Assessment of Very High Order of Accuracy in Implicit LES models

2007 ◽  
Vol 129 (12) ◽  
pp. 1497-1503 ◽  
Author(s):  
Andrew Mosedale ◽  
Dimitris Drikakis
Keyword(s):  
Computational Cost ◽  
Single Mode ◽  
Theoretical Models ◽  
Higher Order ◽  
Second Order ◽  
High Order ◽  
Good Representation ◽  
Order Reconstruction ◽  
Time Required ◽  
Very High

This paper looks at the use of high-resolution and very high-order methods for implicit large-eddy simulation (ILES), with the specific example of simulating the multicomponent two-dimensional single-mode Richtmyer–Meshkov instability for which experimental data is available. The two gases are air and SF6, making stringent demands on the models used in the code. The interface between the two gases is initialized with a simple sinusoidal perturbation over a wavelength of 59mm, and a shock of strength Mach 1.3 is passed through this interface. The main comparison is between the second-order monotone upwind-centered scheme for conservation law methods of van Leer (1979, “Towards the Ultimate Conservative Difference Scheme,” J. Comput. Phys. 32, pp. 101–136) and the current state-of-the-art weighted essentially nonoscillatory interpolation, which is presented to ninth order, concentrating on the effect on resolution of the instability on coarse grids. The higher-order methods as expected provide better resolved and more physical features than the second-order methods on the same grid resolution. While it is not possible to make a definitive statement, the simulations have indicated that the extra time required for the higher-order reconstruction is less than the time saved by being able to obtain the same or better accuracy at lower computational cost (fewer grid points). It should also be noted that all simulations give a good representation of the growth rate of the instability, comparing very favorably to the experimental results, and as such far better than the currently existing theoretical models. This serves to further indicate that the ILES approach is capable of providing accurately physical information despite the lack of any formal subgrid model.

VERY HIGH-ORDER SPATIALLY IMPLICIT SCHEMES FOR COMPUTATIONAL ACOUSTICS ON CURVILINEAR MESHES

2001 ◽  
Vol 09 (04) ◽  
pp. 1259-1286 ◽  
Author(s):  
MIGUEL R. VISBAL ◽  
DATTA V. GAITONDE
Keyword(s):  
Three Dimensional ◽  
Radiation Condition ◽  
Higher Order ◽  
High Order ◽  
High Frequencies ◽  
Decomposition Strategies ◽  
The Impact ◽  
Spurious Modes ◽  
Very High ◽  
Computational Strategy

A high-order compact-differencing and filtering algorithm, coupled with the classical fourth-order Runge–Kutta scheme, is developed and implemented to simulate aeroacoustic phenomena on curvilinear geometries. Several issues pertinent to the use of such schemes are addressed. The impact of mesh stretching in the generation of high-frequency spurious modes is examined and the need for a discriminating higher-order filter procedure is established and resolved. The incorporation of these filtering techniques also permits a robust treatment of outflow radiation condition by taking advantage of energy transfer to high-frequencies caused by rapid mesh stretching. For conditions on the scatterer, higher-order one-sided filter treatments are shown to be superior in terms of accuracy and stability compared to standard explicit variations. Computations demonstrate that these algorithmic components are also crucial to the success of interface treatments created in multi-domain and domain-decomposition strategies. For three-dimensional computations, special metric relations are employed to assure the fidelity of the scheme in highly curvilinear meshes. A variety of problems, including several benchmark computations, demonstrate the success of the overall computational strategy.

scholarly journals Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Reconstructions for Forward-in-Time Schemes

2010 ◽  
Vol 138 (12) ◽  
pp. 4497-4508 ◽  
Author(s):  
William C. Skamarock ◽  
Maximo Menchaca
Keyword(s):  
Volume Transport ◽  
Higher Order ◽  
Second Order ◽  
Time Step ◽  
First Order ◽  
Order Reconstruction ◽  
Flow Test ◽  
Mesh Density ◽  
Original Scheme ◽  
Dependent Flow

Abstract The finite-volume transport scheme of Miura, for icosahedral–hexagonal meshes on the sphere, is extended by using higher-order reconstructions of the transported scalar within the formulation. The use of second- and fourth-order reconstructions, in contrast to the first-order reconstruction used in the original scheme, results in significantly more accurate solutions at a given mesh density, and better phase and amplitude error characteristics in standard transport tests. The schemes using the higher-order reconstructions also exhibit much less dependence of the solution error on the time step compared to the original formulation. The original scheme of Miura was only tested using a nondeformational time-independent flow. The deformational time-dependent flow test used to examine 2D planar transport in Blossey and Durran is adapted to the sphere, and the schemes are subjected to this test. The results largely confirm those generated using the simpler tests. The results also indicate that the scheme using the second-order reconstruction is most efficient and its use is recommended over the scheme using the first-order reconstruction. The second-order reconstruction uses the same computational stencil as the first-order reconstruction and thus does not create any additional parallelization issues.

