237 results on '"Christo I. Christov"'
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2. Identifying the Stationary Viscous Flows Around a Circular Cylinder at High Reynolds Numbers.
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Christo I. Christov, Rossitza S. Marinova, and Tchavdar T. Marinov
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- 2007
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3. An Operator Splitting Scheme for Biharmonic Equation with Accelerated Convergence.
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X. H. Tang and Christo I. Christov
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- 2005
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4. Coefficient Identification in Elliptic Partial Differential Equation.
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Tchavdar T. Marinov, Rossitza S. Marinova, and Christo I. Christov
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- 2005
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5. Numerical implementation of the asymptotic boundary conditions for steadily propagating 2D solitons of Boussinesq type equations.
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Christo I. Christov
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- 2012
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6. Identification of solitary-wave solutions as an inverse problem: Application to shapes with oscillatory tails.
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Christo I. Christov, Tchavdar T. Marinov, and Rossitza S. Marinova
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- 2009
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7. Implicit Scheme for Navier-Stokes Equations in Primitive Variables via Vectorial Operator Splitting.
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Christo I. Christov and Rossitza S. Marinova
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- 1997
8. Non-linear waves of the steady natural convection in a vertical fluid layer: A numerical approach.
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X. H. Tang and Christo I. Christov
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- 2007
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9. Variational imbedding approach to coefficient identification in an elliptic partial differential equation.
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Christo I. Christov, Tchavdar T. Marinov, and Rossitza S. Marinova
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- 2007
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10. A multiunit ADI scheme for biharmonic equation with accelerated convergence.
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X. H. Tang and Christo I. Christov
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- 2007
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11. Novel Numerical Approach to Solitary-wave Solutions Identification of Boussinesq and Korteweg-de Vries Equations.
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Tchavdar T. Marinov, Christo I. Christov, and Rossitza S. Marinova
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- 2005
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12. A Fourier-Series Method for Solving Soliton Problems.
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Christo I. Christov and K. L. Bekyarov
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- 1990
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13. On mechanical waves and Doppler shifts from moving boundaries
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Christo I. Christov and Ivan C. Christov
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Asymptotic analysis ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary problem ,General Engineering ,Boundary (topology) ,Wave equation ,01 natural sciences ,Amplitude modulation ,symbols.namesake ,Amplitude ,0103 physical sciences ,symbols ,0101 mathematics ,010306 general physics ,Mechanical wave ,Doppler effect ,Mathematics - Abstract
We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, Xs(t)=vt is the location of the source with constant velocity v. In the present work, however, we consider a source co-located with a moving boundary x=Xs(t), where Xs(t) can have an arbitrary functional form. For ‘slowly moving’ boundaries (i.e., ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple-scale asymptotic analysis of the moving boundary problem for the linear wave equation. We obtain a closed-form leading-order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i) Xs for non-uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the non-uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated. Copyright © 2017 John Wiley & Sons, Ltd.
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- 2017
14. The Christov-Galerkin spectral method in complex arithmetics
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M. A. Christou, Christo I. Christov, N. C. Papanicolaou, and Christodoulos Sophocleous
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Physics ,symbols.namesake ,Mathematical analysis ,Convergence (routing) ,Phase (waves) ,symbols ,Spectral method ,Galerkin method ,Nonlinear Sciences::Pattern Formation and Solitons ,Nonlinear Schrödinger equation - Abstract
We apply the Christov-Galerkin spectral method for the numerical investigation of the interaction of solitons in the Cubic Nonlinear Schrodinger Equation. The issues of convergence are addressed and an algorithm is devised for the application of the method. Results are obtained for the interaction of solitons with different phase velocities and different carrier frequencies. The interactions are shown to be elastic, save for the phase shifts. The latter are extracted from the numerical solution and discussed.
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- 2017
15. Corrigendum and addendum: modeling weakly nonlinear acoustic wave propagation
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Christo I. Christov, Pedro M. Jordan, and Ivan C. Christov
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Nonlinear system ,Acoustic wave propagation ,Mechanics of Materials ,Applied Mathematics ,Mechanical Engineering ,Addendum ,Statistical physics ,Condensed Matter Physics ,Computer Science::Digital Libraries ,Mathematics - Abstract
This article presents errors, corrections, and additions to the research outlined in the following citation: Christov, I., Christov, C. I., & Jordan, P. M. (2007). Modeling weakly nonlinear acoustic wave propagation. The Quarterly Journal of Mechanics and Applied Mathematics, 60(4), 473-495.
