arXiv subjects

Beth Wingate

Publications and source records attributed to Beth Wingate.

11 recordsLinked to original sources

arXiv2022

Multi-level Parareal algorithm with Averaging for Oscillatory Problems

The present study is an extension of the work done in Parareal convergence for oscillatory pdes with finite time-scale separation (2019), A. G. Peddle, T. Haut, and B. Wingate, [16], and An asymptotic parallel-in-time method for highly oscillatory pdes (2014), T. Haut and B. Wingate, [10], where a two-level Parareal method with averaging is examined. The method proposed in this paper is a multi-level 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 multi-scale 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 computational complexity of the new method is investigated and the efficiency is studied on several examples.

math.NA↗

arXiv2021

Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere

We 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.

math.NA↗

arXiv2021

Higher order phase averaging for highly oscillatory systems

We introduce a higher order phase averaging method for nonlinear oscillatory systems. Phase averaging is a technique to filter fast motions from the dynamics whilst still accounting for their effect on the slow dynamics. Phase averaging is useful for deriving reduced models that can be solved numerically with more efficiency, since larger timesteps can be taken. Recently, Haut and Wingate (2014) introduced the idea of computing finite window numerical phase averages in parallel as the basis for a coarse propagator for a parallel-in-time algorithm. In this contribution, we provide a framework for higher order phase averages that aims to better approximate the unaveraged system whilst still filtering fast motions. Whilst the basic phase average assumes that the solution independent of changes of phase, the higher order method expands the phase dependency in a basis which the equations are projected onto. In this new framework, the original numerical phase averaging formulation arises as the lowest order version of this expansion. Our new projection onto functions that are $k$th degree polynomials in the phase gives rise to higher order corrections to the phase averaging formulation. We illustrate the properties of this method on an ODE that describes the dynamics of a swinging spring due to Lynch (2002). Although idealized, this model shows an interesting analogy to geophysical flows as it exhibits a slow dynamics that arises through the resonance between fast oscillations. 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 a standard phase average, but at higher order it is more accurate in the sense that solutions of the phase averaged model track the solutions of the unaveraged equations more accurately.

math.DS↗

arXiv2019

Quartet, higher order and near resonant interactions in nonlinear wave equations

Motivated by problems arising in geophysical fluid dynamics, we investigate resonant and near resonant wave interactions in nonlinear wave equations with quadratic nonlinearity, We place a special focus on interactions between slow wave modes, with zero frequency in the linear limit, and fast modes. These regularly occur in geophysical fluid systems with conserved potential vorticity or similar conserved quantities. A general multi-scale asymptotic expansion is used to show how the higher order nonlinear interaction coefficients are derived as a combination of the first order terms arising at the triad interaction level. From the general nth-order interaction coefficient we present a proof by induction how the limiting effect for particular combinations of slow and fast modes pushes their interactions to a slower timescale, and we show how this is linked to the form of the conserved quantities. We compare near resonant expansions with exact resonant expansions, and show how near-resonances allow higher order expansions to be reduced to just the most dominant contributions. These contributions then occur at one order higher in the expansion when compared to the analogous exact resonance expansion.

physics.flu-dyn↗

arXiv2018

Fast and slow resonant triads in the two layer rotating shallow water equations

In this paper we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect which cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multi-scale asymptotic expansion based on a small amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored with an emphasis on the parameter regime with small Rossby and Froude numbers.

physics.flu-dyn↗

arXiv2017

Parareal Convergence for Oscillatory PDEs with Finite Time-scale Separation

A variant of the Parareal method for highly oscillatory systems of PDEs was proposed by Haut and Wingate (2014). In that work they proved superlinear conver- gence of the method in the limit of infinite time scale separation. Their coarse solver features a coordinate transformation and a fast-wave averag- ing method inspired by analysis of multiple scales PDEs and is integrated using an HMM-type method. However, for many physical applications the timescale separation is finite, not infinite. In this paper we prove con- vergence for finite timescale separaration by extending the error bound on the coarse propagator to this case. We show that convergence requires the solution of an optimization problem that involves the averaging win- dow interval, the time step, and the parameters in the problem. We also propose a method for choosing the averaging window relative to the time step based as a function of the finite frequencies inherent in the problem.

math.NA↗

arXiv2015

A Computational Investigation of the Finite-Time Blow-Up of the 3D Incompressible Euler Equations Based on the Voigt Regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $\alpha>0$. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be sufficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.

math.AP↗

arXiv2015

A Decentralized Parallelization-in-Time Approach with Parareal

With steadily increasing parallelism for high-performance architectures, simulations requiring a good strong scalability are prone to be limited in scalability with standard spatial-decomposition strategies at a certain amount of parallel processors. This can be a show-stopper if the simulation results have to be computed with wallclock time restrictions (e.g.\,for weather forecasts) or as fast as possible (e.g. for urgent computing). Here, the time-dimension is the only one left for parallelization and we focus on Parareal as one particular parallelization-in-time method. We discuss a software approach for making Parareal parallelization transparent for application developers, hence allowing fast prototyping for Parareal. Further, we introduce a decentralized Parareal which results in autonomous simulation instances which only require communicating with the previous and next simulation instances, hence with strong locality for communication. This concept is evaluated by a prototypical solver for the rotational shallow-water equations which we use as a representative black-box solver.

cs.DC↗

arXiv2013

An asymptotic parallel-in-time method for highly oscillatory PDEs

We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which, alone, can be inefficient for equations that exhibit rapid temporal oscillations). In particular, we use an asymptotic numerical method for computing, in serial, a solution with low accuracy, and a more expensive fine solver for iteratively refining the solutions in parallel. We present examples on the rotating shallow water equations that demonstrate that significant parallel speedup and high accuracy are achievable.

math.NA↗

arXiv2007

Anisotropic small-scale constraints on energy in rotating stratified turbulence

Rapidly rotating, stably stratified three-dimensional inviscid flows conserve both energy and potential enstrophy. We show that in such flows, the forward cascade of potential enstrophy imposes anisotropic constraints on the wavenumber distribution of kinetic and potential energy. The horizontal kinetic energy is suppressed in the large, nearly horizontal wave modes, and should decay with the horizontal wavenumber as $k_h^{-3}$. The potential energy is suppressed in the large, nearly vertical wave modes, and should decay with the vertical wavenumber as $k_z^{-3}$. These results augment the only other exact prediction for the scaling of energy spectra due to constraints by potential enstrophy obtained by Charney (J. Atmos. Sci. 28, 1087 (1971)), who showed that in the quasi-geostrophic approximation for rotating stratified flows, the energy spectra must scale isotropically with total wavenumber as $k^{-3}$. We test our predicted scaling estimates using resolved numerical simulations of the Boussinesq equations in the relevant parameter regimes, and find reasonable agreement.

nlin.CD↗

arXiv2006

On the Two-point Correlation of Potential Vorticity in Rotating and Stratified Turbulence

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Karman-Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between potential vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law for the third-order velocity structure function in turbulence theory.

nlin.CD↗