arXiv subjects

David Goluskin

Publications and source records attributed to David Goluskin.

At least 19 recordsLinked to original sources

arXiv2026

Domain-filling rolls in two-dimensional fixed-flux Rayleigh-B\'enard convection

Rayleigh-B\'enard convection of large horizontal extent sometimes self-organizes into domain-filling structures. In two dimensions, domain-filling rolls persist - from some but not all initial conditions - when velocity boundary conditions are stress-free. When velocity boundary conditions are no-slip, domain-filling rolls are not found with fixed-temperature thermal boundary conditions, but they have not been sought with fixed-flux thermal boundary conditions. Here we explore the latter missing case, which is hard to predict because no-slip boundaries do not encourage domain-filling rolls, but fixed-flux boundaries give domain-filling structures in three dimensions for either boundary condition on velocity. We simulate convection with fixed-flux, no-slip boundaries in two-dimensional domains with horizontal period 20 times their height and at various combinations of the fixed-flux Rayleigh number R and Prandtl number Pr. Domain-filling rolls, which are easily found when they are weakly nonlinear at small R, are continued to other (R,Pr) by changing these parameters slowly in time. The (R,Pr) regime where we find domain-filling rolls is substantial but has a boundary. When R is too large or Pr too small, relative to each other, a domain-filling roll pair breaks up into two pairs. These findings contrast with other combinations of dimension and boundary conditions, where scale selection has not been seen to depend strongly on parameter values. The present case helps disentangle competing effects of dimension and boundary conditions, and it offers a more stringent test for explanations of scale selection that have been proposed.

physics.flu-dyn↗

arXiv2026

Quartic Lyapunov functions for global fluid stability

A fluid system is 'globally stable' if all initial conditions eventually converge to the same state. Since Reynolds (1895) and Orr (1907), the standard way to show global stability has been the energy method, which uses the fluctuation energy as a Lyapunov function. However, the energy method fails whenever transient energy growth is possible, so it often yields overly strict stability criteria. The first broadly applicable alternative has recently been introduced (Goulart & Chernyshenko 2012; Fuentes et al. 2022), using polynomial optimization to construct non-quadratic Lyapunov functions. Unlike the energy method, however, this approach is highly technical, computationally expensive, and hard to interpret physically. Moreover, it treats only one set of parameters at a time; in particular, if it verifies global stability at a certain Reynolds number, it does not imply the same for smaller values. The present work makes progress by connecting this numerical program with new analytical and physical insights. We show how to exploit symmetries of shear flows via convenient complex variable representations, greatly reducing the problem size. We then refine key inequalities, replace several expensive computational steps with simpler analytical alternatives, and show how to prove global stability over a range of Reynolds numbers. Our analysis identifies the simplest class of non-quadratic Lyapunov functions for two-dimensional parallel shear flows: a three-parameter family of quartic polynomials. Using these Lyapunov functions, we verify global stability of 2-D plane Couette flow and plane Poiseuille flow up to higher Reynolds numbers than possible with the energy method. Our work takes a step towards an analytical theory of global fluid stability beyond the energy method, and offers structural insights that should significantly improve future numerical investigations of global stability.

physics.flu-dyn↗

arXiv2025

Computation of attractor dimension and maximal sums of Lyapunov exponents using polynomial optimization

Two approaches are presented for computing upper bounds on Lyapunov exponents and their sums, and on the Lyapunov dimension, among all trajectories of a dynamical system governed by ordinary differential equations. The first approach expresses a sum of Lyapunov exponents as a time average in an augmented dynamical system and then applies methods for bounding time averages. This generalizes the method of Oeri \& Goluskin (Nonlinearity 36:5378--5400, 2023) for bounding the single largest Lyapunov exponent. The second approach considers a different augmented dynamical system, where bounds on sums of Lyapunov exponents are implied by stability of certain sets, and such stability is verified using Lyapunov function methods. Both of our approaches also can be adapted to directly compute bounds on Lyapunov dimension, which in turn imply bounds on the fractal dimension of a global attractor. For systems of ordinary differential equations with polynomial right-hand sides, all of our bounding formulations lead to polynomial optimization problems with sum-of-squares constraints. These sum-of-squares problems can be solved computationally for a chosen system to yield numerical bounds, provided the number of variables and degree of polynomials are not prohibitive. Most of our bounding formulations are proved to be sharp under mild assumptions. In the case of the polynomial optimization problems, sharpness means that upper bounds converge to the quantities being bounded as polynomial degrees are raised. Computational examples demonstrate upper bounds that are sharp to several digits, including for a six-dimensional dynamical system where sums of Lyapunov exponents are maximized on periodic orbits.

