature, z the altitude, η′ the meridional particle displacement and 𝒳 the wave‐induced fluctuation in the diabatic rate of change of potential temperature θ. The formula for V−L is shown to be consistent with the independently derived finite‐amplitude result; and the implication of both results is that, for disturbances dissipated by infrared radiative relaxation in the wintertime lower stratosphere, V−L may well be directed into rather than out of the vortex, though weak outward flow is possible in some cases. There is, in addition, a vertical mean flow w−L controlled by eddy dynamics above the altitude under consideration. This is usually directed downward V−L< 0), and can therefore push mass out of the vortex if the vortex edge has its usual upward equatorward slope. However, under typical parameter conditions for the winter stratosphere, the magnitudes are nowhere near large enough to be consistent, by themselves, with Tuck et al.'s statement that the vortex is ‘flushed several times’ during a single winter.

RELAP5/MOD3 analysis of a ROSA-IV/LSTF loss-of-RHR experiment with a 5% cold leg break

An analysis was performed using the RELAP5/MOD3 version 3.1.2 code for a loss of residual heat removal event during midloop operation after reactor shutdown in a Westinghouse-type pressurized water reactor. An event during a main coolant pump shaft seal removal operation was analyzed using the data of a ROSA-IV LSTF experiment that simulated the event with a 5% cold leg break. The predicted time when boiling started in the core and the peak primary system pressure showed good agreements with the measured data. The late loop seal clearing, however, was calculated because the steam condensation in the U-tubes started early. A large mass error appeared in the transient calculations because of numerical truncation errors. The influence of the mass error was insignificant on the primary pressure prediction in this analysis.

Numerical electronic structure calculations for atoms. I. Generalized variable transformation and nonrelativistic calculations

The radial functions Pi in the nonrelativistic description of the electronic structure of atoms are solutions of eigenvalue equations. These equations are treated as two-point boundary value problems for bound one-electron states and can be solved to high accuracy with finite difference methods. For these numerical techniques the transformation to a suitable new radial variable is essential. The effects and consequences following from an arbitrary suitable variable transformation are studied in the most general way. Important results following from this general analysis are the extension of the range of application of the standard Numerov scheme and the outline for an algorithm of increased overall efficiency and reliability. Especially, the explicit use of transformed solution functions is shown to be unnecessary. Radial functions can be determined for arbitrary effective potentials resulting from the underlying theoretical description. It is demonstrated that all numerical results can be calculated to within a consistent numerical truncation error of order h4 in the grid point spacing. The approach can be extended to handle unbound one-electron states and is applicable in other fields, where differential equations of similar character occur, e.g., in the description of the vibration of a diatomic molecule. A similar analysis for relativistic electronic structure calculations for atoms will be given in Part II. © 1997 John Wiley & Sons, Inc.

Numerical electronic structure calculations for atoms. I. Generalized variable transformation and nonrelativistic calculations

The radial functions Pi in the nonrelativistic description of the electronic structure of atoms are solutions of eigenvalue equations. These equations are treated as two-point boundary value problems for bound one-electron states and can be solved to high accuracy with finite difference methods. For these numerical techniques the transformation to a suitable new radial variable is essential. The effects and consequences following from an arbitrary suitable variable transformation are studied in the most general way. Important results following from this general analysis are the extension of the range of application of the standard Numerov scheme and the outline for an algorithm of increased overall efficiency and reliability. Especially, the explicit use of transformed solution functions is shown to be unnecessary. Radial functions can be determined for arbitrary effective potentials resulting from the underlying theoretical description. It is demonstrated that all numerical results can be calculated to within a consistent numerical truncation error of order h4 in the grid point spacing. The approach can be extended to handle unbound one-electron states and is applicable in other fields, where differential equations of similar character occur, e.g., in the description of the vibration of a diatomic molecule. A similar analysis for relativistic electronic structure calculations for atoms will be given in Part II. © 1997 John Wiley & Sons, Inc.