arXiv subjects

John C. Bowman

Publications and source records attributed to John C. Bowman.

16 recordsLinked to original sources

arXiv2026

Hybrid Dealiasing and Implicit Packing for Real Convolutions

Hybrid dealiasing is an FFT-based method for computing linear convolutions of complex-valued data that reduces the cost of dealiasing by performing zero padding implicitly. We develop two new algorithms that extend hybrid dealiasing to real-valued convolutions. The first algorithm exploits conjugate symmetries in the transformed data and computes each residue contribution directly. The second algorithm employs complex-valued hybrid dealiasing via a new implicit packing technique, which packs real-valued data into complex-valued arrays and performs all unpacking and packing operations implicitly in the transformed domain. Multidimensional versions of both algorithms are obtained by recursive decomposition into lower-dimensional convolutions. Based on these algorithms, efficient routines to compute convolutions of equal-length input arrays are implemented in the open-source FFTW++ library. These routines outperform the standard method of explicit zero padding in one, two, and three dimensions. In many cases, implicit packing without hybrid dealiasing is nearly as fast as the general hybrid dealiasing algorithm, particularly when multithreading.

math.NA↗

arXiv2024

Bounds on the Global Attractor of 2D Incompressible Turbulence in the Palinstrophy-Enstrophy-Energy Space

Analytic bounds on the projection of the global attractor of 2D incompressible turbulence in the palinstrophy--enstrophy plane [Dascaliuc, Foias, and Jolly 2005, 2010] are observed to vastly overestimate the values obtained from numerical simulations. This is due to the lack of a good estimate for the inner product $(\cal{B}(u,u),A^2u)$ of the advection term and the biLaplacian. Sobolev inequalities like Ladyzhenskaya or Agmon's inequalities yield an upper bound that we show is not sharp. In fact, for statistically isotropic turbulence, the expected value of $(\cal{B}(u,u),A^2u)$ is zero. The implications for estimates on the behaviour of the global attractor are discussed.

physics.flu-dyn↗

arXiv2024

Hybrid Dealiasing of Complex Convolutions

Efficient algorithms for computing linear convolutions based on the fast Fourier transform are developed. A hybrid approach is described that combines the conventional practice of explicit dealiasing (explicitly padding the input data with zeros) and implicit dealiasing (mathematically accounting for these zero values). The new approach generalizes implicit dealiasing to arbitrary padding ratios and includes explicit dealiasing as a special case. Unlike existing implementations of implicit dealiasing, hybrid dealiasing tailors its subtransform sizes to the convolution geometry. Multidimensional convolutions are implemented with hybrid dealiasing by decomposing them into lower-dimensional convolutions. Convolutions of complex-valued and Hermitian inputs of equal length are illustrated with pseudocode and implemented in the open-source FFTW++ library. Hybrid dealiasing is shown to outperform explicit dealiasing in one, two, and three dimensions.

math.NA↗

arXiv2023

Schur Decomposition for Stiff Differential Equations

A quantitative definition of numerical stiffness for initial value problems is proposed. Exponential integrators can effectively integrate linearly stiff systems, but they become expensive when the linear coefficient is a matrix, especially when the time step is adapted to maintain a prescribed local error. Schur decomposition is shown to avoid the need for computing matrix exponentials in such simulations, while still circumventing linear stiffness.

math.NA↗

arXiv2023

Hybrid Dealiased Convolutions

This paper proposes a practical and efficient solution for computing convolutions using hybrid dealiasing. It offers an alternative to explicit or implicit dealiasing and includes an optimized hyperparameter tuning algorithm that uses experience to find the optimal parameters. Machine learning algorithms and efficient heuristics are also developed to estimate optimal parameters for larger convolution problems using only small squares/rectangles.

math.NA↗

arXiv2023

Robust Exponential Runge-Kutta Embedded Pairs

Exponential integrators are explicit methods for solving ordinary differential equations that treat linear behaviour exactly. The stiff-order conditions for exponential integrators derived in a Banach space framework by Hochbruck and Ostermann are solved symbolically by expressing the Runge--Kutta weights as unknown linear combinations of phi functions. Of particular interest are embedded exponential pairs that efficiently generate both a high- and low-order estimate, allowing for dynamic adjustment of the time step. A key requirement is that the pair be robust: if the nonlinear source function has nonzero total time derivatives, the order of the low-order estimate should never exceed its design value. Robust exponential Runge--Kutta (3,2) and (4,3) embedded pairs that are well-suited to initial value problems with a dominant linearity are constructed.

physics.comp-ph↗

arXiv2011

Efficient Dealiased Convolutions without Padding

Algorithms are developed for calculating dealiased linear convolution sums without the expense of conventional zero-padding or phase-shift techniques. For one-dimensional in-place convolutions, the memory requirements are identical with the zero-padding technique, with the important distinction that the additional work memory need not be contiguous with the input data. This decoupling of data and work arrays dramatically reduces the memory and computation time required to evaluate higher-dimensional in-place convolutions. The technique also allows one to dealias the higher-order convolutions that arise from Fourier transforming cubic and higher powers. Implicitly dealiased convolutions can be built on top of state-of-the-art fast Fourier transform libraries: vectorized multidimensional implementations for the complex and centered Hermitian (pseudospectral) cases have been implemented in the open-source software FFTW++.