math.DS↗

arXiv2025

Computation of minimal periods for ordinary differential equations

A framework is presented for lower-bounding periods among periodic solutions to an autonomous dynamical system governed by ordinary differential equations. For a chosen dynamical system, lower bounds can be proved by constructing auxiliary functions that, similarly to Lyapunov functions, satisfy a certain inequality pointwise on state space. Different formulations can give bounds applying either to all periodic solutions or to only periodic solutions with chosen symmetry. In the case of differential equations that are polynomial in the state variables, we present computational methods that use semidefinite programming to construct auxiliary functions. Furthermore, we give an algorithm to rigorously validate the numerically computed bounds via rational arithmetic. To illustrate these methods, computations are carried out for two chaotic systems that each have an infinite number of periodic solutions: the Lorenz system, which is dissipative, and the H\'enon-Heiles system, which is Hamiltonian. All computed bounds are validated with rational arithmetic. Separate bounds are computed that apply to all periodic solutions, and to only periodic solutions with certain symmetries. In all cases, our best validated bounds agree with periods of known periodic solutions to at least 5 digits, which strongly suggests exact sharpness of our framework for these examples. The question of how broadly our framework is sharp is discussed, but it remains open.

math.DS↗

arXiv2025

Bounds on dissipation in three-dimensional planar shear flows: reduction to two-dimensional problems

Bounds on turbulent averages in shear flows can be derived from the Navier--Stokes equations by a mathematical approach called the background method. Bounds that are optimal within this method can be computed at each Reynolds number Re by numerically optimizing subject to a spectral constraint, which requires a quadratic integral to be nonnegative for all possible velocity fields. Past authors have eased computations by enforcing the spectral constraint only for streamwise-invariant (2.5D) velocity fields, assuming this gives the same result as enforcing it for three-dimensional (3D) fields. Here we compute optimal bounds over 2.5D fields and then verify, without doing computations over 3D fields, that the bounds indeed apply to 3D flows. One way is to directly check that an optimizer computed using 2.5D fields satisfies the spectral constraint for all 3D fields. A second way uses a criterion we derive that is based on a theorem of Busse (ARMA 47:28, 1972) for energy stability analysis of models with certain symmetry. The advantage of checking this criterion, as opposed to directly checking the 3D constraint, is lower computational cost and natural extrapolation of the criterion to large Re. We compute optimal upper bounds on friction coefficients for the wall-bounded Kolmogorov flow known as Waleffe flow, and for plane Couette flow. This requires lower bounds on dissipation in the first model and upper bounds in the second. For Waleffe flow, all bounds computed using 2.5D fields satisfy our criterion, so they hold for 3D flows. For Couette flow, where bounds have been previously computed using 2.5D fields by Plasting & Kerswell (JFM 477:363, 2003), our criterion holds only up to moderate Re, so at larger Re we directly verify the 3D spectral constraint. Over the Re range of our computations, this confirms the assumption by Plasting & Kerswell that their bounds hold for 3D flows.

physics.flu-dyn↗

arXiv2024

Lifetimes of metastable windy states in two-dimensional Rayleigh-Bénard convection with stress-free boundaries

Two-dimensional horizontally periodic Rayleigh-Bénard convection between stress-free boundaries displays two distinct types of states, depending on the initial conditions. Roll states are composed of pairs of counter-rotating convection rolls. Windy states are dominated by strong horizontal wind (also called zonal flow) that is vertically sheared, precludes convection rolls, and suppresses heat transport. Windy states occur only when the Rayleigh number $Ra$ is sufficiently above the onset of convection. At intermediate $Ra$ values, windy states can be induced by suitable initial conditions, but they undergo a transition to roll states after finite lifetimes. At larger $Ra$ values, where windy states have been observed for the full duration of simulations, it is unknown whether they represent chaotic attractors or only metastable states that would eventually undergo a transition to roll states. We study this question using direct numerical simulations of a fluid with a Prandtl number of 10 in a layer whose horizontal period is 8 times its height. At each of seven $Ra$ values between $9\times10^6$ and $2.25\times10^7$ we have carried out 200 or more simulations, all from initial conditions leading to windy convection with finite lifetimes. The lifetime statistics at each $Ra$ indicate a memoryless process with survival probability decreasing exponentially in time. The mean lifetimes grow with $Ra$ approximately as $Ra^4$. This analysis provides no $Ra$ value at which windy convection becomes stable; it might remain metastable at larger $Ra$ with extremely long lifetimes.

