arXiv subjects

Jon Wilkening

Publications and source records attributed to Jon Wilkening.

At least 19 recordsLinked to original sources

arXiv2026

Local well-posedness of spatially quasiperiodic gravity water waves in two dimensions

We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a conformal mapping formulation of the equations of motion. This allows us to compute the Dirichlet-Neumann operator via the Hilbert transform, which has a simple form in the spatially quasiperiodic setting. We use a Littlewood-Paley decomposition adapted to the quasiperiodic setting and establish multiplicative and commutator estimates in this framework. The key step of the proof is the derivation of quasilinear energy estimates for the linearized water wave equations with quasiperiodic initial data.

math.AP↗

arXiv2025

The semi-analytic theory and computation of finite-depth standing water waves

We propose a Stokes expansion ansatz for finite-depth standing water waves in two dimensions and devise a recursive algorithm to compute the expansion coefficients. We implement the algorithm on a supercomputer using arbitrary-precision arithmetic. The Stokes expansion introduces hyperbolic terms that require exponentiation of power series, which we handle efficiently using Bell polynomials. Although exact resonances occur at a countable dense set of fluid depths, we prove that for almost every depth, the divisors that arise in the recurrence are bounded away from zero by a slowly decaying function of the wave number. A direct connection between small divisors and imperfect bifurcations is observed. They are found to activate secondary standing waves that oscillate non-uniformly in space and time on top of the primary wave, with different amplitudes and phases on each bifurcation branch. We compute new families of standing waves using a shooting method and find that Padé approximants of the Stokes expansion continue to converge to the shooting method solutions at large amplitudes as new small divisors enter the recurrence. Closely spaced poles and zeros of the Padé approximants are observed, which suggests that the bifurcation branches are separated by branch cuts.

physics.flu-dyn↗

arXiv2024

Analyticity and Stable Computation of Dirichlet-Neumann Operators for Laplace's Equation under Quasiperiodic Boundary Conditions in Two and Three Dimensions

Dirichlet-Neumann Operators (DNOs) are important to the formulation, analysis, and simulation of many crucial models found in engineering and the sciences. For instance, these operators permit moving-boundary problems, such as the classical water wave problem (free-surface ideal fluid flow under the influence of gravity and capillarity), to be restated in terms of interfacial quantities, which not only eliminates the boundary tracking problem, but also reduces the problem dimension. While these DNOs have been the object of much recent study regarding their numerical simulation and rigorous analysis, they have yet to be examined in the setting of laterally quasiperiodic boundary conditions. The purpose of this contribution is to begin this investigation with a particular eye towards the problem of more realistically simulating two and three dimensional surface water waves. Here we not only carefully define the DNO with respect to these boundary conditions for Laplace's equation, but we also show the rigorous analyticity of these operators with respect to sufficiently smooth boundary perturbations. These theoretical developments suggest a novel algorithm for the stable and high-order simulation of the DNO, which we implement and extensively test.

math.NA↗

arXiv2023

Spatially quasi-periodic water waves of finite depth

We present a numerical study of spatially quasi-periodic gravity-capillary waves of finite depth in both the initial value problem and traveling wave settings. We adopt a quasi-periodic conformal mapping formulation of the Euler equations, where one-dimensional quasi-periodic functions are represented by periodic functions on a higher-dimensional torus. We compute the time evolution of free surface waves in the presence of a background flow and a quasi-periodic bottom boundary and observe the formation of quasi-periodic patterns on the free surface. Two types of quasi-periodic traveling waves are computed: small-amplitude waves bifurcating from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude periodic traveling waves. We derive weakly nonlinear approximations of the first type and investigate the associated small-divisor problem. We find that waves of the second type exhibit striking nonlinear behavior, e.g., the peaks and troughs are shifted non-periodically from the corresponding periodic waves due to the activation of quasi-periodic modes.

physics.flu-dyn↗

arXiv2023

Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent root-bracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough.

physics.flu-dyn↗

arXiv2022

Numerical Algorithms for Water Waves with Background Flow over Obstacles and Topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

physics.flu-dyn↗

arXiv2021

Spatially quasi-periodic water waves of infinite depth

We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge-Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

physics.flu-dyn↗

arXiv2020

Quasi-periodic traveling gravity-capillary waves

We present a numerical study of spatially quasi-periodic traveling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wave numbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the traveling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least squares problem that we solve using a variant of the Levenberg-Marquardt method. We investigate various properties of quasi-periodic traveling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating, and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.

