arXiv subjects

William Layton

Publications and source records attributed to William Layton.

28 records · Page 2Linked to original sources

arXiv2019

Doubly-Adaptive Artificial Compression Methods for Incompressible Flow

This report presents adaptive artificial compression methods in which the time-step and artificial compression parameter $\varepsilon $ are independently adapted. The resulting algorithms are supported by analysis and numerical tests. The first and second-order methods are embedded. As a result, the computational, cognitive and space complexities of the adaptive $% \varepsilon ,k$ algorithms are negligibly greater than that of the simplest, first-order, constant $\varepsilon ,$ constant $k$ artificial compression method.

math.NA↗

arXiv2019

An Artificial Compression Reduced Order Model

We propose a novel artificial compression, reduced order model (AC-ROM) for the numerical simulation of viscous incompressible fluid flows. The new AC-ROM provides approximations not only for velocity, but also for pressure, which is needed to calculate forces on bodies in the flow and to connect the simulation parameters with pressure data. The new AC-ROM does not require that the velocity-pressure ROM spaces satisfy the inf-sup (Ladyzhenskaya-Babuska-Brezzi) condition and its basis functions are constructed from data that are not required to be weakly-divergence free. We prove error estimates for the reduced basis discretization of the AC-ROM. We also investigate numerically the new AC-ROM in the simulation of a two-dimensional flow between offset cylinders.

math.NA↗

arXiv2019

A time-accurate, adaptive discretization for fluid flow problems

This report presents a low computational and cognitive complexity, stable, time accurate and adaptive method for the Navier-Stokes equations. The improved method requires a minimally intrusive modification to an existing program based on the fully implicit / backward Euler time discretization, does not add to the computational complexity, and is conceptually simple. The backward Euler approximation is simply post-processed with a two-step, linear time filter. The time filter additionally removes the overdamping of Backward Euler while remaining unconditionally energy stable, proven herein. Even for constant stepsizes, the method does not reduce to a standard / named time stepping method but is related to a known 2-parameter family of A-stable, two step, second order methods. Numerical tests confirm the predicted convergence rates and the improved predictions of flow quantities such as drag and lift.

math.NA↗

arXiv2018

A new embedded variable stepsize, variable order family of low computational complexity

Variable Stepsize Variable Order (VSVO) methods are the methods of choice to efficiently solve a wide range of ODEs with minimal work and assured accuracy. However, VSVO methods have limited impact in timestepping methods in complex applications due to their computational complexity and the difficulty to implement them in legacy code. We introduce a family of implicit, embedded, VSVO methods that require only one BDF solve at each time step followed by adding linear combinations of the solution at previous time levels. In particular, we construct implicit and linearly implicit VSVO methods of orders two, three and four with the same computational complexity as variable stepsize BDF3. The choice of changing the order of the method is simple and does not require additional solves of linear or nonlinear systems.

math.NA↗

arXiv2018

Analysis of variable-step/non-autonomous artificial compression methods

A standard artificial compression (AC) method for incompressible flow is $$ \frac{u_{n+1}^{\varepsilon }-u_{n}^{\varepsilon }}{k}+u_{n+1}^{\varepsilon }\cdot \nabla u_{n+1}^{\varepsilon }+{\frac{1}{2}}u_{n+1}^{\varepsilon }\nabla \cdot u_{n+1}^{\varepsilon }+\nabla p_{n+1}^{\varepsilon }-νΔu_{n+1}^{\varepsilon }=f\text{ ,} \\ \varepsilon \frac{p_{n+1}^{\varepsilon }-p_{n}^{\varepsilon }}{k} +\nabla \cdot u_{n+1}^{\varepsilon }=0 $$ for, typically, $\varepsilon =k$ (timestep). It is fast, efficient and stable with accuracy $O(\varepsilon +k)$. For adaptive (and thus variable) timestep $k_{n}$ (and thus $\varepsilon =\varepsilon _{n}$) its long time stability is unknown. For variable $k,\varepsilon $ this report shows how to adapt a standard AC method to recover a provably stable method. For the associated continuum AC model, we prove convergence of the $\varepsilon =\varepsilon (t)$\ artificial compression model to a weak solution of the incompressible Navier-Stokes equations as $\varepsilon =\varepsilon (t)\rightarrow 0$. The analysis is based on space-time Strichartz estimates for a non-autonomous acoustic equation. Variable $\varepsilon ,k$ numerical tests in $2d$ and $3d$ are given for the new AC method.

