Material Stability and Numerical Stability in Peridynamics.
Abstract not provided.
Silling, Stewart A.↗
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Notions of analytic vs numerical stability as applied to the numerical calculation of orbits
This paper deals with the implications of 'stability' as applied to the numerical calculation of orbits. The study was motivated by the recent appearance of several proposed transformations of the classical Newtonian equations of motion which 'analytically stabilize' Cowell's method. This report analyzes the basic properties of such stabilizing transformations and shows the removal of the period as a parameter is the key to these transformations and, that although such transformations do not yield global numerical error bounds, the error propagation properties are more favorable - linear vs quadratic growth.
Velez, C. E.↗
Dynamic State Estimation for Multi-Machine Power System by Unscented Kalman Filter With Enhanced Numerical Stability
In this paper, in order to enhance the numerical stability of the unscented Kalman filter (UKF) used for power system dynamic state estimation, a new UKF with guaranteed positive semidifinite estimation error covariance (UKFGPS) is proposed and compared with five existing approaches, including UKFschol, UKF-kappa, UKFmodified, UKF-Delta Q, and the squareroot UKF (SRUKF). These methods and the extended Kalman filter (EKF) are tested by performing dynamic state estimation on WSCC 3-machine 9-bus system and NPCC 48-machine 140-bus system. For WSCC system, all methods obtain good estimates. However, for NPCC system, both EKF and the classic UKF fail. It is found that UKFschol, UKF-kappa, and UKF-Delta Q do not work well in some estimations while UKFGPS works well in most cases. UKFmodified and SRUKF can always work well, indicating their better scalability mainly due to the enhanced numerical stability.
dynamic state estimation↗
An Overview of the Null-Field Method. II: Convergence and Numerical Stability
In this paper we provide an analysis of the convergence and numerical stability of the null-field method with discrete sources. We show that (i) if the null-field scheme is numerically stable then we can decide whether or not convergence can be achieved; (ii) if the null-field scheme is numerically unstable then we cannot draw any conclusion about the convergence issue; and (iii) the numerical stability is closely related to the property of a tangential system of radiating discrete sources to form a Riesz basis. Our numerical analysis indicates that for prolate spheroids and localized vector spherical wave functions, the null-field scheme is numerically unstable (this system of vector functions does not form a Riesz basis), while for distributed vector spherical wave functions, the numerical instability is not so pronounced (this system of discrete sources almost possesses the property of being a Riesz basis). We also describe an analytical method for computing the surface integrals in the framework of the conventional null-field method with localized vector spherical wave functions which increases the stability of the numerical scheme.
Null-field method↗
Numerical Stability In Hyperbolic Boundary-Value Problems
Technical memorandum discusses stability of numerical solutions involving semidiscrete approximations to hyperbolic partial differential equations in initial-and-boundary-value problems. Topic of practical significance because hyperbolic partial differential equations arise in mathematical modeling of waves and blasts. Solutions often needed over restricted regions of unbounded spaces. Outer boundaries artificial, introduced only to limit domains of numerical solutions. Conditions at such artificial boundaries cause numerical instabilities that degrade computed solutions.
Warming, Robert F.↗
On the numerical stability of computations of stellar evolution
Numerical stability in computations of stellar evolution taking into account hydrostatic equilibrium and thermal processes
Sugimoto, D.↗
On the numerical stability of computations of stellar evolution
Stellar evolution computations numerical instability due to coupling between hydrostatic equilibrium and thermal processes
Sugimoto, D.↗
Analysis of numerical stability and amplification matrices: Fourth-order Runge-Kutta methods
Amplification matrices, numerical kernels, stable, and exponentially stable numerical solutions are examined. The various techniques involved in these concepts are applied to certain systems that have Jordan forms, which are nondiagonal, with particular interest in the case of imaginary or zero eigenvalues.
Kennedy, E. W.↗
Investigating Inherent Numerical Stabilization for the Moist, Compressible, Non‐Hydrostatic Euler Equations on Collocated Grids
This study investigates inherent numerical dissipation due to upwind fluxes and reconstruction strategies for collocated Finite-Volume integration of the Euler equations. Idealized supercell simulations are used without any explicit dissipation. Flux terms are split into: mass flux, pressure, and advected quantities. They are computed with the following upwind strategies: central, advectively upwind, and acoustically upwind. This is performed for third and ninth-order-accurate reconstructions with and without Weighted Essentially Non-Oscillatory limiting. Acoustic-only upwinding for pressure and mass flux terms and advective-only upwinding for advected quantities is the most flexible simplification found. It reduces data movement and computations. Assuming a constant speed of sound in acoustic upwinding gives similar results to using the true speed of sound. Dissipation from upwind adapts automatically to grid spacing, time step, reconstruction accuracy, and flow smoothness. While stability is maintained even at 21st-order spatial accuracy, there is a limit to the spatial order of accuracy for which upwinding alone can create a realizable solution in the conditions of this study. Convex combinations of upwind and central solutions for flux terms also reduced dissipation, but as the central proportion grows, solutions become physically unrealizable. The range of length scales of the kinetic energy spectra can be extended along k −5/3 to smaller spatial scales by reducing dissipation either with higher-order reconstructions or using convex combinations of upwind and central fluxes. However, not all extensions of the length scale range along k −5/3 exhibit physically realizable solutions, even though the spectra appear to be physical.