physics.flu-dyn↗

arXiv2020

Harmonic Stability of Standing Water Waves

A numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory. A Fourier truncation of the monodromy operator is computed by solving the linearized Euler equations about the standing wave with initial conditions ranging over all Fourier modes up to a given wave number. The eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions, which we introduce as a method of retaining only accurately computed Floquet multipliers. The mean wave number matches up with analytical results for the zero-amplitude standing wave and is helpful in identifying which Floquet multipliers collide and leave the unit circle to form unstable eigenmodes or rejoin the unit circle to regain stability. For standing waves in deep water, most waves with crest acceleration below $A_c=0.889$ are found to be linearly stable to harmonic perturbations; however, we find several bubbles of instability at lower values of $A_c$ that have not been reported previously in the literature. We also study the stability of several new or recently discovered time-periodic gravity-capillary or gravity waves in deep or shallow water, finding several examples of large-amplitude waves that are stable to harmonic perturbations and others that are not. A new method of matching the Floquet multipliers of two nearby standing waves by solving a linear assignment problem is also proposed to track individual eigenvalues via homotopy from the zero-amplitude state to large-amplitude standing waves.

math.DS↗

arXiv2019

Computing the Dirichlet-Neumann Operator on a Cylinder

The computation of the Dirichlet-Neumann operator for the Laplace equation is the primary challenge for the numerical simulation of the ideal fluid equations. The techniques used commonly for 2D fluids, such as conformal mapping and boundary integral methods, fail to generalize suitably to 3D. In this study, we address this problem by developing a Transformed Field Expansion method for computing the Dirichlet-Neumann operator in a cylindrical geometry with a variable upper boundary. This technique reduces the problem to a sequence of Poisson equations on a flat geometry. We design a fast and accurate solver for these sub-problems, a key ingredient being the use of Zernike polynomials for the circular cross-section instead of the traditional Bessel functions. This lends spectral accuracy to the method as well as allowing significant computational speed-up. We rigorously analyze the algorithm and prove its applicability to a wide class of problems before demonstrating its effectiveness numerically.

math.NA↗

arXiv2018

Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation

The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.

math.NA↗

arXiv2018

Rigorous Asymptotic Models of Water Waves

We develop a rigorous asymptotic derivation for two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting epsilon denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in epsilon to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water waves system is obtained as an infinite sum of solutions to linear problems at each epsilon^k level, and truncation of this series leads to our two asymptotic models, that we call the quadratic and cubic h-models. Using the growth rate of the Catalan numbers (from number theory), we prove well-posedness of the h-models in spaces of analytic functions, and prove error bounds for solutions of the h-models compared against solutions of the water waves system. We also show that the Craig-Sulem models of water waves can be obtained from our asymptotic procedure and that their WW2 model is well-posed in our functional framework. We then develop a novel numerical algorithm to solve the quadratic and cubic h-models as well as the full water waves system. For three very different examples, we show that the agreement between the model equations and the water waves solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves.

math.AP↗

arXiv2016

Eigenfunctions and the Dirichlet problem for the Classical Kimura Diffusion Operator

We study the classical Kimura diffusion operator defined on the n-simplex, $$L^{Kim}=\sum_{1\leq i,j\leq n+1}x_ix_j\partial_{x_i}\partial_{x_j}$$ We give novel constructions for the basis of eigenpolynomials, and the solution to the inhomogeneous Dirichlet problem, which are well adapted to numerical applications. Our solution of the Dirichlet problem is quite explicit and provides a precise description of the singularities that arise along the boundary.

math.AP↗

arXiv2016

A Fully Discrete Adjoint Method for Optimization of Flow Problems on Deforming Domains with Time-Periodicity Constraints

