arXiv subjects

William Barham

Publications and source records attributed to William Barham.

14 recordsLinked to original sources

arXiv2026

Shock solutions for the one-dimensional information geometric regularization of compressible flow

The information geometric regularization (IGR) is an inviscid regularization of the compressible Euler equations that alters the geometry of Lagrangian characteristics to prevent trajectories from crossing in finite time. Previous work on IGR established global strong solutions in one dimension, explored thermodynamic effects of the model, and enabled large-scale simulations of compressible flow. However, a fundamental question that remains is how this regularization alters the structure and regularity of a shock-like solution. We prove existence, uniqueness modulo translation, and regularity of transonic compressive IGR shock profiles in one spatial dimension. The analysis applies to the full thermodynamic compressible Euler--IGR model with a general equation of state, subject to mild convexity hypotheses. A traveling-wave ansatz reduces the Euler--IGR equations to a degenerate second-order scalar equation for the density profile. At the sonic crossing, the elliptic coefficient degenerates: the density profile remains continuous, but its derivative diverges. The profile is a classical solution away from this single point, while at the degeneracy it retains quantified H\"older and Sobolev regularity. We also analyze the vanishing-regularization limit, showing that the shock width scales like $\sqrt{{\alpha}}$ and that the IGR profiles converge to the entropy-admissible Euler shock.

math.AP↗

arXiv2025

Hamiltonian Information Geometric Regularization of the Compressible Euler Equations

The recently proposed information geometric regularization (IGR) was the first inviscid regularization of the multi-dimensional compressible Euler equations, which enabled the simulation of realistic compressible fluid models at an unprecedented scale. However, the thermodynamic effects of this regularization have not yet been understood in a principled manner. To achieve a proper understanding of the thermodynamic aspects of the IGR, we decompose the regularization into its conservative dynamics, framed as a Hamiltonian subsystem, and its dissipative dynamics. In so doing, we further introduce two more models to compare to IGR, the Hamiltonian regularized Euler (HRE) model, which is the first multi-dimensional, non-dispersive Hamiltonian regularization of the compressible Euler equations with energy, as well as the Hamiltonian IGR (HIGR) model, which modifies the dissipation used in the IGR model to instead utilize a metriplectic dissipative force. Despite having many attractive features, the HRE and HIGR models exhibit notable defects in numerical tests on colliding shock problems, which preclude their use as computational tools without further study of dissipative weak solutions to these models. Additionally, our analysis presents new results on the IGR model itself, including its ability to conserve acoustic waves, as well as local energy transport laws and entropy production rates. By separating the conservative and dissipative dynamics of the IGR, our hope is that subsequent of analysis of the IGR model can benefit from this natural decomposition, such as, for example, rigorous proofs of strong solutions for multi-dimensional IGR for the compressible Euler system with thermodynamics.

math-ph↗

arXiv2025

Diagnosing symplecticity in simulations of high-dimensional Hamiltonian systems

Integrals of the Liouville $1$-form, known as the first Poincar{\'e} integral invariant, provide a computable figure of merit for monitoring the conservation of symplecticity in the numerical integration of Hamiltonian systems. For smooth loop data, these integrals may be approximated with spectral convergence in the number of sample points, with rates limited by regularity. We devise a numerical integral invariant diagnostic for checking preservation of symplecticity in particle-in-cell (PIC) kinetic plasma simulation codes. As a first application of this diagnostic tool, we check the preservation of symplecticity in symplectic electrostatic particle-in-cell (PIC) methods. Surprisingly, such PIC methods fail to have symplectic time-advance maps if the charge is interpolated to the grid using linear shape functions, as is commonly done in practice. It is found that at least quadratic interpolation is needed to avoid this failure of symplecticity preservation.

physics.plasm-ph↗

arXiv2024

A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure

Thermodynamically consistent models in continuum physics, i.e. models which satisfy the first and second laws of thermodynamics, may be expressed using the metriplectic formalism. In this work, we leverage the structures underlying this modeling formalism to preserve thermodynamic consistency in discretizations of a fluid model. The procedure relies (1) on ensuring that the spatial semi-discretization retains certain symmetries and degeneracies of the Poisson and metriplectic 4-brackets, and (2) on the use of an appropriate energy conserving time-stepping method. The minimally simple yet nontrivial example of a one-dimensional thermal-fluid model is treated. It is found that preservation of the requisite symmetries and degeneracies of the 4-bracket is relatively simple to ensure in Galerkin spatial discretizations, suggesting a path forward for thermodynamically consistent discretizations of more complex fluid models using more specialized Galerkin methods.

