'Accelerating time averaging by adding a Lie derivative of an auxiliary function' by Sergei Chernyshenko (ICL)

Duration: 48 mins 23 secs

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'Accelerating time averaging by adding a Lie derivative of an auxiliary function' by Sergei Chernyshenko (ICL)'s image

Description:	Talk given by Prof Sergei Chernyshenko (Imperial College London) at Department of Engineering, University of Cambridge, 4 November 2022, as part of the CUED Fluids seminar series.


Created:	2022-12-06 11:12
Collection:	Cambridge Engineering Dept Fluids Seminars
Publisher:	University of Cambridge
Copyright:	Prof Sergei Chernyshenko
Language:	eng (English)


Abstract:	Obtaining time-averaged quantities with sufficient accuracy can be challenging computationally for systems with a chaotic behaviour. We replace the quantity being averaged with another quantity having the same average but such that it is easier to average. If W(t) is a bounded differentiable function, then the infinite time average of its derivative is zero. Hence, rather than numerically averaging the quantity of interest, which we will denote F, one can average F+dW/dt. We explore first the simplest way of choosing W(t), which is to ensure that the fluctuation amplitude of F+dW/dt is smaller than the fluctuation amplitude of F. For this, F and dW/dt should be correlated. This can often be achieved by taking W(t)=V(x(t)), where x is the state of the dynamical system. The talk will discuss our tests of this idea. (A spoiler: the acceleration is only moderate but is worth doing because it is easy. Further improvement requires progress on interesting and challenging problems.)

Abstract:

Obtaining time-averaged quantities with sufficient accuracy can be challenging computationally for systems with a chaotic behaviour. We replace the quantity being averaged with another quantity having the same average but such that it is easier to average. If W(t) is a bounded differentiable function, then the infinite time average of its derivative is zero. Hence, rather than numerically averaging the quantity of interest, which we will denote F, one can average F+dW/dt. We explore first the simplest way of choosing W(t), which is to ensure that the fluctuation amplitude of F+dW/dt is smaller than the fluctuation amplitude of F. For this, F and dW/dt should be correlated. This can often be achieved by taking W(t)=V(x(t)), where x is the state of the dynamical system. The talk will discuss our tests of this idea. (A spoiler: the acceleration is only moderate but is worth doing because it is easy. Further improvement requires progress on interesting and challenging problems.)

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