A variety of shooting methods for computing fully discrete time-periodic solutions of partial differential equations, including Newton-Krylov and optimization-based methods, are discussed and used to determine the periodic, compressible, viscous flow around a 2D flapping airfoil. The Newton-Krylov method uses matrix-free GMRES to solve the linear systems of equations that arise in the nonlinear iterations, with matrix-vector products computed via the linearized sensitivity evolution equations. The adjoint method is used to compute gradients for the gradient-based optimization shooting methods. The Newton-Krylov method is shown to exhibit superior convergence to the optimal solution for these fluid problems, and fully leverages quality starting data. The central contribution of this work is the derivation of the adjoint equations and the corresponding adjoint method for fully discrete, time-periodically constrained partial differential equations. These adjoint equations constitute a linear, two-point boundary value problem that is provably solvable. The periodic adjoint method is used to compute gradients of quantities of interest along the manifold of time-periodic solutions of the discrete partial differential equation, which is verified against a second-order finite difference approximation. These gradients are then used in a gradient-based optimization framework to determine the energetically optimal flapping motion of a 2D airfoil in compressible, viscous flow over a single cycle, such that the time-averaged thrust is identically zero. In less than 20 optimization iterations, the flapping energy was reduced nearly an order of magnitude and the thrust constraint satisfied to 5 digits of accuracy.

math.OC↗

arXiv2016

The instability of Wilton ripples

Wilton ripples are a type of periodic traveling wave solution of the full water wave problem incorporating the effects of surface tension. They are characterized by a resonance phenomenon that alters the order at which the resonant harmonic mode enters in a perturbation expansion. We compute such solutions using non-perturbative numerical methods and investigate their stability by examining the spectrum of the water wave problem linearized about the resonant traveling wave. Instabilities are observed that differ from any previously found in the context of the water wave problem.

physics.flu-dyn↗

arXiv2016

Optimizing intermittent water supply in urban pipe distribution networks

In many urban areas of the developing world, piped water is supplied only intermittently, as valves direct water to different parts of the water distribution system at different times. The flow is transient, and may transition between free-surface and pressurized, resulting in complex dynamical features with important consequences for water suppliers and users. Here, we develop a computational model of transition, transient pipe flow in a network, accounting for a wide variety of realistic boundary conditions. We validate the model against several published data sets, and demonstrate its use on a real pipe network. The model is extended to consider several optimization problems motivated by realistic scenarios. We demonstrate how to infer water flow in a small pipe network from a single pressure sensor, and show how to control water inflow to minimize damaging pressure gradients.

physics.flu-dyn↗

arXiv2015

Parameter estimation by implicit sampling

Implicit sampling is a weighted sampling method that is used in data assimilation, where one sequentially updates estimates of the state of a stochastic model based on a stream of noisy or incomplete data. Here we describe how to use implicit sampling in parameter estimation problems, where the goal is to find parameters of a numerical model, e.g.~a partial differential equation (PDE), such that the output of the numerical model is compatible with (noisy) data. We use the Bayesian approach to parameter estimation, in which a posterior probability density describes the probability of the parameter conditioned on data and compute an empirical estimate of this posterior with implicit sampling. Our approach generates independent samples, so that some of the practical difficulties one encounters with Markov Chain Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples, are avoided. We describe a new implementation of implicit sampling for parameter estimation problems that makes use of multiple grids (coarse to fine) and BFGS optimization coupled to adjoint equations for the required gradient calculations. The implementation is "dimension independent", in the sense that a well-defined finite dimensional subspace is sampled as the mesh used for discretization of the PDE is refined. We illustrate the algorithm with an example where we estimate a diffusion coefficient in an elliptic equation from sparse and noisy pressure measurements. In the example, dimension\slash mesh-independence is achieved via Karhunen-Loève expansions.

math.NA↗

arXiv2015

A Spectral Transform Method for Singular Sturm-Liouville Problems with Applications to Energy Diffusion in Plasma Physics

We develop a spectrally accurate numerical method to compute solutions of a model partial differential equation used in plasma physics to describe diffusion in velocity space due to Fokker-Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and non-normalizable eigenfunctions via the spectral transform associated with a singular Sturm-Liouville operator. We present a new algorithm for computing the spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh-Weyl $m$-function from the complex upper half-plane to the real axis. The eigenfunctions and density function are rescaled and a new formula for the limiting value of the $m$-function is derived to avoid amplification of roundoff errors when the solution is reconstructed. The complexity of the algorithm is also analyzed, showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired accuracy. A WKB analysis is used to prove that the spectral density function is real analytic. Using this new algorithm, we highlight key properties of the partial differential equation and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.

math.CA↗