physics.comp-ph↗

arXiv2024

A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media

A simple Hamiltonian modeling framework for general models in nonlinear optics is given. This framework is specialized to describe the Hamiltonian structure of electromagnetic phenomena in cubicly nonlinear optical media. The model has a simple Poisson bracket structure with the Hamiltonian encoding all of the nonlinear coupling of the fields. The field-independence of the Poisson bracket facilitates a straightforward Hamiltonian structure-preserving discretization using finite element exterior calculus. The generality and relative simplicity of this Hamiltonian framework makes it amenable for simulating a broad class of time-domain nonlinear optical problems. The main contribution of this work is a finite element discretization of Maxwell's equations in cubicly nonlinear media which is energy-stable and exactly conserves Gauss's laws. Moreover, this approach may be readily adapted to consider more general nonlinear media in subsequent work.

physics.comp-ph↗

arXiv2024

A mimetic discretization of Westervelt's equation

A broad class of nonlinear acoustic wave models possess a Hamiltonian structure in their dissipation-free limit and a gradient flow structure for their dissipative dynamics. This structure may be exploited to design numerical methods which preserve the Hamiltonian structure in the dissipation-free limit, and which achieve the correct dissipation rate in the spatially-discrete dissipative dynamics. Moreover, by using spatial discretizations which preserve the de Rham cohomology, the non-evolving involution constraint for the vorticity may be exactly satisfied for all of time. Numerical examples are given using a mimetic finite difference spatial discretization.

math.NA↗

arXiv2024

On finite-dimensional smoothed-particle Hamiltonian reductions of the Vlasov equation

The inclusion of spatial smoothing in finite-dimensional particle-based Hamiltonian reductions of the Vlasov equation are considered. In the context of the Vlasov-Poisson equation (and other mean-field Lie-Poisson systems), smoothing amounts to a convolutive regularization of the Hamiltonian. This regularization may be interpreted as a change of the inner product structure used to identify the dual space in the Lie-Poisson Hamiltonian formulation. In particular, the shape function used for spatial smoothing may be identified as the kernel function of a reproducing kernel Hilbert space whose inner product is used to define the Lie-Poisson Hamiltonian structure. It is likewise possible to introduce smoothing in the Vlasov-Maxwell system, but in this case the Poisson bracket must be modified rather than the Hamiltonian. The smoothing applied to the Vlasov-Maxwell system is incorporated by inserting smoothing in the map from canonical to kinematic coordinates. In the filtered system, the Lorentz force law and the current, the two terms coupling the Vlasov equation with Maxwell's equations, are spatially smoothed.

math-ph↗

arXiv2023

A self-consistent Hamiltonian model of the ponderomotive force and its structure preserving discretization

In the presence of an inhomogeneous oscillatory electric field, charged particles experience a net force, averaged over the oscillatory timescale, known as the ponderomotive force. We derive a one-dimensional Hamiltonian model which self-consistently couples the electromagnetic field to a plasma which experiences the ponderomotive force. We derive a family of structure preserving discretizations of the model of varying order in space and time using conforming and broken finite element exterior calculus spectral element methods. In all variants of our discretization framework, the method is found to conserve the Casimir invariants of the continuous model to machine precision and the energy to the order of the splitting method used.

physics.comp-ph↗

arXiv2023

A Hamiltonian and geometric formulation of general Vlasov-Maxwell-type models

Three geometric formulations of the Hamiltonian structure of the macroscopic Maxwell equations are given: one in terms of the double de Rham complex, one in terms of L2 duality, and one utilizing an abstract notion of duality. The final of these is used to express the geometric and Hamiltonian structure of kinetic theories in general media. The Poisson bracket so stated is explicitly metric free. Finally, as a special case, the Lorentz covariance of such kinetic theories is investigated. We obtain a Lorentz covariant kinetic theory coupled to nonlinear electrodynamics such as Born-Infeld or Euler-Heisenberg electrodynamics.

math-ph↗

arXiv2022

Dual PIC: a structure preserving method for discretizing Lie-Poisson brackets

We consider a general discretization strategy for Hamiltonian field theories generated by Lie-Poisson brackets which we call dual PIC (DPIC). This method involves prescribing two different discrete representations of the dynamical variable which are constrained as a Casimir invariant of the flow to coincide with one another via an L2 projection throughout the entire simulation. This allows one to leverage the relative advantages of each discrete representation. We begin by describing DPIC as applied to a general Lie-Poisson system and then provide illustrative examples: the discretization of the two-dimensional vorticity equations and the Vlasov-Poisson equation.

