
William J. LaytonUniversity of Pittsburgh | Pitt · Mathematics
William J. Layton
Ph.D.
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251
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Introduction
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September 1980 - April 1986
August 1982 - July 1983
September 1986 - present
Publications
Publications (251)
Data assimilation combines (imperfect) knowledge of a flow's physical laws with (noisy, time-lagged, and otherwise imperfect) observations to produce a more accurate prediction of flow statistics. Assimilation by nudging (from 1964), while non-optimal, is easy to implement and its analysis is clear and well-established. Nudging's uniform in time ac...
The recent 1/2-equation model of turbulence is a simplification of the standard Kolmogorov-Prandtl 1-equation URANS model. Surprisingly, initial numerical tests indicated that the 1/2-equation model produces comparable velocity statistics at reduced cost. It is also a test problem and first step for developing numerical analysis to address a full 1...
In 1-equation URANS models of turbulence the eddy viscosity is given by $\nu_{T}=0.55l(x,t)\sqrt{k(x,t)}$ . The length scale $l$ must be pre-specified and $k(x,t)$ is determined by solving a nonlinear partial differential equation. We show that in interesting cases the spacial mean of $k(x,t)$ satisfies a simple ordinary differential equation. Usin...
Dahlquist, Liniger and Nevanlinna devised a family of one-leg two-step methods (DLN) that is second order, A- and G- stable for arbitrary, non-uniform time steps. The DLN method thus has strong potential for use in adaptive codes, but its adaptive step size selection is little explored. This report develops two approaches for the efficient local er...
Inclusion of a term −γ∇∇⋅u, forcing ∇⋅u to be pointwise small, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added grad-div term couples all velocity components decreasing sparsity and increasing the condition number in the linear systems that must be solved every time step. To address...
Penalizing incompressibility in the Stokes problem leads, under mild assumptions, to matrices with condition numbers $\kappa =\mathcal{O} (\varepsilon ^{-1}h^{-2})$, $\varepsilon =$ penalty parameter $<<1$, and $ h= $ mesh width $<1$. Although $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solv...
We prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. The ratio of the near wall average viscosity to the effective global viscosity is the key parameter in the estimate. This result is then applied to the 1-equation, URANS model of turbulence for which this ratio depend...
In simulations of fluid motion time accuracy has proven to be elusive. We seek highly accurate methods with strong enough stability properties to deal with the richness of scales of many flows. These methods must also be easy to implement within current complex, possibly legacy codes. Herein we develop, analyze and test new time stepping methods ad...
Inclusion of a grad-div term, $-\gamma\nabla\nabla\cdot u$, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added term $-\gamma\nabla\nabla\cdot u$ couples all velocity components, decreases sparsity and increases the condition number in the linear systems that must be solved every time...
The one-leg, two-step time-stepping scheme proposed by Dahlquist, Liniger and Nevanlinna has clear advantages in complex, stiff numerical simulations: unconditional G-stability for variable time-steps and second-order accuracy. Yet it has been underutilized due, partially, to its complexity of direct implementation. We prove herein that this method...
Artificial compression methods are used in computational fluid dynamics as a cost-effective way of solving for the velocity and pressure in a flow. However, relaxation of compressibility in these algorithms yields nonphysical oscillations in the pressure. This report presents analysis and computational tests of time filters to reduce nonphysical ac...
Clipping refers to adding 1 line of code A=min{A,B} to force the variable A to stay below a present bound B. Phenomenological clipping also occurs in turbulence models to correct for over dissipation caused by the action of eddy viscosity terms in regions of small scales. Herein we analyze eddy viscosity model energy dissipation rates with 2 phenom...
The one‐leg, two‐step time discretization proposed by Dahlquist, Liniger and Nevanlinna is second order and variable step G‐stable. G‐stability for systems of ordinary differential equations (ODEs) corrresponds to unconditional, long time energy stability when applied to the Navier–Stokes equations (NSEs). In this report, we analyze the method of D...
The one-leg, two-step time-stepping scheme proposed by Dahlquist, Liniger and Nevanlinna has clear advantages in complex, stiff numerical simulations: unconditional $G$-stability for variable time-steps and second-order accuracy. Yet it has been underutilized due, partially, to its complexity of direct implementation. We prove herein that this meth...
We prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. For general eddy viscosity models, we show that the ratio of the near wall average viscosity to the effective global viscosity is the key parameter. This result is then applied to the 1-equation, URANS model of turbul...
In simulations of fluid motion time accuracy has proven to be elusive. We seek highly accurate methods with strong enough stability properties to deal with the richness of scales of many flows. These methods must also be easy to implement within current complex, possibly legacy codes. Herein we develop, analyze and test new time stepping methods ad...
This report gives a summary of some recent developments in the mathematical foundations of eddy viscosity models of turbulence.