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- 2014
16. Collision dynamics of elliptically polarized solitons in Coupled Nonlinear Schrödinger Equations
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Michail D. Todorov and Christo I. Christov
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Physics ,Numerical Analysis ,General Computer Science ,Applied Mathematics ,Phase (waves) ,Elliptical polarization ,Collision ,Polarization (waves) ,Theoretical Computer Science ,Schrödinger equation ,symbols.namesake ,Nonlinear system ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Collision dynamics ,Classical mechanics ,Modeling and Simulation ,Quantum mechanics ,symbols ,Jump ,Nonlinear Sciences::Pattern Formation and Solitons - Abstract
We investigate numerically the collision dynamics of elliptically polarized solitons of the System of Coupled Nonlinear Schrodinger Equations (SCNLSE) for various different initial polarizations and phases. General initial elliptic polarizations (not sech-shape) include as particular cases the circular and linear polarizations. The elliptically polarized solitons are computed by a separate numerical algorithm. We find that, depending on the initial phases of the solitons, the polarizations of the system of solitons after the collision change, even for trivial cross-modulation. This sets the limits of practical validity of the celebrated Manakov solution. For general nontrivial cross-modulation, a jump in the polarization angles of the solitons takes place after the collision ('polarization shock'). We study in detail the effect of the initial phases of the solitons and uncover different scenarios of the quasi-particle behavior of the solution. In majority of cases the solitons survive the interaction preserving approximately their phase speeds and the main effect is the change of polarization. However, in some intervals for the initial phase difference, the interaction is ostensibly inelastic: either one of the solitons virtually disappears, or additional solitons are born after the interaction. This outlines the role of the phase, which has not been extensively investigated in the literature until now.
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- 2012
17. Frame indifferent formulation of Maxwell’s elastic-fluid model and the rational continuum mechanics of the electromagnetic field
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Christo I. Christov
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Electromagnetic field ,Physics ,35Q61, 74A05, 76A10, 78A02, 78A25 ,Mechanical Engineering ,Mathematical analysis ,Moving magnet and conductor problem ,FOS: Physical sciences ,Inhomogeneous electromagnetic wave equation ,Mathematical Physics (math-ph) ,Maxwell stress tensor ,Condensed Matter Physics ,symbols.namesake ,Classical mechanics ,Maxwell's equations ,Mechanics of Materials ,Electromagnetism ,symbols ,General Materials Science ,Matrix representation of Maxwell's equations ,Mathematical Physics ,Civil and Structural Engineering ,Electromagnetic tensor - Abstract
We show that the linearized equations of the incompressible elastic medium admit a `Maxwell form' in which the shear component of the stress vector plays the role of the electric field, and the vorticity plays the role of the magnetic field. Conversely, the set of dynamic Maxwell equations are strict mathematical corollaries from the governing equations of the incompressible elastic medium. This suggests that the nature of `electromagnetic field' may actually be related to an elastic continuous medium. The analogy is complete if the medium is assumed to behave as fluid in shear motions, while it may still behave as elastic solid under compressional motions. Then the governing equations of the elastic fluid are re-derived in the Eulerian frame by replacing the partial time derivatives by the properly invariant (frame indifferent) time rates. The `Maxwell from' of the frame indifferent formulation gives the frame indifferent system that is to replace the Maxwell system. This new system comprises terms already present in the classical Maxwell equations, alongside terms that are the progenitors of the Biot--Savart, Oersted--Ampere's, and Lorentz--force laws. Thus a frame indifferent (truly covariant) formulation of electromagnetism is achieved from a single postulate that the electromagnetic field is a kind of elastic (partly liquid partly solid) continuum., accepted
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- 2011
18. Perturbation solution for the 2D Boussinesq equation
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Christo I. Christov and Jayanta Choudhury
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Mechanical Engineering ,Mathematical analysis ,Ode ,Linearity ,Perturbation (astronomy) ,Condensed Matter Physics ,symbols.namesake ,Singularity ,Mechanics of Materials ,symbols ,General Materials Science ,Soliton ,Phase velocity ,Boussinesq approximation (water waves) ,Bessel function ,Civil and Structural Engineering ,Mathematics - Abstract
Boussinesq equation arises in shallow water flows and in elasticity of rods and shells. It contains non- linearity and fourth-order dispersion and has been one of the main soliton models in 1D. To find its 2D solutions, a perturbation series with respect to the small parameter e = c 2 is developed in the present work, where c is the phase speed of the localized wave. Within the order O(e 2 )= O(c 4 ), a hierarchy is derived consisting of one-dimensional fourth-order equations. The Bessel operators involved are refor- mulated to facilitate the creation of difference schemes for the ODEs from the hierarchy. The numerical scheme uses a special approximation for the behavioral condition in the singularity point (the origin). The results of this work show that at infinity the stationary 2D wave shape decays algebraically, rather than exponentially as in the 1D cases. The new result can be instrumental for understanding the interaction of 2D Boussinesq solitons, and for creating more efficient numerical algorithms explicitly acknowledging the asymptotic behavior of the solution.