physics.comp-ph↗

arXiv2021

A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus

A Hamiltonian field theory for the macroscopic Maxwell equations with fully general polarization and magnetization is stated in the language of differential forms. The precise procedure for translating the vector calculus formulation into differential forms is discussed in detail. We choose to distinguish between straight and twisted differential forms so that all integrals be taken over densities (i.e. twisted top forms). This ensures that the duality pairings, which are stated as integrals over densities, are orientation independent. The relationship between functional differentiation with respect to vector fields and with respect to differential forms is established using the chain rule. The theory is developed such that the Poisson bracket is metric and orientation independent with all metric dependence contained in the Hamiltonian. As is typically seen in the exterior calculus formulation of Maxwell's equations, the Hodge star operator plays a key role in modeling the constitutive relations. As a demonstration of the kind of constitutive models this theory accommodates, the paper concludes with several examples.

math-ph↗

arXiv2021

A mimetic discretization of the macroscopic Maxwell equations in Hamiltonian form

A mimetic spectral element discretization, utilizing a novel Galerkin projection Hodge star operator, of the macroscopic Maxwell equations in Hamiltonian form is presented. The idea of splitting purely topological and metric dependent quantities is natural in the Hamiltonian modeling framework as the Poisson bracket is metric free with the Hamiltonian containing all metric information. This idea may be incorporated into the mimetic spectral element method by directly discretizing the Poincar\'e duality structure. This "split exterior calculus mimetic spectral element method" yields spatially discretized Maxwell's equations which are Hamiltonian and exactly and strongly conserve Gauss's laws. Moreover, the new discrete Hodge star operator is itself of interest as a partition of the purely topological and metric dependent portions of the Hodge star operator. As a simple test case, the numerical results of applying this method to a one-dimensional version of Maxwell's equations are given.

physics.comp-ph↗

arXiv2018

Exact instantaneous optimals in the non-geostrophic Eady problem and the detrimental effects of discretization

We derive exact analytical expressions for flow configurations that optimize the instantaneous growth rate of energy in the linear Eady problem, along with the associated growth rates. These optimal perturbations are relevant linear stability analysis, but, more importantly, they are relevant for understanding the energetics of fully nonlinear baroclinic turbulence. The optimal perturbations and their growth rates are independent of the Richardson number. The growth rates of the optimal perturbations grow linearly as the horizontal wavelength of the perturbation decreases. Perturbation energy growth at large scales is driven by extraction of potential energy from the mean flow, while at small scales it is driven by extraction of kinetic energy from the mean shear. We also analyze the effect of spatial discretization on the optimal perturbations and their growth rates. A second order energy-conserving discretization on the Arakawa B grid generally has too-weak growth rates at small scales and is less accurate than two second order discretizations on the Arakawa C grid. The two C grid discretizations, one that conserves energy and another that conserves both energy and enstrophy, yield very similar optimal perturbation growth rates that are significantly more accurate than the B grid discretization at small scales.

physics.ao-ph↗

arXiv2017

An eddifying Stommel model: Fast eddy effects in a two-box ocean

A system of stochastic differential equations is formulated describing the heat and salt content of a two-box ocean. Variability in the heat and salt content and in the thermohaline circulation between the boxes is driven by fast Gaussian atmospheric forcing and by ocean-intrinsic, eddy-driven variability. The eddy forcing of the slow dynamics takes the form of a colored, non-Gaussian noise. The qualitative effects of this non-Gaussianity are investigated by comparing to two approximate models: one that includes only the mean eddy effects (the `averaged model'), and one that includes an additional Gaussian white-noise approximation of the eddy effects (the `Gaussian model'). Both of these approximate models are derived using the methods of fast averaging and homogenization. In the parameter regime where the dynamics has a single stable equilibrium the averaged model has too little variability. The Gaussian model has accurate second-order statistics, but incorrect skew and rare-event probabilities. In the parameter regime where the dynamics has two stable equilibria the eddy noise is much smaller than the atmospheric noise. The averaged, Gaussian, and non-Gaussian models all have similar stationary distributions, but the jump rates between equilibria are too small for the averaged and Gaussian models.

math.DS↗