physics.flu-dyn↗

arXiv2023

Convex computation of maximal Lyapunov exponents

We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent, provided that trajectories of interest remain in a compact set. The minimization is over auxiliary functions that are defined on the state space and subject to a pointwise inequality. In the polynomial case -- i.e., when the ODE's right-hand side is polynomial, the set of interest can be specified by polynomial inequalities or equalities, and auxiliary functions are sought among polynomials -- the minimization can be relaxed into a computationally tractable polynomial optimization problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimization problems of increasing computational cost whose infima converge from above to the maximal Lyapunov exponent, at least when the set of interest is compact. For illustration, we carry out such polynomial optimization computations for two chaotic examples: the Lorenz system and the Hénon-Heiles system. The computed upper bounds converge as polynomial degrees are raised, and in each example we obtain a bound that is sharp to at least five digits. This sharpness is confirmed by finding trajectories whose leading Lyapunov exponents approximately equal the upper bounds.

math.DS↗

arXiv2022

Steady Rayleigh--B\'enard convection: strongly nonlinear high-wavenumber rolls

In Rayleigh--B\'{e}nard convection, two-dimensional steady rolls bifurcate supercritically at a Rayleigh number $Ra$ that depends on their horizontal-to-vertical aspect ratio $\Gamma$, and they exist at all larger $Ra$ despite being unstable. Heat transport by certain rolls---quantified by the Nusselt number $Nu$---closely resembles turbulent transport, yet $Nu$ scalings of rolls are understood only for specific boundary conditions and $\Gamma$--$Ra$ limits. Here we investigate the high-wavenumber limit $\Gamma = O(Ra^{-1/4})$ as $Ra \to \infty$, using numerics and matched asymptotic analysis. We compute steady rolls between stress-free boundaries for Prandtl numbers $10^{-1} \leq Pr \leq 10^{3/2}$ and $Ra$ reaching $10^{19}$. While the $\Gamma = O(Ra^{-1/4})$ limit gives smaller $Nu$ than when $\Gamma = O(1)$, we identify prefactors $c$ in $\Gamma = c\,Ra^{-1/4}$ that locally maximize $Nu$. These locally $Nu$-maximizing rolls display approximate scalings $Nu \propto Ra^{0.29}$ and $Re \propto Ra^{0.40}$, with the Reynolds number $Re$ defined using root-mean-square velocity. Our asymptotic analysis reveals a vertically stacked four-layer structure near each boundary, predicting $Nu = O(Ra^{3/10})$ and $Re = O(Ra^{2/5})$. This asymptotic construction largely follows that of Taylor vortices by \cite{Deguchi2023}, but we identify a thin plume region within the middle boundary layer whose inclusion eliminates the logarithmic factors in Deguchi's predictions. Asymptotic arguments and numerics suggest the same scalings for stress-free or no-slip boundaries, unlike in other $\Gamma$--$Ra$ limits. Our asymptotics extend the weakly nonlinear analysis of Blennerhassett \& Bassom (1994) into the strongly nonlinear regime and complement the asymptotics of Chini \& Cox (2009) for $\Gamma = O(1)$ rolls.

physics.flu-dyn↗

arXiv2022

Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization

We present a method for finding lower bounds on the global infima of integral variational problems, wherein $\int_Ωf(x,u(x),\nabla u(x)){\rm d}x$ is minimized over functions $u\colonΩ\subset\mathbb{R}^n\to\mathbb{R}^m$ satisfying given equality or inequality constraints. Each constraint may be imposed over $Ω$ or its boundary, either pointwise or in an integral sense. These global minimizations are generally non-convex and intractable. We formulate a particular convex maximization, here called the pointwise dual relaxation (PDR), whose supremum is a lower bound on the infimum of the original problem. The PDR can be derived by dualizing and relaxing the original problem; its constraints are pointwise equalities or inequalities over finite-dimensional sets, rather than over infinite-dimensional function spaces. When the original minimization can be specified by polynomial functions of $(x,u,\nabla u)$, the PDR can be further relaxed by replacing pointwise inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS program is computationally tractable when the dimensions $m,n$ and number of constraints are not too large. The framework presented here generalizes an approach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat. Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given by the PDR is sharp for several classes of problems, whose special cases include leading eigenvalues of Sturm-Liouville problems and optimal constants of Poincaré inequalities. For these same classes, we prove that SOS relaxations of the PDR converge to the sharp lower bound as polynomial degrees are increased. Convergence of SOS computations in practice is illustrated for several examples.

math.OC↗

arXiv2022

Transition between Boundary-Limited Scaling and Mixing-Length Scaling of Turbulent Transport in Internally Heated Convection

Heat transport in turbulent thermal convection increases with the thermal forcing, but in almost all studies the rate of this increase is slower than it would be if transport became independent of the molecular diffusivities -- the heat transport scaling is slower than the mixing-length (or `ultimate') scaling. In configurations driven by either thermal boundary conditions or internal heating, thermal boundary layers instead lead to a boundary-limited (or `classical') scaling. With net-zero internal heating and cooling in different regions, mixing-length scaling can occur because heat need not cross a boundary. We report numerical simulations in which heating and cooling are unequal, as in various natural systems, at a Prandtl number of unity. As heating and cooling rates are made closer, the scaling exponent of heat transport varies from its boundary-limited value to its mixing-length value.

physics.flu-dyn↗

arXiv2022

Global stability of fluid flows despite transient growth of energy

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct non-quadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the energy stability threshold found by Orr in 1907. This is the first global stability result for any flow that surpasses the energy method.

physics.flu-dyn↗

arXiv2022

Steady Rayleigh--Bénard convection between no-slip boundaries

The central open question about Rayleigh--Bénard convection -- buoyancy-driven flow in a fluid layer heated from below and cooled from above -- is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to\infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations, and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed `classical' $Nu \sim Ra^{1/3}$ or `ultimate' $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$, we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to\infty$, it cannot achieve the ultimate scaling.

physics.flu-dyn↗

arXiv2021

A study of the double pendulum using polynomial optimization

In dynamical systems governed by differential equations, a guarantee that trajectories emanating from a given set of initial conditions do not enter another given set can be obtained by constructing a barrier function that satisfies certain inequalities on phase space. Often these inequalities amount to nonnegativity of polynomials and can be enforced using sum-of-squares conditions, in which case barrier functions can be constructed computationally using convex optimization over polynomials. To study how well such computations can characterize sets of initial conditions in a chaotic system, we use the undamped double pendulum as an example and ask which stationary initial positions do not lead to flipping of the pendulum within a chosen time window. Computations give semialgebraic sets that are close inner approximations to the fractal set of all such initial positions.

nlin.CD↗

arXiv2020

Steady Rayleigh--Bénard convection between stress-free boundaries

Steady two-dimensional Rayleigh--Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios $π/5\leΓ\le4π$, where $Γ$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\le Ra\le10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\le Pr\le10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $Γ$ and is largest at $Γ\approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $Γ\approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini \& Cox (2009). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $Γ$ is smaller.

physics.flu-dyn↗

arXiv2020

Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization

Many monostable reaction-diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction-diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction-diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction-diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar RD equations.

math.DS↗

arXiv2020

Heat transport bounds for a truncated model of Rayleigh-Bénard convection via polynomial optimization

Upper bounds on time-averaged heat transport are obtained for an eight-mode Galerkin truncation of Rayleigh's 1916 model of natural thermal convection. Bounds for the ODE model---an extension of Lorenz's three-ODE system---are derived by constructing auxiliary functions that satisfy sufficient conditions wherein certain polynomial expressions must be nonnegative. Such conditions are enforced by requiring the polynomial expressions to admit sum-of-squares representations, allowing the resulting bounds to be minimized using semidefinite programming. Sharp or nearly sharp bounds on mean heat transport are computed numerically for numerous values of the model parameters: the Rayleigh and Prandtl numbers and the domain aspect ratio. In all cases where the Rayleigh number is small enough for the ODE model to be quantitatively close to the PDE model, mean heat transport is maximized by steady states. In some cases at larger Rayleigh number, time-periodic states maximize heat transport in the truncated model. Analytical parameter-dependent bounds are derived using quadratic auxiliary functions, and they are sharp for sufficiently small Rayleigh numbers.

physics.flu-dyn↗

arXiv2020

Bounding extrema over global attractors using polynomial optimisation

We describe a framework for bounding extreme values of quantities on global attractors of differential dynamical systems. A global attractor is the minimal set that attracts all bounded sets; it contains all forward-time limit points. Our approach uses (generalised) Lyapunov functions to find attracting sets, which must contain the global attractor, and the choice of Lyapunov function is optimised based on the quantity whose extreme value one aims to bound. We also present a non-global framework for bounding extrema over the minimal set that is attracting in a specified region of state space. If the dynamics are governed by ordinary differential equations, and the equations and quantities of interest are polynomial, then our methods can be implemented computationally using polynomial optimisation. In particular, we enforce nonnegativity of certain polynomial expressions by requiring them to be representable as sums of squares, leading to a convex optimisation problem that can be recast as a semidefinite program and solved computationally. This computer assistance lets one construct complicated polynomial Lyapunov functions. Computations are illustrated using three examples. The first is the chaotic Lorenz system, where we bound extreme values of various monomials of the coordinates over the global attractor. In the second example we bound extreme values in a nine-mode truncation of fluid dynamics which displays long-lived chaotic transients. The third example has two locally stable limit cycles, each with its own basin of attraction, and we apply our non-global framework to construct bounds for one basin that do not apply to the other. For each example we compute Lyapunov functions of polynomial degrees up to at least eight. In cases where we can judge the sharpness of our bounds, they are sharp to at least three digits when the polynomial degree is at least four or six.

math.DS↗

arXiv2020

Bounding extreme events in nonlinear dynamics using convex optimization

We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.

math.DS↗