One-space Dimension Research Articles

TRAC method in dissipative media—a first analysis in frequency domain and homogeneous media

We propose to explore the time-reversed absorbing condition (TRAC) method in the case of dissipative homogeneous media. In previous work, the TRAC method was derived from the time-reversibility of the (undamped) wave equation and proved to be efficient in both the time-domain and the frequency-domain. Namely, two main utilisations of the TRAC method have been probed: (a) redatuming, i.e. moving virtually the measurements by reconstructing the wavefield and (b) tracking down the location of a possible inclusion inside the domain. In this paper, we focus on the redatuming application and investigate the feasibility of the TRAC method in the case of dissipation. In particular, we will see that performing the classical TRAC method, i.e. ignoring the dissipation, may give satisfactory results, even for larger values of dissipation. An analysis is provided in the frequency-domain and one-space dimension and shows satisfactory updated versions of the TRAC method. Moreover, a systematic error study in two-space dimension is illustrated via numerical examples.

Structure-preserving finite difference scheme for 1D thermoviscoelastoplastic equations under uniformly distributed temperature

In this article, we investigate a structure-preserving finite difference scheme to the thermo-visco-elasto-plastic system in a one-space dimension under a perfect elasto-plasticity represented by the stop operator. We introduce the structure-preserving finite difference scheme for the system which satisfies the energy conservation law, momentum conservation law and the increasing law of entropy in the discrete sense. The main purpose of this article is to give mathematical treatment such as the existence of solution and error estimate for the scheme, under the assumption of uniformly distributed temperature with respect to space. In addition, several numerical simulations are also demonstrated as examples.

BV solutions for a hydrodynamic model of flocking-type with all-to-all interaction kernel

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions with concentration to the Cauchy problem for any [Formula: see text] initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions with concentration admit time-asymptotic flocking.

Parametric interpolation framework for scalar conservation laws

In this paper we present a novel framework for obtaining high-order numerical methods for scalar conservation laws in one-space dimension for both the homogeneous and non-homogeneous cases (or balance laws). The numerical schemes for these two settings are somewhat different in the presence of shocks, however at their core they both rely heavily on the solution curve being represented parametrically. By utilizing high-order parametric interpolation techniques we succeed to obtain fifth order accuracy (in space) everywhere in the computation domain, including the shock location itself. In the presence of source terms a slight modification is required, yet the spatial order is maintained but with an additional temporal error appearing. We provide a detailed discussion of a sample scheme for non-homogeneous problems which obtains fifth order in space and fourth order in time even in the presence of shocks.

Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

Dynamic phenomena in social and biological sciences can often be modeled by reaction-diffusion equations. When addressing the control from a mathematical viewpoint, one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [0,1])). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady states of spatially homogeneous monostable and bistable semilinear heat equations, with constraints in the state, and using boundary controls. We prove that controlling the system to a constant steady state may become impossible when the diffusivity is too small due to the existence of barrier functions. We build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves, and some connectivity of the set of steady states to ensure controllability whenever it is possible. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain.

Nonuniqueness for the Kinetic Fokker–Planck Equation with Inelastic Boundary Conditions

We describe the structure of solutions of the kinetic Fokker–Planck equations in domains with boundaries near the singular set in one-space dimension. We study in particular the behaviour of the solutions of this equation for inelastic boundary conditions which are characterized by means of a coefficient r describing the amount of energy lost in the collisions of the particles with the boundaries of the domain. A peculiar feature of this problem is the onset of a critical exponent rc which follows from the analysis of McKean (J Math Kyoto Univ 2:227–235 1963) of the properties of the stochastic process associated to the Fokker–Planck equation under consideration. In this paper, we prove rigorously that the solutions of the considered problem are nonunique if r < rc and unique if $${r_{c} < r \leqq 1.}$$ In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the Fokker–Planck equation. In the proof of the results of this paper we use several asymptotic formulas and computations in the companion paper (Hwang in Q Appl Math 2018).

