160 research outputs found
Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere
No full textWe describe a proof-of-concept development and application of a phase
averaging technique to the nonlinear rotating shallow water equations on the
sphere, discretised using compatible finite element methods. Phase averaging
consists of averaging the nonlinearity over phase shifts in the exponential of
the linear wave operator. Phase averaging aims to capture the slow dynamics in
a solution that is smoother in time (in transformed variables) so that larger
timesteps may be taken. We overcome the two key technical challenges that stand
in the way of studying the phase averaging and advancing its implementation: 1)
we have developed a stable matrix exponential specific to finite elements and
2) we have developed a parallel finite averaging proceedure. Following Peddle
et al (2019), we consider finite width phase averaging windows, since the
equations have a finite timescale separation. In our numerical implementation,
the averaging integral is replaced by a Riemann sum, where each term can be
evaluated in parallel. This creates an opportunity for parallelism in the
timestepping method, which we use here to compute our solutions. Here, we focus
on the stability and accuracy of the numerical solution. We confirm there is an
optimal averaging window, in agreement with theory. Critically, we observe that
the combined time discretisation and averaging error is much smaller than the
time discretisation error in a semi-implicit method applied to the same spatial
discretisation. An evaluation of the parallel aspects will follow in later
work.Comment: Author Beth Wingate added for submitted versio
On the Love Affair between Computing and Maths
No full textThis is the final version. Available from Institute of Mathematics and its Applications via the link in this recordThe 2017 IMA Lighthill Lecture, was given by Professor Beth
Wingate at the British Applied Mathematics Colloquium,
University of Surrey. The lecture is given in memory of Sir
James Lighthill, founder President of the IMA. This article is
based on Beth’s Lighthill Lecture
On the Love Affair between Computing and Maths
No full textThe 2017 IMA Lighthill Lecture, was given by Professor Beth
Wingate at the British Applied Mathematics Colloquium,
University of Surrey. The lecture is given in memory of Sir
James Lighthill, founder President of the IMA. This article is
based on Beth’s Lighthill Lecture
Passive Advection in a Stommel Gyre
No full textthis paper we have presented a solution procedure to the passive tracer advection problem in a Stommel Gyre which is independant of the grid and can be computed to machine accuracy. The procedure can be used for any problem where the stream function is fixed in time. A computer program which reads in an arbitrary grid and writes out the solution can be obtained by sending email to [email protected]. Reference
Multilevel Parareal Algorithm with Averaging for Oscillatory Problems
No full texthe present study is an extension of the work done by Peddle, Haut, and Wingate [SIAM J. Sci. Comput., 41 (2019), pp. A3476–A3497] and Haut and Wingate [SIAM J. Sci. Comput., 36 (2014), pp. A693–A713], where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples
Multilevel Parareal Algorithm with Averaging for Oscillatory Problems
No full textThe present study is an extension of the work done by Peddle, Haut, and Wingate [SIAM J. Sci. Comput., 41 (2019), pp. A3476–A3497] and Haut and Wingate [SIAM J. Sci. Comput., 36 (2014), pp. A693–A713], where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples
n-dimensional Quadrature
No full textThis is the author accepted manuscript. The final version is available from Wiley via the DOI in this recordThis article discusses n‐dimensional quadrature. To show how dimensionality complicates integration rules this article focus on polynomial integration over squares and triangles where quadrature points are required to be on the boundary. In the case of a square, high quality formulae, called Gauss–Lobatto quadrature, are available as tensor products of 1‐dimensional quadrature. In triangles it has been shown that analogous Gauss–Lobatto formulae do not even exist
Components of Nonlinear Oscillation and Optimal Averaging for Stiff PDEs
No full textA novel solver which uses finite wave averaging to mitigate oscillatory stiffness is proposed and
analysed. We have found that triad resonances contribute to the oscillatory stiffness of the problem and
that they provide a natural way of understanding stability limits and the role averaging has on reducing
stiffness. In particular, an explicit formulation of the nonlinearity gives rise to a stiffness regulator function
which allows for analysis of the wave averaging.