This paper develops, analyzes and tests a time-accurate partitioned method for the Stokes–Darcy equations. The method combines a time filter and Backward Euler scheme, is second order accurate and provides, at no extra complexity, an estimate of the temporal error. This approach post-processes the solutions of Backward Euler scheme by adding three...
The standard 1-equation model of turbulence was first derived by Prandtl and has evolved to be a common method for practical flow simulations. Five fundamental laws that any URANS model should satisfy are This report proves that a kinematic specification of the model’s turbulence lengthscale by where is the time filter window, results in a 1-equati...
Classical eddy viscosity models add a viscosity term with turbulent viscosity coefficient whose specification varies from model to model. Turbulent viscosity coefficient approximations of unknown accuracy are typically constructed by solving associated systems of nonlinear evolution equations or by data driven approaches such as deep neural network...
Artificial compression methods are used in computational fluid dynamics as a
cost-effective way of solving for the velocity and pressure in a flow.
However, relaxation of compressibility in these algorithms yields
nonphysical oscillations in the pressure. This\ report\ presents tests of\
time filters to reduce nonphysical acoustic waves in artifici...
The two-step time discretization proposed by Dahlquist, Liniger and Nevanlinna is variable step $G$-stable. (In contrast, for increasing time steps, the BDF2 method loses $A$-stability and suffers non-physical energy growth in the approximate solution.) While unexplored, it is thus ideal for time accurate approximation of the Navier-Stokes equation...
This report presents adaptive artificial compression methods in which the time-step and artificial compression parameter ε 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 adaptiv...
Classical eddy viscosity models add a viscosity term with turbulent viscosity coefficient whose specification varies from model to model. Turbulent viscosity coefficient approximations of unknown accuracy are typically constructed by solving associated systems of nonlinear evolution equations or by data driven approaches such as deep neural network...
The standard $1-$equation model \ of turbulence was first derived by Prandtl and has evolved to be a common method for practical flow simulations. Five fundamental laws that any URANS model should satisfy are \[ \begin{array} [c]{ccc} \textbf{1.} & \text{Time window:} & \begin{array} [c]{c} \tau\downarrow 0\text{ implies }v_{\text{{\small URANS}}}\...
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 o...
This report presents adaptive artificial compression methods in which the time-step and artificial compression parameter ε 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 adaptiv...
Complex turbulence not at statistical equilibrium is impossible to simulate using eddy viscosity models due to a backscatter. This research presents the way to correct the Baldwin–Lomax model for nonequilibrium effects and gives an analysis of the energy evolution in the corrected model. Furthermore, a finite element approximation of the corrected...
A standard artificial compression (AC) method for incompressible flow is un+1ε-unεk+un+1ε·∇un+1ε+12un+1ε∇·un+1ε+∇pn+1ε-νΔun+1ε=f,εpn+1ε-pnεk+∇·un+1ε=0for, typically, ε= k (timestep). It is fast, efficient and stable with accuracy O(ε+ k). For adaptive (and thus variable) timestep k n (and thus ε= ε n ) its long time stability is unknown. For variab...
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 ne...
Artificial compression methods create nonphysical acoustic waves. Time filters, often used in geophysical fluid dynamics, are shown in this paper to selectively damp these acoustics. We analyze the stability of a two‐step artificial compression method with the Robert–Asselin (RA) time filter, and provide tests delineating the filter's positive effe...
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...
This report presents a low complexity, stable and time accurate 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...
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 }-\nu \Delta u_{n+1}^{\varepsilon }=f\text{ ,} \\ \varepsilon \...
There has been a surge of work on models for coupling surface‐water with groundwater flows which is at its core the Stokes–Darcy problem, as well as methods for uncoupling the problem into subdomain, subphysics solves. The resulting (Stokes–Darcy) fluid velocity is important because the flow transports contaminants. The numerical analysis and algor...
Magnetohydrodynamics (MHD) is the study of the interaction of electrically conducting fluids in the presence of magnetic fields. MHD applications require substantially more efficient numerical methods than currently exist. In this paper, we construct two decoupled methods based on the artificial compression method (uncoupling the pressure and veloc...
The problem of accurate and reliable simulation of turbulent flows is a central and intractable challenge that crosses disciplinary boundaries. As the needs for accuracy increase and the applications expand beyond flows where extensive data is available for calibration, the importance of a sound mathematical foundation that addresses the needs of p...
The problem of accurate and reliable prediction of turbulent flows is a central and intractable challenge that crosses disciplinary boundaries. [...]
This report considers the effect of adding a simple time filter to the fully implicit or backward Euler method. The approach is modular and requires the addition of only one line of additional code. Error estimation and variable time step are straightforward and the individual effect of each step is conceptually clear. The backward Euler method wit...
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. S...
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. S...
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...