math.NA↗

arXiv2017

Robust and Efficient Modular Grad-Div Stabilization

This paper presents two modular grad-div algorithms for calculating solutions to the Navier-Stokes equations (NSE). These algorithms add to an NSE code a minimally intrusive module that implements grad-div stabilization. The algorithms do not suffer from either breakdown (locking) or debilitating slow down for large values of grad-div parameters. Stability and optimal-order convergence of the methods are proven. Numerical tests confirm the theory and illustrate the benefits of these algorithms over a fully coupled grad-div stabilization.

math.NA↗

arXiv2017

Analysis of the effect of Time Filters on the implicit method: increased accuracy and improved stability

This report considers linear multistep methods through time filtering. The approach has several advantages. It is modular and requires the addition of only one line of additional code. Error estimation and variable timesteps is straightforward and the individual effect of each step\ is conceptually clear. We present its development for the backward Euler method and a curvature reducing time filter leading to a 2-step, strongly A-stable, second order linear multistep method.

math.NA↗

arXiv2016

Energy Dissipation in the Smagorinsky Model of Turbulence

The Smagorinsky model, unmodified, is often reported to severely overdiffuse flows. Previous estimates of the energy dissipation rate of the Smagorinsky model for shear flows reflect a blow up of model energy dissipation as Re increases. This blow up is consistent with the numerical evidence and leads to the question: Is the over dissipation due to the influence of the turbulent viscosity in boundary layers alone or is its action on small scales generated by the nonlinearity through the cascade also a contributor? This report develops model dissipation estimates for body force driven flow under periodic boundary conditions (and thus only with nonlinearity generated small scales). It is proven that the model's time averaged energy dissipation rate satisfies the same upper bound as for the NSE plus one additional term that vanishes uniformly in the Reynolds number as the Smagorinsky length scale decreases. Since this estimate is consistent with that observed for the NSE, it establishes that, without boundary layers, the Smagorinsky model does not over dissipate.

math.NA↗

arXiv2015

Algorithms and Models for Turbulence Not at Statistical Equilibrium

Standard eddy viscosity models, while robust, cannot represent backscatter and have severe difficulties with complex turbulence not at statistical equilibrium. This report gives a new derivation of eddy viscosity models from an equation for the evolution of variance in a turbulent flow. The new derivation also shows how to correct eddy viscosity models. The report proves the corrected models preserve important features of the true Reynolds stresses. It gives algorithms for their discretization including a minimally invasive modular step to adapt an eddy viscosity code to the extended models. A numerical test is given with the usual and over diffusive Smagorinsky model. The correction (scaled by $10^{-8}$ ) does successfully exhibit intermittent backscatter.

math.NA↗

arXiv2007

A high accuracy Leray-deconvolution model of turbulence and its limiting behavior

In 1934 J. Leray proposed a regularization of the Navier-Stokes equations whose limits were weak solutions of the NSE. Recently, a modification of the Leray model, called the Leray-alpha model, has atracted study for turbulent flow simulation. One common drawback of Leray type regularizations is their low accuracy. Increasing the accuracy of a simulation based on a Leray regularization requires cutting the averaging radius, i.e., remeshing and resolving on finer meshes. This report analyzes a family of Leray type models of arbitrarily high orders of accuracy for fixed averaging radius. We establish the basic theory of the entire family including limiting behavior as the averaging radius decreases to zero, (a simple extension of results known for the Leray model). We also give a more technically interesting result on the limit as the order of the models increases with fixed averaging radius. Because of this property, increasing accuracy of the model is potentially cheaper than decreasing the averaging radius (or meshwidth) and high order models are doubly interesting.

math-ph↗