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- 2011
19. The influence of thermal relaxation on the oscillatory properties of two-gradient convection in a vertical slot
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P.M. Jordan, N. C. Papanicolaou, and Christo I. Christov
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Physics ,Natural convection ,Heat flux ,Time derivative ,Heat transfer ,General Physics and Astronomy ,Absolute value ,Boundary value problem ,Mechanics ,Thermal conduction ,Galerkin method ,Mathematical Physics - Abstract
We study the effects of the Maxwell–Cattaneo (MC) law of heat conduction on the flow of a Newtonian fluid in a vertical slot subject to both vertical and horizontal temperature gradients. Working in one spatial dimension (1D), we employ a spectral expansion involving Rayleigh’s beam functions as the basis set, which are especially well-suited to the fourth order boundary value problem (b.v.p.) considered here, and the stability of the resulting dynamical system for the Galerkin coefficients is investigated. It is shown that the absolute value of the (negative) real parts of the eigenvalues are reduced, while the absolute values of the imaginary parts are somewhat increased, under the MC law. This means that the presence of the time derivative of the heat flux increases the order of the system, thus leading to more oscillatory regimes in comparison with the usual Fourier case. Moreover, no eigenvalues with positive real parts were found, which means that in this particular situation, the inclusion of thermal relaxation does not lead to destabilization of the motion.
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- 2011
20. Identification of solitary-wave solutions as an inverse problem: Application to shapes with oscillatory tails
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Rossitza S. Marinova, Christo I. Christov, and Tchavdar T. Marinov
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Numerical Analysis ,General Computer Science ,Tension (physics) ,Applied Mathematics ,Mathematical analysis ,Type (model theory) ,Inverse problem ,Theoretical Computer Science ,Identification (information) ,Trivial solution ,Modeling and Simulation ,Boundary value problem ,Restoring force ,Bifurcation ,Mathematics - Abstract
The propagation of stationary solitary waves on an infinite elastic rod on elastic foundation equation is considered. The asymptotic boundary conditions admit the trivial solution along with the solution of type of solitary wave, which is a bifurcation problem. The bifurcation is treated by prescribing the solution in the origin and introducing an unknown coefficient in the equation. Making use of the method of variational imbedding, the inverse problem for the coefficient identification is reformulated as a higher-order boundary value problem. The latter is solved by means of an iterative difference scheme, which is thoroughly validated. Solitary waves with oscillatory tails are obtained for different values of tension and linear restoring force. Special attention is devoted to the case with negative tension, when the solutions have oscillatory tails.