This report presents a new artificial compression method for incompressible, viscous flows. The method has second order consistency error and is unconditionally, long time, energy stable for the velocity and, weighted by the timestep, for the pressure. It uncouples the pressure and velocity and requires no artificial pressure boundary conditions. W...
Grad-div stabilization, adding a term −γgraddivu to penalize violation of incompressibility, has proven to be a useful tool in the simulation of incompressible flows. Such a term requires a choice of the coefficient γ and studies have begun appearing with various suggestions for its value. We give an analysis herein that provides a restricted range...
Grad-div stabilization, adding a term -gamma grad div u, has proven to be a useful tool in the simulation of incompressible flows. Such a term requires a choice of the coefficient gamma and studies have begun appearing with various suggestions for its value. We give an analysis herein that provides a restricted range of possible values for the coef...
This textbook was designed for senior undergraduates in mathematics, engineering and the sciences with diverse backgrounds and goals. It presents modern tools from numerical linear algebra with supporting theory along with examples and exercises, both theoretical and computational with MATLAB. The major topics of numerical linear algebra covered ar...
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 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...
This report proves that under the time step condition ∆t|Λ| < 1 (| · | = Euclidean norm) suggested by root condition analysis and necessary for stability, all modes of the Crank-Nicolson Leap-Frog (CNLF) approximate solution to the system (formula presented) where A + AT is symmetric positive definite and Λ is skew symmetric, are asymptotically sta...
There has been a surge of work on models for coupling surface-water with groundwater flows which is at its core the Stokes-Darcy problem. The resulting (Stokes-Darcy) fluid velocity is important because the flow transports contaminants. The analysis of models including the transport of contaminants has, however, focused on a quasi-static Stokes-Dar...
This report presents a summary of the numerical analysis of time filters used to control the unstable mode in the Crank-Nicolson-Leapfrog discretization of evolution equations.
The Boussinesq assumption that turbulent fluctuations have a dissipative effect on the mean flow is the basis for most turbulence models used in practical flow simulations. Data from computational tests and experiments has indicated that in 2d fluid flows an inverse energy cascade is expected. However, the Boussinesq assumption has recently been pr...
We propose and analyze a linear stabilization of the Crank–Nicolson Leapfrog (CNLF) method that removes all time step/CFL conditions for stability and controls the unstable mode. It also increases the SPD part of the linear system to be solved at each time step while increasing solution accuracy. We give a proof of unconditional stability of the me...
This report develops an ensemble or statistical eddy viscosity model.
The model is parameterized by an ensemble of solutions of an ensemble-Leray
regularization. The combined approach of ensemble time stepping and ensemble
eddy viscosity modeling allows direct parametrization of the turbulent
viscosity coefficient. We prove unconditional stability...
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 mo...
In this work, we present a comprehensive study of several partitioned methods for the coupling of flow and mechanics. We derive energy estimates for each method for the fully-discrete problem. We write the obtained stability conditions in terms of a key control parameter defined as a ratio of the coupling strength and the speed of propagation. Depe...
Geophysical flow simulations have evolved sophisticated implicit-explicit time stepping methods (based on fast-slow wave splittings) followed by time filters to control any unstable models that result. Time filters are modular and parallel. Their effect on stability of the overall process has been tested in numerous simulations. In this paper, we s...
This report analyzes an efficient ensemble regularization algorithm for under-resolved and convection dominated flows (including ones at higher Reynolds numbers). Computing an ensemble simultaneously allows each realization to access ensemble data. This allows use of means and fluctuations in regularizations used for each realization. The combined...
We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1: Advance the NSE one time step, Step 2: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for simple time stepping methods in Step 1 and simple stabilizations in S...
This report presents an algorithm for computing an ensemble of p solutions of the Navier-Stokes equations. The solutions are found, at each timestep, by solving a linear system with one shared coefficient matrix and p right hand sides, reducing both storage required and computational cost of the solution process. The price that must be paid is a ti...
We present the design of a novel framework for the visual integration, comparison, and exploration of correlations in spatial and non-spatial geriatric research data. These data are in general high-dimensional and span both the spatial, volumetric domain - through magnetic resonance imaging volumes - and the non-spatial domain, through variables su...
MHD flows are governed by the Navier-Stokes equations coupled with the Maxwell equations. Broadly, MHD flows in astrophysics occur at large magnetic Reynolds numbers while those in terrestrial applications, such as liquid metals, occur at small magnetic Reynolds numbers, the case considered herein. The physical processes of fluid flows and electric...
Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear. We consider nonlinear filters which select eddies for damping (simulating breakdown) based on knowledge of how nonlinearity acts in real flow proble...
We study analytically and numerically the relaxation time of flow evolution governed by the Navier-Stokes-Voigt NSV model. We first show that for the Taylor–Green vortex decay problem, NSV admits an exact solution which evolves slower than true fluid flow. Secondly, we show numerically for a channel flow test problem using standard discretisation m...