A practical application of such a solver is also presented. As this method provides large timesteps at
comparable computational cost but with some additional error when compared to a full solution, it is a
natural choice for the coarse solver in a Parareal-style parallel-in-time method
Effects of Varying Load During A Wingate Test
No full textEffects of Varying Load During A Wingate Test Author: Bobbie Cooks Faculty Sponsor: J.R. Wilson, Ph.D. INTRODUCTION: The Wingate Anaerobic Test (WAnT) also known as Wingate test is used for athletes to assess peak anaerobic power, anaerobic fatigue, and total anaerobic capacity. It is crucial in determining physiological profiles, training programs, and assessing human muscle capacity. The test requires the subject to pedal a mechanically braked bicycle ergometer for 30 seconds at an “all out” pace. They are advised to complete a 3-5 minute warm-up followed by a 1-2 minute cool down. Research shows that, depending on the study, and subjects, there are differences in results based on the resistance they pedal against. PURPOSE: The specific purpose of this research study was to determine the effects of two different breaking loads during a Wingate test. METHODS: Five women (W; age 22 +3yrs) students of The University of Texas at Arlington volunteered to participate in this study. Each subject came to the research laboratory on two different occasions at the scheduled time and the test was explained to them. Data was collected including: height, weight, and age. The resistances were a torque factor of 0.60 and 0.67. The order of the two different resistances was randomized among the subjects. The seat on the cycle was adjusted for each subject’s leg length. Subjects pedaled for 1 min during a warmup of easy cycling before the resistance was increased. With the command “start,” the subject pedaled as fast as possible against the resistance and was encouraged to pedal as hard and fast as they could for 30 seconds. They were not allowed to stand up during the 30 second test. The resistance was automatically removed at the end of 30 seconds. Resistance was removed and the subject continued to pedal as long as needed to cool down. During each test peak power, minimum power, fatigue slope, mean power, and peak power (body mass) were recorded. They were then scheduled to return to perform the WAnT using the second resistance. RESULTS: At a torque factor of 0.60 [Nm/Kg] the means of each variable were calculated as follows: Peak Power 552±105.4, Mean Power 310±42.2, Minimum Power 135.8±60.5, Peak Power/ Body Mass 8.5±0.3, and Fatigue slope 2.1±3.1. At the torque of 0.67 [Nm/Kg] Peak Power 523.3±102.5, Mean Power 288.9±41.6, Minimum Power 111±52.7, Peak Power/Body mass 8±0.59, Fatigue slope 2.6±2. Results show no significant difference between the two resistances (p\u3e0.05). CONCLUSION: The results of this study indicated that there was no significant difference in peak power, minimum power, mean power, peak power/BM, and fatigue slope when changing the resistance
The Correlational Study of the Vertical Jump Test and Wingate Cycle Test as a Method To Assess Anaerobic Power in Road Cyclists
No full textRoad Cycling is an important sport that uses anaerobic and aerobic metabolism and especially sprinter cyclists have higher anaerobic capacity. The assessment of anaerobic power in cyclists often involves the use of the vertical jump and Wingate cycle tests. The lack of research in the field of cycling-specific tests to assess anaerobic performance has led to the improvement of existing research. The objective of this research was to investigate the correlation between the vertical jump test and the Wingate anaerobic cycling tests, both of which are often used to assess anaerobic power in road cyclists. A correlation study was conducted on 15 athletes of the Turkish national road cycling team in the 14-16 age group (15.107 ± 0.717 (SD)). The sample of the study was determined by using the convenient sampling method. On the first day, anthropometric measurements and the vertical jump test were conducted. The Wingate cycle ergometer test, lasting for a duration of 30 seconds, was administered to the participants on the second day. The computer application was used to determine the 30-second peak and average anaerobic power during the test. The results acquired from the study revealed a statistically significant positive relationship between the vertical jump performance and the peak power production measured during the Wingate cycle test (r=0.321, p<0.05). The findings indicate that vertical jump tests may serve as suitable field measurements of anaerobic power for road cyclists, as an alternative to the laboratory-based Wingate anaerobic test. © The Author(s) 2023
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