arXiv subjects

William Layton

Publications and source records attributed to William Layton.

At least 19 recordsLinked to original sources

arXiv2026

Data Assimilation with Sparse Observations

Data assimilation by nudging (also called CDA) yields exponentially decaying errors and an infinite predictability horizon if the method parameter is large enough and the observations are frequent enough in time and dense enough in space. We consider the complementary case of moderate parameters and sparse and infrequent observations. We prove that assimilation with any data and any(positive) parameter strictly decreases errors and strictly increases the (now finite) predictability horizon.

math.NA↗

arXiv2026

Energy dissipation rates of ensemble eddy viscosity models of turbulence: the periodic box

Classical eddy viscosity models of turbulence add an eddy viscosity term based on the Kolmogorov-Prandtl parameterization by a turbulent length scale $l$ and a turbulent kinetic energy $k^{\prime }$. Approximations of the unknowns $l,k^{\prime }$ are typically constructed by solving multi-parameter systems of nonlinear convection-diffusion-reaction equations. Often these over-diffuse so additional fixes are added. Alternately, one can solve an ensemble of NSE's with perturbed data and simply compute directly $k^{\prime }$(without modeling). The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? We prove herein that for turbulence in a periodic box it does not.

math.NA↗

arXiv2026

On the energy dissipation rate of ensemble eddy viscosity models of turbulence: Shear flows

Classical eddy viscosity models add a viscosity term with turbulent viscosity coefficient developed beginning with the Kolmogorov-Prandtl parameterization. Approximations of unknown accuracy of the unknown mixing lengths and turbulent kinetic energy are typically constructed by solving associated systems of nonlinear convection-diffusion-reaction equations with nonlinear boundary conditions. Often these over-diffuse so additional fixes are added such as wall laws or using different approximations in different regions (which must also be specified). Alternately, one can solve an ensemble of NSE's with perturbed data, compute the ensemble mean and fluctuation and simply compute directly the turbulent viscosity parameterization. This idea is recent, seems to be of lower complexity and greater accuracy and produces parameterizations with the correct near wall asymptotic behavior. The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? This question is addressed herein.

math.NA↗

arXiv2025

Modular data assimilation for flow prediction

This report develops several modular, 2-step realizations (inspired by Kalman filter algorithms) of nudging-based data assimilation $$Step \ 1 \quad \frac{\widetilde {v}^{n+1}-v^{n}}{k}+v^{n}\cdot \nabla \widetilde {v}^{n+1}-\nu \triangle \widetilde {v}^{n+1}+\nabla q^{n+1}=f(x)$$ $$\nabla \cdot \widetilde {v}^{n+1}=0$$ $$Step \ 2 \quad \frac{v^{n+1}-\widetilde {v}^{n+1}}{k}-\chi I_{H}(u(t^{n+1})-v^{n+1})=0.$$ Several variants of this algorithm are developed. Three main results are developed. The first is that if $I_{H}^{2}=I_{H}$, then Step 2 can be rewritten as the explicit step $$v^{n+1}=\widetilde {v}^{n+1}+\frac{k\chi }{1+k\chi }[I_{H}u(t^{n+1})-I_{H} \widetilde {v}^{n+1}].$$ This means Step 2 has the greater stability of an implicit update and the lesser complexity of an explicit analysis step. The second is that the basic result of nudging (that for $H$ small enough and $\chi$ large enough predictability horizons are infinite) holds for one variant of the modular algorithm. The third is that, for any $H>0$ and any $\chi>0$, one step of the modular algorithm decreases the next step's error and increases (an estimate of) predictability horizons. A method synthesizing assimilation with eddy viscosity models of turbulence is also presented. Numerical tests are given, confirming the effectiveness of the modular assimilation algorithm. The conclusion is that the modular, 2-step method overcomes many algorithmic inadequacies of standard nudging methods and retains a robust mathematical foundation.

math.NA↗

arXiv2025

Data assimilation with model errors

Nudging is a data assimilation method amenable to both analysis and implementation. It also has the (reported) advantage of being insensitive to model errors compared to other assimilation methods. However, nudging behavior in the presence of model errors is little analyzed. This report gives an analysis of nudging to correct model errors. The analysis indicates that the error contribution due to the model error decays as the nudging parameter $\chi \to \infty$ like $\mathcal{O}(\chi^{-\frac{1}{2}})$, Theorem 3.2. Numerical tests verify the predicted convergence rates and validate the nudging correction to model errors.

math.NA↗

arXiv2024

Assessing the Viability of Quantum-Resistant IKEv2 over Constrained and Internet-Scale Networks