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- 2009
21. The concept of a quasi-particle and the non-probabilistic interpretation of wave mechanics
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Christo I. Christov
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Physics ,Numerical Analysis ,General Computer Science ,Field (physics) ,Applied Mathematics ,Duality (optimization) ,Lorentz covariance ,Theoretical Computer Science ,Schrödinger equation ,Gravitation ,Dispersive partial differential equation ,symbols.namesake ,Classical mechanics ,Wave–particle duality ,Modeling and Simulation ,symbols ,Schrödinger's cat ,Mathematical physics - Abstract
In recent works of the author [Found. Phys. 36 (2006) 1701-1717; Math. Comput. Simul. 74 (2007) 93-103], the argument has been made that Hertz's equations of electrodynamics reflect the material invariance (indifference) of the latter. Then the principle of material invariance was postulated in lieu of Lorentz covariance, and the respective absolute medium was named the metacontinuum. Here, we go further to assume that the metacontinuum is a very thin but very stiff 3D hypershell in the 4D space. The equation for the deflection of the shell along the fourth dimension is the ''master'' nonlinear dispersive equation of wave mechanics whose linear part (Euler-Bernoulli equation) is nothing else but the Schrodinger wave equation written for the real or the imaginary part of the wave function. The wave function has a clear non-probabilistic interpretation as the actual amplitude of the flexural deformation. The ''master'' equation admits solitary-wave solutions/solitons that behave as quasi-particles (QPs). We stipulate that particles are our perception of the QPs (schaumkommen in Schrodinger's own words). We show the passage from the continuous Lagrangian of the field to the discrete Lagrangian of the centers of QPs and introduce the concept of (pseudo)mass. We interpret the membrane tension as an attractive (gravitational?) force acting between the QPs. Thus, a self-consistent unification of electrodynamics, wave mechanics, gravitation, and the wave-particle duality is achieved.
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- 2009
22. Impact of the large cross-modulation parameter on the collision dynamics of quasi-particles governed by vector NLSE
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Michail D. Todorov and Christo I. Christov
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Physics ,Numerical Analysis ,Work (thermodynamics) ,General Computer Science ,Linear polarization ,Applied Mathematics ,Collision ,Theoretical Computer Science ,Schrödinger equation ,Momentum ,symbols.namesake ,Nonlinear system ,Classical mechanics ,Modeling and Simulation ,symbols ,Soliton ,Complex number - Abstract
For the Coupled Nonlinear Schrodinger Equations (CNLSE) we construct a conservative fully implicit scheme (in the vein of the scheme with internal iterations proposed in [C.I. Christov, S. Dost, G.A. Maugin, Inelasticity of soliton collisions in system of coupled nls equations, Physica Scripta 50 (1994) 449-454.]). Our scheme makes use of complex arithmetic which allows us to reduce the computational time fourfold. The scheme conserves the ''mass'', momentum, and energy. We investigate collisions of solitary waves (quasi-particles or QPs) with linear polarization in the initial configuration. We elucidate numerically the role of nonlinear coupling on the quasi-particle dynamics. We find that the initially linear polarizations of the QPs change after the collision to elliptic polarizations. For large values of cross-modulation parameter, an additional QP is created during the collision. We find that although the total energy is positive and conserved, the energy only of the system of identifiable after the collision QPs is negative, i.e., the different smaller excitations and radiation carry away part of the energy. The effects found in the present work shed light on the intimate mechanisms of interaction of QPs.
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- 2009
23. On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction
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Christo I. Christov
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Physics ,Field (physics) ,Mechanical Engineering ,Frame (networking) ,Mathematical analysis ,Derivative ,Relativistic heat conduction ,Condensed Matter Physics ,Thermal conduction ,Classical mechanics ,Heat flux ,Mechanics of Materials ,Relaxation rate ,Single equation ,General Materials Science ,Civil and Structural Engineering - Abstract
A material-invariant (frame indifferent) version of the Maxwell–Cattaneo law is proposed in which the relaxation rate of the heat flux is given by Oldroyd’s upper-convected derivative. It is shown that the new formulation allows for the elimination of the heat flux, thus yielding a single equation for the temperature field. This feature is to be expected from a truly frame indifferent description.
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- 2009
24. Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot
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N. C. Papanicolaou, Christo I. Christov, and George M. Homsy
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Floquet theory ,Series (mathematics) ,Basis (linear algebra) ,Applied Mathematics ,Mechanical Engineering ,Computational Mechanics ,Geometry ,Computer Science Applications ,Nonlinear system ,Harmonic function ,Rate of convergence ,Mechanics of Materials ,Applied mathematics ,Asymptotic expansion ,Galerkin method ,Mathematics - Abstract
A Fourier–Galerkin spectral technique for solving coupled higher-order initial-boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so-called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived. The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth-order algebraic for both linear and nonlinear problems. Though algebraic, the fifth-order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems. An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement. The new technique is applied to the G-jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters. Copyright © 2008 John Wiley & Sons, Ltd.
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- 2009
25. On the evolution of localized wave packets governed by a dissipative wave equation
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Christo I. Christov
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Physics ,Applied Mathematics ,Wave packet ,Gaussian ,General Physics and Astronomy ,Dissipation ,Wave equation ,Redshift ,Computational Mathematics ,symbols.namesake ,Classical mechanics ,Apodization ,Electromagnetism ,Modeling and Simulation ,symbols ,Dissipative system - Abstract
The present paper deals with the effect of dissipation on the propagation of wave packets governed by a wave equation of Jeffrey type. We show that all packets undergo a shift of the central frequency (the mode with maximal amplitude) towards the lower frequencies (“redshift” in theory of light or “baseshift” in acoustics). Packets with Gaussian apodization function do not change their shape and remain Gaussian but undergo redshift and spread. The possible applications of the results are discussed.
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- 2008
26. Modeling weakly nonlinear acoustic wave propagation
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Ivan C. Christov, Pedro M. Jordan, and Christo I. Christov
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Conservation law ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Finite difference method ,Condensed Matter Physics ,Wave equation ,Euler equations ,Nonlinear system ,symbols.namesake ,Mechanics of Materials ,Inviscid flow ,symbols ,Boundary value problem ,Hyperbolic partial differential equation ,Mathematics - Abstract
Summary Three weakly nonlinear models of lossless, compressible fluid flow—a straightforward weakly nonlinear equation (WNE), the inviscid Kuznetsov equation (IKE) and the Lighthill–Westervelt equation (LWE)—are derived from first principles and their relationship to each other is established. Through a numerical study of the blow-up of acceleration waves, the weakly nonlinear equations are compared to the ‘exact’ Euler equations, and the ranges of applicability of the approximate models are assessed. By reformulating these equations as hyperbolic systems of conservation laws, we are able to employ a Godunov-type finite-difference scheme to obtain numerical solutions of the approximate models for times beyond the instant of blow-up (that is, shock formation), for both density and velocity boundary conditions. Our study reveals that the straightforward WNE gives the best results, followed by the IKE, with the LWE’s performance being the poorest overall.
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- 2007
27. On Boussinesq's paradigm in nonlinear wave propagation
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Gérard A. Maugin, Alexey V. Porubov, and Christo I. Christov
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Marketing ,Strategy and Management ,Fluid layer ,Linear dispersion ,Wave equation ,Nonlinear wave propagation ,Nonlinear system ,Classical mechanics ,Surface wave ,Media Technology ,Dissipative system ,General Materials Science ,Boussinesq approximation (water waves) ,Mathematics - Abstract
Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of ‘Boussinesq's paradigm’. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the ‘mechanics’ of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered. To cite this article: C.I. Christov et al., C. R. Mecanique 335 (2007).
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- 2007
28. Maxwell–Lorentz electrodynamics as a manifestation of the dynamics of a viscoelastic metacontinuum
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Christo I. Christov
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Physics ,Numerical Analysis ,General Computer Science ,Displacement current ,Applied Mathematics ,Lorentz transformation ,Constitutive equation ,Vorticity ,Physics::Classical Physics ,Theoretical Computer Science ,Magnetic field ,Physics::Fluid Dynamics ,symbols.namesake ,Classical mechanics ,Maxwell's equations ,Modeling and Simulation ,Quantum electrodynamics ,symbols ,Stochastic electrodynamics ,Lorentz force - Abstract
We prove that, when linearized, the governing equations of an incompressible viscoelastic continuum can be rendered into a form identical to that of Maxwell's equations of electrodynamics. The divergence of deviator stress tensor is analogous to the electric field, while the vorticity (the curl of velocity field) is interpreted as the magnetic field. The elastic part of constitutive relation explains Maxwell's displacement current, and is responsible for the propagation of gradient (shear) waves. In turn, the viscous part is associated with the Ampere's and Ohm's laws for the current. This analogy is extended further and the nonlinearity of the material time derivative (the advective part of acceleration) is interpreted as the Lorentz force. The classical wave equations of electrodynamics are also derived as corollaries. Thus an interesting and far reaching analogy between the viscoelastic continuum and the electrodynamics is established.