This report analyzes a multirate, decoupling algorithm, which allows different time steps in the fluid region and the porous region for the nonstationary Stokes–Darcy problem. The method presented requires only one, uncoupled Stokes and Darcy subphysics and subdomain solve per time step. Under a time step restriction of the form △t ≤ C (physical pa...
This paper presents the stabilities for both two modular, projection-based variational multiscale (VMS) methods and the error analysis for only first one for the incompressible Naiver-Stokes equations, expanding the analysis in [39] to include nonlinear eddy viscosities. In VMS methods, the influence of the unresolved scales onto the resolved small...
We give an energy stability analysis of a first order, 2 step partitioned time discretization of systems of evolution equations. The method requires only uncoupled solutions of sub-systems at every time step without iteration, is long time stable and applies to general system couplings. We give a proof of long time energy stability under a time ste...
The most effective simulations of the multiphysics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes. Partitioned methods, which solve the coupled problem by successively solving the subphysics problems, have recently been studied for the Stokes-Darcy coupling with convergenc...
An approximate deconvolution operator denoted by D is an approximate filter inverse that is accurate on the smooth velocity components and does not magnify the rough components.
The fundamental requirement for a successful turbulent flow simulation is to truncate scales to those representable on a computationally feasible mesh without substantially changing the large flow structures.
In LES one solves a system whose solution is an approximation to local spacial averages of the NSE. In regularization modeling one solves a system similar to the NSE which has better qualitative properties for numerical simulation than the underlying NSE.
Consider the NSE in rotational form: \(\begin{array}{rrr}\rm{u}_t+(\nabla\times \rm{u})\times\rm{u}-\rm{v}\triangle\rm{u}+\nabla P=f(\rm{x},t), \\ \nabla \cdot = 0, \\ \text{where} P = p + 1/2|u|_2\end{array}\)
At high Reynolds number the fluid velocity is exponentially sensitive to perturbations of the problem data. This sensitivity, however, is not uniform. The large structures (large eddies) evolve deterministically and are thus not sensitive [BFG02]. The small eddies are sensitive because they have a random character.
This report analyzes the long time stability of four methods for non-iterative, sub-physics, uncoupling for the evolutionary Stokes–Darcy problem. The four methods uncouple each timestep into separate Stokes and Darcy solves using ideas from splitting methods. Three methods uncouple sequentially while one is a parallel uncoupling method. We prove l...
Stability is proven for two second order, two step methods for uncoupling a system of two evolution equations with exactly skew symmetric coupling: the Crank-Nicolson Leap Frog (CNLF) combination and the BDF2-AB2 combination. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. For CNLF we prove stability...
We reconsider the error in van Cittert deconvolution. We show that without any extra boundary conditions on higher derivatives of u, away from the boundary the error in van Cittert deconvolution attains the high order of accuracy seen in the periodic problem. This error result is important for differential filters and approximate deconvolution mode...
The great challenge in simulation of turbulent flows from applications ranging from geophysics to biomedical device design is that equations for the pointwise flow quantities are well-known but intractable to computational solution and sensitive to uncertainties and perturbation in problem data. On the other hand, closed equations for the averages...
We study a new regularization of the Navier-Stokes equations, the NS-ω model. This model has similarities to the NS-α model but its struc-ture is more amenable to be used as a basis for numerical simulations of turbulent flows. In this report we present the model and prove existence and uniqueness of strong solutions as well as convergence (modulo...
Variational multiscale methods have proven to be an accurate and systematic approach to the simulation of turbulent flows. Many turbulent flows are solved by legacy codes or by ones written by a team of programmers and of great complexity so implementing a new approach to turbulence in such cases can be daunting. We propose a new approach to induci...
Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking
down resolved scales to unresolved scales. This process is highly nonlinear and yet current models for it use linear filters
to select the eddies that will be damped. In this report we consider for the first time nonlinear fi...
When ltering through a wall with constant averaging radius, in addition to the sublter scale stresses, a non-closed commutator term arises. We consider a proposal of Das and Moser to close the commutator error term by embedding it in an optimization probem. This report shows that this optimization based closure, with a small modication, leads to a...
This article shows that so called general Green–Taylor solutions, also called Taylor solutions or eddy solutions, of the Navier–Stokes equations are also exact solutions to approximate deconvolution models of turbulence. Thus, these special structures in flows exist as exact features in the models studied and their persistence/transient behavior is...
This study considers Pao's transfer theory of turbulence for the family of Approximate Decon-volution Models (ADMs). By taking a different representation of the persistent input of energy into the large scales of the turbulent flow, the Pao theory simplifies somewhat. Analysis of the resulting model is given and it is verified that (after the simpl...