cs.CE↗

arXiv2010

Surface Parametrization of Nonsimply Connected Planar Bézier Regions

A technique is described for constructing three-dimensional vector graphics representations of planar regions bounded by cubic Bézier curves, such as smooth glyphs. It relies on a novel algorithm for compactly partitioning planar Bézier regions into nondegenerate Coons patches. New optimizations are also described for Bézier inside-outside tests and the computation of global bounds of directionally monotonic functions over a Bézier surface (such as its bounding box or optimal field-of-view angle). These algorithms underlie the three-dimensional illustration and typography features of the TeX-aware vector graphics language Asymptote.

cs.CG↗

arXiv2006

Links between dissipation, intermittency, and helicity in the GOY model revisited

High-resolution simulations within the GOY shell model are used to study various scaling relations for turbulence. A power-law relation between the second-order intermittency correction and the crossover from the inertial to the dissipation range is confirmed. Evidence is found for the intermediate viscous dissipation range proposed by Frisch and Vergassola. It is emphasized that insufficient dissipation-range resolution systematically drives the energy spectrum towards statistical-mechanical equipartition. In fully resolved simulations the inertial-range scaling exponents depend on both model parameters; in particular, there is no evidence that the conservation of a helicity-like quantity leads to universal exponents.

nlin.CD↗

arXiv2004

Large-scale energy spectra in surface quasi-geostrophic turbulence

The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation $$\partial_t(-Δ)^{1/2}ψ+J(ψ,(-Δ)^{1/2}ψ) =μΔψ+f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities $Ψ_1=<[(-Δ)^{1/4}ψ]^2>/2$ and $Ψ_2=<[(-Δ)^{1/2}ψ]^2>/2$ (kinetic energy), where $<\cdot>$ denotes a spatial average. The energy density $Ψ_2$ is bounded and its spectrum $Ψ_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $Ψ_2(k)$ in the low-wavenumber region can be bounded by $Ck$ where $C$ is a constant independent of $k$ but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented.

nlin.CD↗

arXiv2002

On the dual cascade in two-dimensional turbulence

We study the dual cascade scenario for two-dimensional turbulence driven by a spectrally localized forcing applied over a finite wavenumber range $[k_\min,k_\max]$ (with $k_\min > 0$) such that the respective energy and enstrophy injection rates $ε$ and $η$ satisfy $k_\min^2ε\leη\le k_\max^2ε$. The classical Kraichnan--Leith--Batchelor paradigm, based on the simultaneous conservation of energy and enstrophy and the scale-selectivity of the molecular viscosity, requires that the domain be unbounded in both directions. For two-dimensional turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the across-channel direction, a direct enstrophy cascade is not possible. In the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers, constant spectral slopes must satisfy $β>5$ and $α+β\ge8$, where $-α$ ($-β$) is the asymptotic slope of the range of wavenumbers lower (higher) than the forcing wavenumber. The influence of a large-scale dissipation on the realizability of a dual cascade is analyzed. We discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain.

nlin.CD↗

arXiv2002

An Exactly Conservative Integrator for the n-Body Problem

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the computed value of the total energy and angular momentum. Even symplectic integration schemes exactly conserve only an approximate Hamiltonian. We present an algorithm that conserves the true Hamiltonian and the total angular momentum to machine precision. It is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables. We develop the method first for the restricted circular three-body problem, then for the general two-dimensional three-body problem, and finally for the planar n-body problem. Jacobi coordinates are used to reduce the two-dimensional n-body problem to an (n-1)-body problem that incorporates the constant linear momentum and center of mass constraints. For a four-body choreography, we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators.

physics.comp-ph↗

arXiv2002

Non-white noise and a multiple-rate Markovian closure theory for turbulence

Markovian models of turbulence can be derived from the renormalized statistical closure equations of the direct-interaction approximation (DIA). Various simplifications are often introduced, including an assumption that the two-time correlation function is proportional to the renormalized infinitesimal propagator (Green's function), i.e. the decorrelation rate for fluctuations is equal to the decay rate for perturbations. While this is a rigorous result of the fluctuation--dissipation theorem for thermal equilibrium, it does not necessarily apply to all types of turbulence. Building on previous work on realizable Markovian closures, we explore a way to allow the decorrelation and decay rates to differ (which in some cases affords a more accurate treatment of effects such as non-white noise), while retaining the computational advantages of a Markovian approximation. Some Markovian approximations differ only in the initial transient phase, but the multiple-rate Markovian closure (MRMC) presented here could modify the steady-state spectra as well. Markovian models can be used directly in studying turbulence in a wide range of physical problems (including zonal flows, of recent interest in plasma physics), or they may be a useful starting point for deriving subgrid turbulence models for computer simulations.

physics.flu-dyn↗

arXiv1999

Spectral reduction: a statistical description of turbulence

A method is described for predicting statistical properties of turbulence. Collections of Fourier amplitudes are represented by nonuniformly spaced modes with enhanced coupling coefficients. The statistics of the full dynamics can be recovered from the time-averaged predictions of the reduced model. A Liouville theorem leads to inviscid equipartition solutions. Excellent agreement is obtained with two-dimensional forced-dissipative pseudospectral simulations. For the two-dimensional enstrophy cascade, logarithmic corrections to the high-order structure functions are observed.

chao-dyn↗

arXiv1995

Exactly Conservative Integrators

Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants. We illustrate the general method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator--prey model, and the Kepler problem. This method is discussed in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.

chao-dyn↗

arXiv1995

On inertial-range scaling laws

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorov's $k^{-5/3}$ scaling is derived for the energy inertial range. A related modification is found to Kraichnan's logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.

chao-dyn↗