scholarly journals TECHNIQUE FOR VERY HIGH ORDER NONLINEAR SIMULATION AND VALIDATION

2002 ◽  
Vol 10 (02) ◽  
pp. 211-229 ◽  
Author(s):  
RODGER W. DYSON
Keyword(s):  
Euler Equations ◽  
Harmonic Wave ◽  
Computational Cost ◽  
High Order ◽  
Mean Flow ◽  
Order Accuracy ◽  
Nonlinear Simulation ◽  
Fixed Design ◽  
The Mean ◽  
Very High

Finding the sources of sound in large nonlinear fields via direct simulation currently requires excessive computational cost. This paper describes a simple technique for efficiently solving the multidimensional nonlinear Euler equations that significantly reduces this cost and demonstrates a useful approach for validating high order nonlinear methods. Up to 15th order accuracy in space and time methods were compared and it is shown that an algorithm with a fixed design accuracy approaches its maximal utility and then its usefulness exponentially decays unless higher accuracy is used. It is concluded that at least a 7th order method is required to efficiently propagate a harmonic wave using the nonlinear Euler equations to a distance of five wavelengths while maintaining an overall error tolerance that is low enough to capture both the mean flow and the acoustics.

Propagating Skewness and Kurtosis Through Engineering Models for Low-Cost, Meaningful, Nondeterministic Design

2012 ◽  
Vol 134 (10) ◽  
Author(s):  
Travis V. Anderson ◽  
Christopher A. Mattson
Keyword(s):  
Order Statistics ◽  
Taylor Series ◽  
Computational Cost ◽  
Higher Order ◽  
Second Order ◽  
System Model ◽  
Higher Order Statistics ◽  
Actual System ◽  
System Output ◽  
Skewness And Kurtosis

System models help designers predict actual system output. Generally, variation in system inputs creates variation in system outputs. Designers often propagate variance through a system model by taking a derivative-based weighted sum of each input’s variance. This method is based on a Taylor-series expansion. Having an output mean and variance, designers typically assume the outputs are Gaussian. This paper demonstrates that outputs are rarely Gaussian for nonlinear functions, even with Gaussian inputs. This paper also presents a solution for system designers to more meaningfully describe the system output distribution. This solution consists of using equations derived from a second-order Taylor series that propagate skewness and kurtosis through a system model. If a second-order Taylor series is used to propagate variance, these higher-order statistics can also be propagated with minimal additional computational cost. These higher-order statistics allow the system designer to more accurately describe the distribution of possible outputs. The benefits of including higher-order statistics in error propagation are clearly illustrated in the example of a flat-rolling metalworking process used to manufacture metal plates.

scholarly journals Signal Denoise Method Based on the Higher Order Cumulant and Local Tangent Space Mean Reconstruction

2011 ◽  
Vol 2-3 ◽  
pp. 188-192
Author(s):  
Guang Bin Wang ◽  
Xue Jun Li ◽  
Ke Wang
Keyword(s):  
Covariance Matrix ◽  
Tangent Space ◽  
Higher Order ◽  
Second Order ◽  
High Order ◽  
Eigenvalue Decomposition ◽  
Order Moment ◽  
Dimensional Manifold ◽  
Intrinsic Dimension ◽  
Local Tangent

In signal denoise method to nonlinear time series based on principle manifold learning, reduction target dimension is chosen at random, which cause low efficiency. Local low dimensional manifold is obtained by the eigenvalue decomposition to the covariance matrix, but covariance belongs to the second order statistics and cannot reflect the nonlinear essential structure of signal, these reduce denoise efficiency and effect. In order to solve these problem, a new denoise algorithm based on the higher order cumulant and local tangent space mean reconstruction is proposed in this reserch. First, the signal's intrinsic dimension is obtained as dimension of reduction targets by maximum likelihood estimation. And then making use of restraining character to colored noise of high order cumulan,covariance matrix is constructed by high order cumulant function instead of second order moment function. The data outside intrinsic dimension space will be regarded as noise signal to be eliminated. Finanly the process of global array by affine transformation will be replaced by mean reconstruction,the data after denoise may be obtained in the inverse process of the phase space reconstruction. The effectiveness of the algorithm is verified through the denoise experiment in fan vibration signal with noise.