Crime modeling with truncated Lévy flights for residential burglary models

Statistical agent-based models for crime have shown that repeat victimization can lead to predictable crime hotspots (see e.g. M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267.), then a recent study in one-space dimension (S. Chaturapruek, J. Breslau, D. Yazdi, T. Kolokolnikov and S. G. McCalla, Crime modeling with Lévy flights, SIAM J. Appl. Math. 73 (2013) 1703–1720.) shows that the hotspot dynamics changes when movement patterns of the criminals involve long-tailed Lévy distributions for the jump length as opposed to classical random walks. In reality, criminals move in confined areas with a maximum jump length. In this paper, we develop a mean-field continuum model with truncated Lévy flights (TLFs) for residential burglary in one-space dimension. The continuum model yields local Laplace diffusion, rather than fractional diffusion. We present an asymptotic theory to derive the continuum equations and show excellent agreement between the continuum model and the agent-based simulations. This suggests that local diffusion models are universal for continuum limits of this problem, the important quantity being the diffusion coefficient. Law enforcement agents are also incorporated into the model, and the relative effectiveness of their deployment strategies are compared quantitatively.

Energy-transport systems for optical lattices: Derivation, analysis, simulation

Energy-transport equations for the transport of fermions in optical lattices are formally derived from a Boltzmann transport equation with a periodic lattice potential in the diffusive limit. The limit model possesses a formal gradient-flow structure like in the case of the energy-transport equations for semiconductors. At the zeroth-order high-temperature limit, the energy-transport equations reduce to the whole-space logarithmic diffusion equation which has some unphysical properties. Therefore, the first-order expansion is derived and analyzed. The existence of weak solutions to the time-discretized system for the particle and energy densities with periodic boundary conditions is proved. The difficulties are the nonstandard degeneracy and the quadratic gradient term. The main tool of the proof is a result on the strong convergence of the gradients of the approximate solutions. Numerical simulations in one-space dimension show that the particle density converges to a constant steady state if the initial energy density is sufficiently large, otherwise the particle density converges to a nonconstant steady state.

Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data

Let u be a solution to a quasi-linear Klein-Gordon equation in one-space dimension, $\Box u + u = P (u, ∂_t u, ∂_x u; ∂_t ∂_x u, ∂^2_x u)$ , where P is a homogeneous polynomial of degree three, and with smooth Cauchy data of size $e → 0$. It is known that, under a suitable condition on the nonlinearity, the solution is global-in-time for compactly supported Cauchy data. We prove in this paper that the result holds even when data are not compactly supported but just decaying as $\langle x \rangle^ {−1}$ at infinity, combining the method of Klainerman vector fields with a semiclassical normal forms method introduced by Delort. Moreover, we get a one term asymptotic expansion for u when $t → +∞$.

Adjoint-Based, Superconvergent Galerkin Approximations of Linear Functionals

We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where $$J(\cdot )$$ is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if $$u_h$$ is the approximation given by the continuous Galerkin method with piecewise polynomials of degree $$k>0$$ , then, as a direct consequence of its property of Galerkin orthogonality, the functional $$J(u_h)$$ converges to J(u) with a rate of order $$h^{2k}$$ . We show how to define approximations to J(u), with a computational effort about twice of that of computing $$J(u_h)$$ , which converge with a rate of order $$h^{4k}$$ . The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for $$k=1,2,3$$ in one-space dimension and for $$k=1,2$$ in two-space dimensions, which show that $$J(u_h)$$ converges to J(u) with order $$h^{2k+1}$$ and that the new approximations converges with order $$h^{4k}$$ . The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of $$J(u_h)$$ .

Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory

This paper is concerned with the long-time dynamics of the nonlinear wave equation in one-space dimension, $$ u_{tt} - \delta^2 u_{xx} +V'(u) =0 \qquad x\in [0,1] $$ where $\delta>0$ is a parameter and $V(u)$ is a potential bounded from below and growing at least like $u^2$ as $|u|\to\infty$. Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure and when the potential is double-welled, for example when $V(u) = \tfrac14(1-u^2)^2$, there is a regime such that two small disjoint sets in the system's phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here we quantify this phenomenon by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter $\delta$ in small enough. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger $\delta$, however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency, indicating that successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.

Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient

We are concerned with fully-discrete schemes for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension. More precisely, we show the convergence of approximate solutions, generated by the scheme corresponding to vanishing diffusive-dispersive scalar conservation laws with a discontinuous coefficient, to the corresponding scalar conservation law with discontinuous coefficient. Finally, the convergence is illustrated by several examples. In particular, it is delineated that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kružkov−Oleĭnik entropy solutions, but contain nonclassical undercompressive shock waves.

A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws

For a nonlinear scalar conservation law in one-space dimension, we develop a locally conservative semi-Lagrangian finite difference scheme based on weighted essentially non-oscillatory reconstructions (SL-WENO). This scheme has the advantages of both WENO and semi-Lagrangian schemes. It is a locally mass conservative finite difference scheme, it is formally high-order accurate in space, it has small time truncation error, and it is essentially non-oscillatory. The scheme is nearly free of a CFL time step stability restriction for linear problems, and it has a relaxed CFL condition for nonlinear problems. The scheme can be considered as an extension of the SL-WENO scheme of Qiu and Shu (2011) [2] developed for linear problems. The new scheme is based on a standard sliding average formulation with the flux function defined using WENO reconstructions of (semi-Lagrangian) characteristic tracings of grid points. To handle nonlinear problems, we use an approximate, locally frozen trace velocity and a flux correction step. A special two-stage WENO reconstruction procedure is developed that is biased to the upstream direction. A Strang splitting algorithm is used for higher-dimensional problems. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy. Included are applications to the Vlasov–Poisson and guiding-center models of plasma flow.

Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension

We design and analyze the first hybridizable discontinuous Galerkin methods for stationary, third-order linear equations in one-space dimension. The methods are defined as discrete versions of characterizations of the exact solution in terms of local problems and transmission conditions. They provide approximations to the exact solution u u and its derivatives q := u ′ q:=u’ and p := u p:=u which are piecewise polynomials of degree k u k_u , k q k_q and k p k_p , respectively. We consider the methods for which the difference between these polynomial degrees is at most two. We prove that all these methods have superconvergence properties which allows us to prove that their numerical traces converge at the nodes of the partition with order at least 2 k + 1 2\,k+1 , where k k is the minimum of k u , k q k_u,k_q , and k p k_p . This allows us to use an element-by-element post-processing to obtain new approximations for u , q u, q and p p converging with order at least 2 k + 1 2k+1 uniformly. Numerical results validating our error estimates are displayed.

Trusted frequency region of convergence for the enclosure method in thermal imaging

Abstract This study deals with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponentially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using the theorems provided in this paper and is numerically implemented for various cases.

Theory of Phase Separation and Polarization for Pure Ionic Liquids.

Room temperature ionic liquids are attractive to numerous applications and particularly, to renewable energy devices. As solvent free electrolytes, they demonstrate a paramount connection between the material morphology and Coulombic interactions: the electrode/RTIL interface is believed to be a product of both polarization and spatiotemporal bulk properties. Yet, theoretical studies have dealt almost exclusively with independent models of morphology and electrokinetics. Introduction of a distinct Cahn-Hilliard-Poisson type mean-field framework for pure molten salts (i.e., in the absence of any neutral component), allows a systematic coupling between morphological evolution and the electrokinetic phenomena, such as transient currents. Specifically, linear analysis shows that spatially periodic patterns form via a finite wavenumber instability and numerical simulations demonstrate that while labyrinthine type patterns develop in the bulk, lamellar structures are favored near charged surfaces. The results demonstrate a qualitative phenomenology that is observed empirically and thus, provide a physically consistent methodology to incorporate phase separation properties into an electrochemical framework.