Within 1-2 decades, quantum computers may become powerful enough to break current public-key cryptography, prompting authorities such as the IETF and NIST to push for adopting quantum-resistant cryptography (QRC) in ecosystems like Internet Protocol Security (IPsec). Yet, IPsec struggles to adopt QRC, primarily because Internet Key Exchange Protocol Version 2 (IKEv2), which sets up IPsec sessions, cannot easily tolerate the large public keys and digital signatures of QRC. Many IETF RFCs have been proposed to integrate QRC into IKEv2, but their performance and interplay remain largely untested in practice. In this paper, we measure the performance of these RFCs over constrained links by developing a flexible, reproducible measurement testbed for IPsec with quantum-resistant IKEv2 proposals. Deploying our testbed in lossy wireless links and on the internationally distributed FABRIC testbed for Internet scenarios, we reveal that bottlenecks arise with quantum-resistant IKEv2 under high round-trip times, non-trivial packet loss, or other constraints. Our results, including the revelation of a 400-1000-fold increase in data overhead over high-loss wireless links, expose the shortcomings of today's RFCs and call for further work in this vital area of post-quantum network security.

cs.NI↗

arXiv2024

Adaptive Parameter Selection in Nudging Based Data Assimilation

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 accuracy has even been established under conditions on the nudging parameter $\chi$ and the density of observational locations, $H$, Larios, Rebholz, and Zerfas [1]. One remaining issue is that nudging requires the user to select a key parameter. The conditions required for this parameter, derived through \'a priori (worst case) analysis are severe (Section 2.1 herein) and far beyond those found to be effective in computational experience. One resolution, developed herein, is self-adaptive parameter selection. This report develops, analyzes, tests, and compares two methods of self-adaptation of nudging parameters. One combines analysis and response to local flow behavior. The other is based only on response to flow behavior. The comparison finds both are easily implemented and yield effective values of the nudging parameter much smaller than those of \'a priori analysis.

math.NA↗

arXiv2024

Numerical analysis of a 1/2-equation model of turbulence

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-equation model. This report begins the numerical analysis of the 1/2 equation model. Stability, convergence and error estimates are proven for a semi-discrete and fully discrete approximation. Finally, numerical tests are conducted to validate our convergence theory.

math.NA↗

arXiv2024

The Ramshaw-Mesina Hybrid Algorithm applied to the Navier Stokes Equations

In 1991, Ramshaw and Mesina proposed a novel synthesis of penalty methods and artificial compression methods. When the two were balanced they found the combination was 3-4 orders more accurate than either alone. This report begins the study of their interesting method applied to the Navier-Stokes equations. We perform stability analysis, semi-discrete error analysis, and tests of the algorithm. Although most of the results for implicit time discretizations of our numerical tests comply with theirs for explicit time discretizations, the behavior in damping pressure oscillations and violations of incompressibility are different from their findings and our heuristic analysis.

math.NA↗

arXiv2024

Convergence of a Ramshaw-Mesina Iteration

In 1991 Ramshaw and Mesina introduced a clever synthesis of penalty methods and artificial compression methods. Its form makes it an interesting option to replace the pressure update in the Uzawa iteration. The result, for the Stokes problem, is \begin{equation} \left\{ \begin{array} [c]{cc} Step\ 1: & -\triangle u^{n+1}+\nabla p^{n}=f(x),\ {\rm in}\ \Omega,\ u^{n+1}|_{\partial\Omega}=0,\\ Step\ 2: & p^{n+1}-p^{n}+\beta\nabla\cdot(u^{n+1}-u^{n})+\alpha ^{2}\nabla\cdot u^{n+1}=0. \end{array} \right. \end{equation} For saddle point problems, including Stokes, this iteration converges under a condition similar to the one required for Uzawa iteration.

math.NA↗

arXiv2023

On a 1/2-equation model of turbulence

In 1-equation URANS models of turbulence the eddy viscosity is given by $ν_{T}=0.55l(x,t)\sqrt{k(x,t)}$ . 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. Using its solution in $ν_{T}$ results in a 1/2-equation model. This model has attractive analytic properties. Further, in comparative tests in 2d and 3d the velocity statistics produced by the 1/2-equation model are comparable to those of the full 1-equation model.

math.NA↗

arXiv2022

Conditioning of linear systems arising from penalty methods

Penalizing incompressibility in the Stokes problem leads, under mild assumptions, to matrices with condition numbers $κ=\mathcal{O} (\varepsilon ^{-1}h^{-2})$, $\varepsilon =$ penalty parameter $<<1$, and $ h= $ mesh width $<1$. Although $κ=\mathcal{O}(\varepsilon ^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solving these systems. In the SPD case, using the conjugate gradient method, this is usually explained by spectral gaps occurring in the penalized coefficient matrix. Herein we point out a second contributing factor. Since the solution is approximately incompressible, solution components in the eigenspaces associated with the penalty terms can be small. As a result, the effective condition number can be much smaller than the standard condition number.

math.NA↗

arXiv2022

Stability in 3d of a sparse grad-div approximation of the Navier-Stokes equations