An Upwind-Biased Transport Scheme Using a Quadratic Reconstruction on Spherical Icosahedral Grids

2013 ◽  
Vol 141 (2) ◽  
pp. 832-847 ◽  
Author(s):  
Hiroaki Miura ◽  
William C. Skamarock
Keyword(s):  
Fourth Order ◽  
Piecewise Linear ◽  
Computational Cost ◽  
Second Order ◽  
Quadratic Approximation ◽  
Piecewise Linear Approximation ◽  
Reconstruction Method ◽  
Reconstruction Scheme ◽  
Order Reconstruction ◽  
Least Squares Fit

Abstract Several transport schemes developed for spherical icosahedral grids are based on the piecewise linear approximation. The simplest one among them uses an algorithm where the tracer distribution in the upwind side of a cell face is reconstructed using a linear surface. Recently, it was demonstrated that using second- or fourth-order reconstructions instead of the linear one produces better results. The computational cost of the second-order reconstruction method was not much larger than the linear one, while that of the fourth-order one was significantly larger. In this work, the authors propose another second-order reconstruction scheme on the spherical icosahedral grids, motivated by some ideas from the piecewise parabolic method. The second-order profile of a tracer is reconstructed under two constraints: (i) the area integral of the profile is equal to the cell-averaged value times the cell area and (ii) the profile is the least squares fit to the cell-vertex values. The new scheme [the second upwind-biased quadratic approximation (UQA-2)] is more accurate than the preceding second-order reconstruction scheme [the first upwind-biased quadratic approximation (UQA-1)] in most of the tests in this work. Solutions of UQA-2 are sharper than those of UQA-1, although with slightly larger phase errors. The computational cost of UQA-2 is comparable to UQA-1.

scholarly journals On Various High-Order Newton-Like Power Flow Methods for Well and Ill-Conditioned Cases

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2019
Author(s):  
Talal Alharbi ◽  
Marcos Tostado-Véliz ◽  
Omar Alrumayh ◽  
Francisco Jurado
Keyword(s):  
Power Flow ◽  
Order Of Convergence ◽  
Computational Cost ◽  
High Order ◽  
Sixth Order ◽  
Power Networks ◽  
Power Flow Calculation ◽  
Flow Problems ◽  
High Computational Cost ◽  
Very High

Recently, the high-order Newton-like methods have gained popularity for solving power flow problems due to their simplicity, versatility and, in some cases, efficiency. In this context, recent research studied the applicability of the 4th order Jarrat’s method as applied to power flow calculation (PFC). Despite the 4th order of convergence of this technique, it is not competitive with the conventional solvers due to its very high computational cost. This paper addresses this issue by proposing two efficient modifications of the 4th order Jarrat’s method, which present the fourth and sixth order of convergence. In addition, continuous versions of the new proposals and the 4th order Jarrat’s method extend their applicability to ill-conditioned cases. Extensive results in multiple realistic power networks serve to sow the performance of the developed solvers. Results obtained in both well and ill-conditioned cases are promising.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

scholarly journals Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Applied Sciences ◽  
Degree Of Approximation ◽  
Higher Order ◽  
Second Order ◽  
Dimensional Case ◽  
Moduli Of Continuity ◽  
One Dimensional ◽  
Multidimensional Generalization ◽  
The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

Higher Order Language: Risk, Promotive, and Risk-Based Protective Associations with Youth Offending

2021 ◽  
pp. 001112872110077
Author(s):  
Stavroola A. S. Anderson ◽  
David J. Hawes ◽  
Pamela C. Snow
Keyword(s):  
Oral Language ◽  
Figurative Language ◽  
Protective Factor ◽  
Language Skills ◽  
Theoretical Models ◽  
Higher Order ◽  
Protective Effects ◽  
Youth Offenders ◽  
Order Language ◽  
Nonverbal Skills

Research has implicated oral language deficits as risk factors for antisocial behavior. The aim of this study was to investigate the relationship between higher order language skills and youth offending through a risk, promotive and risk-based protective factor paradigm. In a sample of adolescent males ( n = 130; 13 to 20 years; 62% youth offenders) skills in understanding ambiguity, making inferences, and understanding figurative language were demonstrated to have risk and promotive effects in association with youth offending. Figurative language also met criteria for having risk-based protective effects for youth at high offending risk due to poor nonverbal skills. Conceptualization of higher order language skills through this paradigm promotes a broader frame of reference for considering theoretical models and practical interventions.

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