Numerical Errors In Computation Research Articles

Do even computers make mistakes in calculations?: Accuracy and errors in numerical computation

Do even computers make mistakes in calculations?: Accuracy and errors in numerical computation

Non-linear instability analysis of the two-dimensional Navier-Stokes equation: The Taylor-Green vortex problem

An enstrophy-based non-linear instability analysis of the Navier-Stokes equation for two-dimensional (2D) flows is presented here, using the Taylor-Green vortex (TGV) problem as an example. This problem admits a time-dependent analytical solution as the base flow, whose instability is traced here. The numerical study of the evolution of the Taylor-Green vortices shows that the flow becomes turbulent, but an explanation for this transition has not been advanced so far. The deviation of the numerical solution from the analytical solution is studied here using a high accuracy compact scheme on a non-uniform grid (NUC6), with the fourth-order Runge-Kutta method. The stream function-vorticity (ψ, ω) formulation of the governing equations is solved here in a periodic square domain with four vortices at t = 0. Simulations performed at different Reynolds numbers reveal that numerical errors in computations induce a breakdown of symmetry and simultaneous fragmentation of vortices. It is shown that the actual physical instability is triggered by the growth of disturbances and is explained by the evolution of disturbance mechanical energy and enstrophy. The disturbance evolution equations have been traced by looking at (a) disturbance mechanical energy of the Navier-Stokes equation, as described in the work of Sengupta et al., “Vortex-induced instability of an incompressible wall-bounded shear layer,” J. Fluid Mech. 493, 277–286 (2003), and (b) the creation of rotationality via the enstrophy transport equation in the work of Sengupta et al., “Diffusion in inhomogeneous flows: Unique equilibrium state in an internal flow,” Comput. Fluids 88, 440–451 (2013).

An efficient linear scheme to approximate nonlinear diffusion problems

This paper deals with nonlinear diffusion problems including the Stefan problem, the porous medium equation and cross-diffusion systems. A linear discrete-time scheme was proposed by Berger, Brezis and Rogers [RAIRO Anal. Numer. 13 (1979) 297–312] for degenerate parabolic equations and was extended to cross-diffusion systems by Murakawa [Math. Mod. Numer. Anal. 45 (2011) 1141–1161]. There is a constant stability parameter \(\mu \) in the linear scheme. In this paper, we propose a linear discrete-time scheme replacing the constant \(\mu \) with given functions depending on time, space and species. After discretizing the scheme in space, we obtain an easy-to-implement numerical method for the nonlinear diffusion problems. Convergence rates of the proposed discrete-time scheme with respect to the time increment are analyzed theoretically. These rates are the same as in the case where \(\mu \) is constant. However, actual errors in numerical computation become significantly smaller if varying \(\mu \) is employed. Our scheme has many advantages even though it is very easy-to-implement, e.g., the ensuing linear algebraic systems are symmetric, it requires low computational cost, the accuracy is comparable to that of the well-studied nonlinear schemes, the computation is much faster than the nonlinear schemes to obtain the same level of accuracy.

Non-Minimal Order Model of Mechanical Systems With Redundant Constraints for Simulations and Controls

This technical note presents a non-minimal order dynamics model for many analysis, simulation, and control problems of constrained mechanical systems passing through singular configurations during their motion by making use of linear projection operator. The distinct features of this model describing dynamics of the dependent coordinates are: i) The mass matrix $\bar{M}(q)$ is always positive definite even at singular configurations; ii) matrix $\dot{\bar{M}}-2\bar{C}$ is skew symmetric, where all nonlinear terms are lumped into vector $\bar{C}(q,\dot{q})\dot{q}$ after elimination of constraint forces. Eigenvalue analysis shows that the condition number of the constraint mass matrix can be minimized upon adequate selection of a scalar parameter called “virtual mass” thereby reducing the sensitivity to round-off errors in numerical computation. It follows by derivation of two oblique projection matrices for computation of constraint forces and actuation forces. It is shown that projection-based model allows feedback control of dependent coordinates which, unlike reduced-order dependent coordinates, uniquely define spatial configuration of constrained systems.

One-step downward continuation of potential fields in a wavenumber domain

Based on the iteration method for downward continuation of potential fields, we derive an equivalent one-step downward continuation wavenumber domain operator. We analyse the influence of numerical computation errors on the operator and provide an effective way to depress them. The one-step downward continuation method is tested on both model and real data. The results show that it yields the same effective outcome as the iteration method, with a much higher computation speed comparable to the traditional fast Fourier transform continuation method.

The ‘Drop of Honey Effect’. A Note on Chaos in Economics

Relationships in non-linear systems are unstable. Considering that, chaos theory aims to understand and to explain the unpredictable aspects of nature, social life, uncertainties, nonlinearities, disorders and confusion. Small differences in initial conditions - such as those due to rounding errors in numerical computation – may perform diverging outcomes tracking to systems’ chaotic behaviour. In these circumstances, long-term predictions become impossible in general. A brilliant metaphor on this is the so-called butterfly effect, about the way how the “flapping of the wings of a butterfly” may cause huge phenomena either they are physical, atmospheric or social. In this work, the drop of honey effect metaphor is proposed for representing this kind of butterfly effect for chaos in social phenomena, in particular in economics and politics.

Multiscale analysis of gas dynamic fields

On the basis of numerical data processing, it is shown that a joint analysis of singularities detected in an initial gas dynamic field and at the first level of its wavelet decomposition allows one to remove the most part of numerical artifacts caused by numerical errors in computations. The application of asymmetric wavelets for this decomposition leads to a displacement of discontinuities. It is also shown that the values of the field can be identified with the coefficients of its wavelet decomposition.

