Accelerating time averaging by adding a Lie derivative of an auxiliary function

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.)

This talk is part of the Fluids Group Seminar (CUED) series.