Unconditionally stable higher-order accurate collocation time-step integration algorithms for first-order equations

2000 ◽  
Vol 190 (13-14) ◽  
pp. 1651-1662 ◽  
Author(s):  
T.C. Fung
Keyword(s):  
Higher Order ◽  
Time Step ◽  
Unconditionally Stable ◽  
First Order ◽  
Integration Algorithms

Construction of Higher-Order Accurate Time-Step Integration Algorithms by Equal-Order Polynomial Projection

2005 ◽  
Vol 11 (1) ◽  
pp. 19-49 ◽  
Author(s):  
T. C. Fung
Keyword(s):  
Integral Relation ◽  
Higher Order ◽  
Numerical Results ◽  
Numerical Dissipation ◽  
Time Interval ◽  
Time Step ◽  
Higher Order Terms ◽  
Optimal Polynomial ◽  
Integration Algorithms ◽  
New Framework

This paper presents a new framework to construct higher-order accurate time-step integration algorithms based on weakly enforcing the differential/integral relation. The dependent variable and its time derivatives are assumed to be polynomials of equal order. A differential equation is then transformed into an algebraic equation directly. The main issue is how to approximate the integral of a polynomial by another polynomial of the same degree. Various methods to determine the optimal representation (or projection) are considered. It is shown that to reproduce numerical results equivalent to the Padé or generalized Padé approximations, the coefficients of the optimal polynomial representation are related to the weighting parameters derived previously for time-step integration algorithms with predetermined coefficients. A special feature for the present formulation is that the same procedure can be used to solve first-, second-, and even higher-order non-homogeneous initial value problems in a unified manner. The resultant algorithms are higher-order accurate and unconditionally A-stable with controllable numerical dissipation. It is also shown that for the numerical results to maintain higher-order accuracy at the end of a time interval, the higher-order terms in the excitation have to be projected as polynomials of lower degree within the present framework as well. Numerical examples are given to illustrate the validity of the present formulations.

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.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Challenging the Multidimensional Conception of Perceived Person-Environment Fit

2020 ◽  
pp. 1-9
Author(s):  
Julian M. Etzel ◽  
Gabriel Nagy
Keyword(s):  
Higher Education ◽  
University Students ◽  
Educational Outcomes ◽  
Factor Model ◽  
Higher Order ◽  
Substantial Proportion ◽  
Unique Variance ◽  
First Order ◽  
Person Environment Fit ◽  
Order Factor

Abstract. In the current study, we examined the viability of a multidimensional conception of perceived person-environment (P-E) fit in higher education. We introduce an optimized 12-item measure that distinguishes between four content dimensions of perceived P-E fit: interest-contents (I-C) fit, needs-supplies (N-S) fit, demands-abilities (D-A) fit, and values-culture (V-C) fit. The central aim of our study was to examine whether the relationships between different P-E fit dimensions and educational outcomes can be accounted for by a higher-order factor that captures the shared features of the four fit dimensions. Relying on a large sample of university students in Germany, we found that students distinguish between the proposed fit dimensions. The respective first-order factors shared a substantial proportion of variance and conformed to a higher-order factor model. Using a newly developed factor extension procedure, we found that the relationships between the first-order factors and most outcomes were not fully accounted for by the higher-order factor. Rather, with the exception of V-C fit, all specific P-E fit factors that represent the first-order factors’ unique variance showed reliable and theoretically plausible relationships with different outcomes. These findings support the viability of a multidimensional conceptualization of P-E fit and the validity of our adapted instrument.

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