54 ENVIRONMENTAL SCIENCES↗
RELAP-7 Numerical Stabilization: Entropy Viscosity Method
The RELAP-7 code is the next generation nuclear reactor system safety analysis code being developed at the Idaho National Laboratory (INL). The code is based on the INL's modern scientific software development framework, MOOSE (Multi-Physics Object Oriented Simulation Environment). The overall design goal of RELAP-7 is to take advantage of the previous thirty years of advancements in computer architecture, software design, numerical integration methods, and physical models. The end result will be a reactor systems analysis capability that retains and improves upon RELAP5's capability and extends the analysis capability for all reactor system simulation scenarios. RELAP-7 utilizes a single phase and a novel seven-equation two-phase flow models as described in the RELAP-7 Theory Manual (INL/EXT-14-31366). The basic equation systems are hyperbolic, which generally require some type of stabilization (or artificial viscosity) to capture nonlinear discontinuities and to suppress advection-caused oscillations. This report documents one of the available options for this stabilization in RELAP-7 -- a new and novel approach known as the entropy viscosity method. Because the code is an ongoing development effort in which the physical sub models, numerics, and coding are evolving, so too must the specific details of the entropy viscosity stabilization method. Here the fundamentals of the method in their current state are presented.
42 ENGINEERING↗
Numerical stability in problems of linear algebra.
Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.
Babuska, I.↗
Comparison of the Numerical Stability between CMFD and GCMR with Stabilization Techniques
The Coarse Mesh Finite Difference (CMFD) method and the Generalized Coarse Mesh Rebalance (GCMR) method have been applied to many reactor physics codes and these methods contribute for improvement of calculation speed. On the other hand, the numerical instability of these methods have been a crucial issue. In order to overcome this issue, some stabilization techniques such as damping factor and correction techniques of diffusion coefficient have been developed. The systematic comparison of the calculation stability and efficiency between GCMR and CMFD with various stabilization techniques has not been reported and these results are important for the improvement of nuclear reactor analysis. The present summary focuses on this point. Namely, calculation stability and efficiency of the CMFD and the GCMR are compared with investigation of the effect of stabilization techniques on these methods. We applied these methods and techniques to 2D geometries with and without voided regions. (authors)
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Numerical Stability of Detonations in White Dwarf Simulations
Some simulations of SNe Ia feature self-consistent thermonuclear detonations. However, these detonations are not meaningful if the simulations are not resolved, so it is important to establish the requirements for achieving a numerically converged detonation. In this study we examine a test detonation problem inspired by collisions of white dwarfs. This test problem demonstrates that achieving a converged thermonuclear ignition requires a spatial resolution much finer than 1 km in the burning region. Current computational resource constraints place this stringent resolution requirement out of reach for multidimensional supernova simulations. Consequently, contemporary simulations that self-consistently demonstrate detonations are possibly not converged and should be treated with caution.
79 ASTRONOMY AND ASTROPHYSICS↗
Improving the Numerical Stability of Fast Matrix Multiplication
Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently implemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical algorithm, they are not used even in cases where they provide a performance benefit. We argue in this study that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoretically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulating the algorithms, and reducing input anomalies by various forms of diagonal scaling. In conclusion, we benchmark performance and demonstrate our improved numerical accuracy.
97 MATHEMATICS AND COMPUTING↗
Investigating the Numerical Stability of Using an Impedance Boundary Condition to Model Broadband Noise Scattering With Acoustic Liners
Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for achieving aircraft noise reduction and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model will be discussed in which the liner impedance is specified along a wide range of frequencies. The liner impedance boundary condition will be derived and coupled with a time-domain boundary integral equation to model acoustic scattering by a flat plate consisting of both un-lined and lined surfaces. The scattering solution will be obtained iteratively using both spatial and temporal basis functions and the stability will be demonstrated through eigenvalue analysis. Stability will be assessed for its dependence on time step, spatial discretization, as well as temporal basis function order. Both second- and third-order Lagrange temporal basis functions are considered.
acoustics↗
Investigating the Numerical Stability of Using an Impedance Boundary Condition to Model Broadband Noise Scattering With Acoustic Liners
Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for achieving aircraft noise reduction and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model will be discussed in which the liner impedance is specified along a wide range of frequencies. The liner impedance boundary condition will be derived and coupled with a time-domain boundary integral equation to model acoustic scattering by a flat plate consisting of both un-lined and lined surfaces. The scattering solution will be obtained iteratively using both spatial and temporal basis functions and the stability will be demonstrated through eigenvalue analysis. Stability will be assessed for its dependence on time step, spatial discretization, as well as temporal basis function order. Both second- and third-order Lagrange temporal basis functions are considered.
acoustics↗
Numerical stability in multifluid gas dynamics with implicit drag forces
Not Available
Ramshaw, J. D.↗