Divergent expansion, Borel summability and three-dimensional Navier–Stokes equation

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier-Stokes (NS) system and show how the IE approach can give rise to local existence proofs. In this approach, the global existence problem in three-dimensional NS systems, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/t or some positive power of 1/t). Furthermore, the errors in numerical computations in the associated IE can be controlled rigorously, which is very important for nonlinear PDEs such as NS when solutions are not known to exist globally.Moreover, computation of the solution of the IE over an interval [0,p0] provides sharper control of its p-->infinity behaviour. Preliminary numerical computations give encouraging results.

A relation on round-off error, attractor size and its dynamics in driven or coupled logistic map system

In this paper, we investigate three kinds of numerical artifacts: period-like, strange-nonchaotic-attractor-like, and chaos-like behaviors in an extended logistic map system. These artificial behaviors appear in double precision and change into other real attractors in high-precision simulations. All of them are generated by a complicated dynamical process of the system and round-off truncation errors in numerical computations. A quantity beta, which is closely related to the local Lyapunov exponent, is proposed to measure the extremum of large expansion or contraction dynamical capability. Eventually, we find the artifacts will emerge if the relation is not kept: alphabeta<gamma, where gamma is the attractor size of the system and alpha is the computational precision digit, for instance, alpha=2 x 10(-16) for double precision, which has a unit round-off of 2 x 10(-16).

Intrinsic Compiler Support for Interval Arithmetic

Interval arithmetic provides an efficient method for monitoring errors in numerical computations and for solving problems that cannot be efficiently solved with floating-point arithmetic. To support interval arithmetic, several software tools have been developed including interval arithmetic libraries, extended scientific programming languages, and interval-enhanced compilers. The main disadvantage of these software tools is their speed, since interval operations are implemented using function calls. In this paper, compiler support for interval arithmetic is investigated. In particular, the performance benefits of having the compiler inline interval operations to eliminate function call overhead is researched. Interval operations are inlined with the GNU gcc compiler and the performance of interval arithmetic is evaluated on a superscalar architecture. To implement interval operations with compiler support, the compiler produces sequences of instructions that use existing floating point hardware. Simulation results show that the compiler implementation of interval arithmetic is approximately 4 to 5 times faster than a functionally equivalent interval arithmetic software implementation with function call overhead and approximately 1.2 to 1.5 times slower than a dedicated interval arithmetic hardware implementation.

EFFECTS OF BASE POINTS AND NORMALIZATION SCHEMES ON THE NON-LINEAR NORMAL MODES OF CONSERVATIVE SYSTEMS

In this paper, two factors that affect the behaviors of the non-linear normal modes (NNMs) of conservative vibratory systems are investigated. The first factor is the base points (which are equivalent to Taylor series expanding points) of the non-linear normal modes and the second one is the normalization schemes of the corresponding linear modes. For non-linear systems, in general only the approximated NNM manifolds are obtainable in practice, so different base points may lead to different forms of NNM oscillators and different normalization schemes lead to different forward and backward transformations which in turn affect the numerical computation errors. Three different kinds of base points and two different normalization schemes are adopted for comparison respectively. Two examples of non-linear systems with two and three degrees of freedom, respectively, are given as illustration. Simulations for various cases are made. The analysis and the simulation results indicated that, the best base points are the abstract base points determined via the linear normal mode, which would eliminate the third order terms containing velocity (for cubic systems) or quadratic terms (for quadratic systems) in equations of the NNM oscillators. A better invariance of the NNMs would also be maintained with such base points. The best scheme of normalization is the norm-one scheme that would minimize the numerical errors.

Low redundancy polynomial checks for numerical computation

This paper presents a new approach to detection and correction of errors in numerical computations caused by both software and/or hardware faults. This approach does not depend on the form of representation or on the specific features of the implementation of a program or a device computing the given function. The described approach results in a substantial reduction of the hardware overhead required for multiple error detection and correction as compared to the check sum method or other methods previously known.

Analytical modeling of glacier dynamics

Flow velocities and stresses within a glacier are determined by inverting known surface velocities with a specified glacier geometry. The surface velocities depend only weakly on the unknown velocities at the bed of a glacier, so the inversion is illposed and unstable. This instability causes both numerical computation errors and data errors to grow dramatically with depth, usually masking the actual velocity and stress solutions. To control the numerical errors, an analytical modeling scheme is presented which modifies the method of mean weighted residuals (used in finite element techniques). The resulting scheme impairs convergence by producing powerseries solutions, but in an advantageous tradeoff, the coefficients to the power series can be determined analytically rather than numerically. This leads to arbitrary order analytical powerseries solutions to the internal stress state of glaciers. The symbolic powerseries solutions can be evaluated at any point in the glacier with negligible roundoff and discretization errors. Analytical model accuracy is confirmed with known stress solutions for several widely used constitutive relations for ice.

Calibration procedure and error analysis for a violin radiation measuring system

The system referred to in the preceding paper involves measurement of acoustic pressure, both in amplitude and in phase, on two concentric spheres of known radii. Among possible sources of error are: the finite number of measurements; inaccuracy in microphone positioning; scattering by the moving boom system; electrical and acoustical noise in the microphone system; spurious phase shifts and gain inequalities in the microphone and amplifier channels; timing errors and overtone sensitivity of the phase‐sensitive detectors; offsets and quantizing errors in the analog‐to‐digital converter; and numerical computation errors. Effects due to the nonideal nature of the anechoic chamber are in a different category, since our system measures the reflected wave directory. [Work supported by NSF.]