Finite element or Galerkin type semidiscrete schemes

A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented

Solute transport within porous biofilms: diffusion or dispersion?

Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behaviour by controllling nutrient supply, evacuation of waste products and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilmscale. We show that solute transport may be described via two coupled partial differential equations for the averaged concentrations, or telegrapher’s equations. These models are particularly relevant for chemical species, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterised by a second-order tensor whose components depend on: (1) the topology of the channels’ network; (2) the solute’s diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion-dominated, this analysis shows that dispersion effects may significantly contribute to transport

High-frequency homogenization of zero frequency stop band photonic and phononic crystals

We present an accurate methodology for representing the physics of waves, for periodic structures, through effective properties for a replacement bulk medium: This is valid even for media with zero frequency stop-bands and where high frequency phenomena dominate. Since the work of Lord Rayleigh in 1892, low frequency (or quasi-static) behaviour has been neatly encapsulated in effective anisotropic media. However such classical homogenization theories break down in the high-frequency or stop band regime. Higher frequency phenomena are of significant importance in photonics (transverse magnetic waves propagating in infinite conducting parallel fibers), phononics (anti-plane shear waves propagating in isotropic elastic materials with inclusions), and platonics (flexural waves propagating in thin-elastic plates with holes). Fortunately, the recently proposed high-frequency homogenization (HFH) theory is only constrained by the knowledge of standing waves in order to asymptotically reconstruct dispersion curves and associated Floquet-Bloch eigenfields: It is capable of accurately representing zero-frequency stop band structures. The homogenized equations are partial differential equations with a dispersive anisotropic homogenized tensor that characterizes the effective medium. We apply HFH to metamaterials, exploiting the subtle features of Bloch dispersion curves such as Dirac-like cones, as well as zero and negative group velocity near stop bands in order to achieve exciting physical phenomena such as cloaking, lensing and endoscope effects. These are simulated numerically using finite elements and compared to predictions from HFH. An extension of HFH to periodic supercells enabling complete reconstruction of dispersion curves through an unfolding technique is also introduced

Hydrodynamic dispersion within porous biofilms

Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behavior by controlling nutrient supply, evacuation of waste products, and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilm-scale in the case where the width of the channels is significantly smaller than the thickness of the biofilm. We show that solute transport may be described via two coupled partial differential equations or telegrapher's equations for the averaged concentrations. These models are particularly relevant for chemicals, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterized by a second-order tensor whose components depend on (1) the topology of the channels' network; (2) the solute's diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion dominated, this analysis shows that hydrodynamic dispersion effects may significantly contribute to transport

Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties $\operatorname{curl}^\alpha \operatorname{grad}^\alpha = \mathbf{0}$ and $\operatorname{div}^\alpha \operatorname{curl}^\alpha = 0$ hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.Comment: 25 pages, 4 figure

Numerical bifurcation analysis for multi-section semiconductor lasers

We investigate the dynamics of a multi-section laser resembling a delayed feedback experiment where the length of the cavity is comparable to the length of the laser. Firstly, we reduce the traveling-wave model with gain dispersion (a hyperbolic system of partial differential equations) to a system of ordinary differential equations (ODEs) describing the semiflow on a local center manifold. Then, we analyse the dynamics of the system of ODEs using numerical continuation methods (AUTO). We explore the plane of the two parameters feedback phase and feedback strength to obtain a complete bifurcation diagram for small and moderate feedback strength. This diagram allows to understand the roots of a variety of nonlinear phenomena like, e. g., self-pulsations, excitability, hysteresis or chaos, and to locate them in the parameter plane

Wave Solutions

In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be understood as a specific type of solution of an appropriate mathematical equation modeling the underlying physics. Typical models consist of partial differential equations that exhibit certain general properties, e.g., hyperbolicity. This, in turn, leads to the possibility of wave solutions. Various analytical techniques (integral transforms, complex variables, reduction to ordinary differential equations, etc.) are available to find wave solutions of linear partial differential equations. Furthermore, linear hyperbolic equations with higher-order derivatives provide the mathematical underpinning of the phenomenon of dispersion, i.e., the dependence of a wave's phase speed on its wavenumber. For systems of nonlinear first-order hyperbolic equations, there also exists a general theory for finding wave solutions. In addition, nonlinear parabolic partial differential equations are sometimes said to posses wave solutions, though they lack hyperbolicity, because it may be possible to find solutions that translate in space with time. Unfortunately, an all-encompassing methodology for solution of partial differential equations with any possible combination of nonlinearities does not exist. Thus, nonlinear wave solutions must be sought on a case-by-case basis depending on the governing equation.Comment: 22 pages, 3 figures; to appear in the Mathematical Preliminaries and Methods section of the Encyclopedia of Thermal Stresses, ed. R.B. Hetnarski, Springer (2014), to appea

High frequency homogenisation for elastic lattices

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice near the edges of the Brillouin zone. The leading order solution is expressed as a product between the standing wave solution and long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure

Time-Fractional KdV Equation: Formulation and Solution using Variational Methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure