A notion of monitored recurrence for discrete-time quantum processes was recently introduced in [15] taking the initial state as an absorbing one. We extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension.The generating function approach leads to a connection with the well-known theory of operator-valued Schur functions. This is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in [15]. The spectral decomposition of the unitary step operator driving the evolution yields a spectral measure, which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent, and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time.This notion of subspace recurrence also links the concept of expected return time to an Aharonov-Anandan phase that, in contrast to the case of state recurrence, can be non-integer. Even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning, so that the averaged expected return time is always a rational number. Moreover, state recurrence can occasionally give higher return probabilities than subspace recurrence, a fact that reveals once more the counterintuitive behavior of quantum systems.All these phenomena are illustrated with explicit examples, including as a natural application the analysis of site recurrence for coined walks.Key words and phrases. Quantum dynamical systems, quantum walks, recurrence, matrix Schur functions, degree of a function, Aharonov-Anandan phase, matrix measures and orthogonal polynomials on the unit circle.

A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton's ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counterpropagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth.In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Efficiency is achieved in the trust-region approach by parallelizing the Jacobian computation so the setup cost of computing the Dirichlet-to-Neumann operator in the variational equation is not repeated for each column. Updates of the Jacobian are also delayed until the previous Jacobian ceases to be useful. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high order Runge-Kutta or spectral deferred correction method in time, and quadruple-precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for GPU acceleration are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.

We study the limiting behavior of large-amplitude standing waves on deep water using highresolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokes's sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation patterns that occur. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t = 0.We use our method to study global paths of nontrivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our J Nonlinear Sci (2010) 20: 277-308 numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached. By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.

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

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.

Abstract. Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they capture successfully the nonlinear features of the flows, such as shocks and rarefaction waves for the isentropic Euler equations. We also show how to design higher order methods for these problems in the optimal transport setting using backward differentiation formula (BDF) multi-step methods or diagonally implicit Runge-Kutta methods.Mathematics Subject Classification. 35L65, 49J40, 76M30, 76M28.

Abstract. We develop a spectrally accurate numerical method to compute solutions of a model PDE 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 nonnormalizable 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 PDE and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations. 1. Introduction. Partial differential equations involving singular Sturm-Liouville operators with continuous spectra arise frequently in computational physics. Common approaches to solving them include domain truncation, which often regularizes the operator and makes the spectrum discrete, or projection onto finite dimensional orthogonal polynomial or finite element subspaces, which also leads to discrete spectra. Here we develop an alternative approach in which the continuous spectrum is treated analytically via a spectral transform, and the numerical challenge is in accurately representing and evaluating the integrals giving the exact solution.While the methods developed in this paper to diagonalize singular Sturm-Liouville operators are quite general, we will describe them in the context of velocity-space diffusion in one dimension,