Inclusion of a term $-γ\nabla\nabla\cdot u$, forcing $\nabla\cdot u$ 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, decreases sparsity and increases the condition number in the linear systems that must be solved every time step. To address these three issues various sparse grad-div regularizations and a modular grad-div method have been developed. We develop and analyze herein a synthesis of a fully decoupled, parallel sparse grad-div method of Guermond and Minev with the modular grad-div method. Let $G^{\ast}=-diag(\partial_{x}^{2},\partial_{y}^{2},\partial_{z}^{2})$ denote the diagonal of $G=-\nabla\nabla\cdot$, and $α\geq0$ an adjustable parameter. The 2-step method considered is $$\begin{eqnarray} 1 &:&\frac{\widetilde{u}^{n+1}-u^{n}}{k}+u^{n}\cdot \nabla \widetilde{u}^{n+1}+\nabla p^{n+1}-νΔ\widetilde{u}^{n+1}=f\text{ & }\nabla \cdot \widetilde{u}^{n+1}=0,\\ 2 &:&\left[ \frac{1}{k}I+(γ+α)G^{\ast }\right] u^{n+1}=\frac{1}{k }\widetilde{u}^{n+1}+\left[ (γ+α)G^{\ast }-γG\right] u^{n}. \end{eqnarray}$$ We prove its unconditional, nonlinear, long time stability in $3d$ for $α\geq0.5γ$. The analysis also establishes that the method controls the persistent size of $||\nabla\cdot u||$ in general and controls the transients in $||\nabla\cdot u||$ when $u(x,0)=0$ and $f(x,t)\neq0$ provided $α>0.5γ$. Consistent numerical tests are presented.

math.NA↗

arXiv2021

Clipping over dissipation in turbulence models

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 phenomenological clipping strategies. Since the true Reynolds stresses are O(d^2) (d= wall normal distance) in the near wall region, the first is to force this near wall behavior in the eddy viscosity by clipping the turbulent viscosity. The second is Escudier's early proposal to clip the turbulence length scale, reducing too large values in the interior of the flow. Analyzing respectively shear flow turbulence and turbulence in a box (i.e., periodic boundary conditions), we show that both clipping strategies do prevent aggregate over dissipation of model solutions.

physics.flu-dyn↗

arXiv2021

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

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 is equivalent to the backward Euler method with pre- and post arithmetic steps added. This refactorization eases implementation in complex, possibly legacy codes. The realization we develop reduces complexity, including cognitive complexity and increases accuracy over both first order methods and constant time steps second order methods.

math.NA↗

arXiv2021

On the Prandtl-Kolmogorov 1-equation model of turbulence

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 turbulence for which this ratio depends on the specification of the turbulence length scale. The model, which was derived by Prandtl in 1945, is a component of a 2-equation model derived by Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models for prediction of turbulent flows. Away from walls, interpreting an early suggestion of Prandtl, we set \begin{equation*} l=\sqrt{2}k^{+1/2}τ, \hspace{50mm} \end{equation*} where $τ=$ selected time scale. In the near wall region analysis suggests replacing the traditional $l=0.41d$ ($d=$ wall normal distance) with $l=0.41d\sqrt{d/L}$ giving, e.g., \begin{equation*} l=\min \left\{ \sqrt{2}k{}^{+1/2}τ,\text{ }0.41d\sqrt{\frac{d}{L}} \right\} . \hspace{50mm} \end{equation*} This $l(\cdot )$ results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct $O(U^{3}/L)$, balancing energy input with energy dissipation.

physics.flu-dyn↗

arXiv2020

Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

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 equations. This report presents an analysis, for variable time-steps, of the method's stability and convergence rates when applied to the NSE. It is proven that the method is variable step, unconditionally, long time stable and second order accurate. Variable step error estimates are also proven. The results are supported by several numerical tests.

math.NA↗

arXiv2019

On URANS Congruity with Time Averaging: Analytical laws suggest improved models

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 \[ \begin{array} [c]{ccc} \textbf{1.} & \text{Time window:} & \begin{array} [c]{c} τ\downarrow 0\text{ implies }v_{\text{\small URANS}}\rightarrow u_{\text{\small NSE}}\text{ \&}\\ \text{ }τ\uparrow\text{implies }ν_{T}\uparrow \end{array} \\ \textbf{2.} & l(x)=0\ \text{at walls:} & l(x)\rightarrow 0\text{ as }x\rightarrow walls,\\ \textbf{3.} & \text{ Bounded energy:} & \sup_{t}\int\frac{1} {2}|v(x,t)|^{2}+k(x,t)dx<\infty\\ \textbf{4.} & \begin{array} [c]{c} \text{Statistical }\\ \text{equilibrium:} \end{array} & \lim\sup_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\varepsilon _{\text{model}}(t)dt=\mathcal{O}\left( \frac{U^{3}}{L}\right) \\ \textbf{5.} & \begin{array} [c]{c} \text{Backscatter}\\ \text{possible:} \end{array} & \text{(without negative viscosities)} \end{array} \] This report proves that a \textit{kinematic} specification of the model's turbulence lengthscale by \[ l(x,t)=\sqrt{2}k^{1/2}(x,t)τ\text{ }, \] where $τ$\ is the time filter window, results in a $1-$equation model satisfying Conditions 1,2,3,4 without model tweaks, adjustments or wall damping multipliers.

math.NA↗