Chaotic Oscillations of Second Order Linear Hyperbolic Equations with Nonlinear Boundary Conditions: A Factorizable but Noncommutative Case

If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a certain regime [Chen et al., 2014]. However, if the commutativity of the two first order operators fails to hold, then the treatment in [Chen et al., 2014] no longer works and significant new challenges arise in determining nonlinear boundary conditions that engenders chaos. In this paper, we show that by incorporating a linear memory effect, a nonlinear van der Pol boundary condition can cause chaotic oscillations when the parameter enters a certain regime. Numerical simulations illustrating chaotic oscillations are also presented.

REMARKS ON NONLINEAR DIRAC EQUATIONS IN ONE SPACE DIMENSION

Abstract. This paper reviews recent mathematical progresses made onthe study of the initial-value problem for nonlinear Dirac equations in onespace dimension. We also prove the global existence of solutions to somenonlinear Dirac equations and propose a model problem (3.6). 1. IntroductionWe are interested in the following initial value problem for the one dimen-sional nonlinear Dirac equationsi(∂ t U 1 +∂ x U 1 ) +mU 2 = ∂ U¯ 1 W(U 1 , U 2 ),i(∂ t U 2 −∂ x U 2 ) +mU 1 = ∂ U¯ 2 W(U 1 , U 2 ),U j (x, 0) = u j (x),(1.1)where U j : R 1+1 → C for j = 1, 2 and m(≥ 0) is a mass. U¯ is a complexconjugate of U. The potential W satisfies the following properties:1. Symmetry: W(U 1 , U 2 ) = W(U 2 , U 1 ).2. Gauge invariance: W(e iθ U 1 , e iθ U 2 ) = W(U 1 , U 2 ) for any θ ∈ R.3. Polynomial in (U 1 , U 2 ) and (U¯ 1 , U¯ 2 ).It is known [11] that fourth order homogeneous polynomial satisfying theabove properties takes the formW = a 1 |U 1 | 2 |U 2 | 2 +a 2 (U¯ 1 U 2 +U¯ 2 U 1 ) 2 +a 3 (|U 1 | 4

Hamiltonian, path integral and BRST formulations of large $$N$$ N scalar QCD $$_{2}$$ 2 on the light-front and spontaneous symmetry breaking

Recently Grinstein, Jora, and Polosa have studied a theory of large-$N$ scalar quantum chromodynamics in one-space one-time dimension. This theory admits a Bethe-Salpeter equation describing the discrete spectrum of quark-antiquark bound states. They consider gauge fields in the adjoint representation of $SU(N)$ and scalar fields in the fundamental representation. The theory is asymptotically free and linearly confining. The theory could possibly provide a good field theoretic framework for the description of a large class of diquark-antidiquark (tetra-quark) states. Recently we have studied the light-front quantization of this theory without a Higgs potential. In the present work, we study the light-front Hamiltonian, path integral and BRST formulations of the theory in the presence of a Higgs potential. The light-front theory is seen to be gauge-invariant, possessing a set of first-class constraints. The explicit occurrence of spontaneous symmetry breaking in the theory is shown in unitary gauge as well as in the light-front 't Hooft gauge.

Bounded weak solutions to a matrix drift–diffusion model for spin-coherent electron transport in semiconductors

The global-in-time existence and uniqueness of bounded weak solutions to a spinorial matrix drift–diffusion model for semiconductors is proved. Developing the electron density matrix in the Pauli basis, the coefficients (charge density and spin-vector density) satisfy a parabolic 4 × 4 cross-diffusion system. The key idea of the existence proof is to work with different variables: the spin-up and spin-down densities as well as the parallel and perpendicular components of the spin-vector density with respect to the precession vector. In these variables, the diffusion matrix becomes diagonal. The proofs of the L∞ estimates are based on Stampacchia truncation as well as Moser- and Alikakos-type iteration arguments. The monotonicity of the entropy (or free energy) is also proved. Numerical experiments in one-space dimension using a finite-volume discretization indicate that the entropy decays exponentially fast to the equilibrium state.