Higher order phase averaging for big timesteps

<p>We introduce a higher order phase averaging method for nonlinear PDEs. Our method is suitable for highly oscillatory systems of nonlinear PDEs that generate slow motion through resonance between fast frequencies, such as is the case for rotating fluids with small but finite Rossby number. Phase averaging is a technique to filter fast motions from the dynamics whilst still accounting for their effect on the slow dynamics. In the small Rossby number limit of the phase averaged rotating shallow water equations, one recovers the quasi-geostrophic equations (as shown by Schochet, Majda and others). Peddle et al. 2017, Haut and Wingate 2014, have shown that phase averaging at finite Rossby number allows to take larger timesteps than would otherwise be possible. This was used as a coarse propagator (large timesteps at lower accuracy) for a Parareal method where corrections were made using a standard timestepping method with small timesteps.</p><p>In this contribution, we introduce an additional phase variable in the exponential time integrator that allows us to derive arbitrary order averaging methods that can be used as more accurate corrections to the basic phase averaged model, without needing small timesteps. We envisage their use as part of a time-parallel algorithm based on deferred corrections to the basic average. We illustrate the properties of this method on an ODE that describes the dynamics of a swinging spring, a model due to Peter Lynch. Although idealized, this model shows an interesting analogy to geophysical flows as it exhibits a high sensitivity of small scale oscillation on the large scale dynamics. On this example, we show convergence to the non-averaged (exact) solution with increasing approximation order also for finite averaging windows. At zeroth order, our method coincides with that in Peddle et al. 2017, Haut and Wingate 2014, but at higher order it is more accurate in the sense that it better approximates the faster oscillations around the slow manifold.</p>

Model Verification and Error Sensitivity of Turbulence-Related Tensor Characteristics in Pulsatile Blood Flow Simulations

Model verification, validation, and uncertainty quantification are essential procedures to estimate errors within cardiovascular flow modeling, where acceptable confidence levels are needed for clinical reliability. While more turbulent-like studies are frequently observed within the biofluid community, practical modeling guidelines are scarce. Verification procedures determine the agreement between the conceptual model and its numerical solution by comparing for example, discretization and phase-averaging-related errors of specific output parameters. This computational fluid dynamics (CFD) study presents a comprehensive and practical verification approach for pulsatile turbulent-like blood flow predictions by considering the amplitude and shape of the turbulence-related tensor field using anisotropic invariant mapping. These procedures were demonstrated by investigating the Reynolds stress tensor characteristics in a patient-specific aortic coarctation model, focusing on modeling-related errors associated with the spatiotemporal resolution and phase-averaging sampling size. Findings in this work suggest that attention should also be put on reducing phase-averaging related errors, as these could easily outweigh the errors associated with the spatiotemporal resolution when including too few cardiac cycles. Also, substantially more cycles are likely needed than typically reported for these flow regimes to sufficiently converge the phase-instant tensor characteristics. Here, higher degrees of active fluctuating directions, especially of lower amplitudes, appeared to be the most sensitive turbulence characteristics.

Benchmark Sea Trials on a 6-Meter Boat Powered by Kite

This paper presents sea trials on a 6-m boat specifically designed for kite propulsion. The kite control was automatic or manual, dynamic or static, depending on the point of sailing. The measurement system recorded boat motion and load generated by the kite. A particular attention was paid to wind measurement with several fixed and mobile locations directly on the kiteboat or in the vicinity. A high resolution weather modelling showed that a classical power law, describing the wind gradient, was not satisfactory to get the wind at kite location. 5-min measurement phases were systematically recorded. In the end, 101 runs were carried out. Data were processed with the phase-averaging method in order to produce reliable and accurate results.