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- 2007
29. Fourier–Galerkin method for 2D solitons of Boussinesq equation
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Christo I. Christov and M. A. Christou
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Numerical Analysis ,General Computer Science ,Applied Mathematics ,Mathematical analysis ,Wave equation ,Theoretical Computer Science ,Exponential function ,symbols.namesake ,Fourier transform ,Rate of convergence ,Modeling and Simulation ,symbols ,Orthonormal basis ,Boussinesq approximation (water waves) ,Galerkin method ,Mathematics - Abstract
We develop a Fourier-Galerkin spectral technique for computing the stationary solutions of 2D generalized wave equations. To this end a special complete orthonormal system of functions in L^2(-~,~) is used for which product formula is available. The exponential rate of convergence is shown. As a featuring example we consider the Proper Boussinesq Equation (PBE) in 2D and obtain the shapes of the stationary propagating localized waves. The technique is thoroughly validated and compared to other numerical results when possible.
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- 2007
30. Stress retardation versus stress relaxation in linear viscoelasticity
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Ivan C. Christov and Christo I. Christov
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Continuum (measurement) ,Mechanical Engineering ,Fluid Dynamics (physics.flu-dyn) ,FOS: Physical sciences ,Fluid mechanics ,Physics - Fluid Dynamics ,Strain rate ,Condensed Matter Physics ,01 natural sciences ,Viscoelasticity ,010305 fluids & plasmas ,Functional relation ,Mathematics - Analysis of PDEs ,Classical mechanics ,Mechanics of Materials ,0103 physical sciences ,Stress relaxation ,FOS: Mathematics ,General Materials Science ,010306 general physics ,Civil and Structural Engineering ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We present a preliminary examination of a new approach to a long-standing problem in non-Newtonian fluid mechanics. First, we summarize how a general implicit functional relation between stress and rate of strain of a continuum with memory is reduced to the well-known linear differential constitutive relations that account for "relaxation" and "retardation." Then, we show that relaxation and retardation are asymptotically equivalent for small Deborah numbers, whence causal pure relaxation models necessarily correspond to ill-posed pure retardation models. We suggest that this dichotomy could be a possible way to reconcile the discrepancy between the theory of and certain experiments on viscoelastic liquids that are conjectured to exhibit only stress retardation., Comment: 5 pages, 1 figure, elsarticle format, v2 corrects some typograpgical errors, published in Mechanics Research Communications
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- 2015
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31. On the Material Invariant Formulation of Maxwell’s Displacement Current
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Christo I. Christov
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Electromagnetic field ,Physics ,Displacement current ,General Physics and Astronomy ,Invariant (physics) ,Physics::Classical Physics ,Viscoelasticity ,Physics::Fluid Dynamics ,symbols.namesake ,Classical mechanics ,Maxwell's equations ,Time derivative ,symbols ,Stochastic electrodynamics ,Lorentz force ,Mathematical physics - Abstract
Maxwell accounted for the apparent elastic behavior of the electromagnetic field by augmenting Ampere’s law with the so-called displacement current, in much the same way that he treated the viscoelasticity of gases. Maxwell’s original constitutive relations for both electrodynamics and fluid dynamics were not material invariant. In the theory of viscoelastic fluids, the situation was later corrected by Oldroyd, who introduced the upper-convective derivative. Assuming that the electromagnetic field should follow the general requirements for a material field, we show that if the upper convected derivative is used in place of the partial time derivative in the displacement current term, Maxwell’s electrodynamics becomes material invariant. Note, that the material invariance of Faraday’s law is automatically established if the Lorentz force is admitted as an integral part of the model. The new formulation ensures that the equation for conservation of charge is also material invariant in vacuo. The viscoelastic medium whose apparent manifestation are the known phenomena of electrodynamics is called here the metacontinuum.
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- 2006
32. Nonlinear acoustic propagation in homentropic perfect gases: A numerical study
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Ivan C. Christov, Pedro M. Jordan, and Christo I. Christov
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Physics ,Conservation law ,Mathematical analysis ,General Physics and Astronomy ,Context (language use) ,Perfect gas ,Acoustic wave ,Euler equations ,Nonlinear system ,symbols.namesake ,Nonlinear acoustics ,Classical mechanics ,Euler's formula ,symbols - Abstract
A study of nonlinear acoustic waves in a homentropic perfect gas is presented. Conservation laws for the Euler and Lighthill–Westervelt equations are constructed and solved numerically using a Godunov-type finite-difference scheme. Simulations are carried out in the context of two initial-boundary-value problems (IBVP)s—one resulting in finite-time, and the other in infinite-time, blow-up at the wavefront. Additionally, analytical results are presented to support the numerical findings.
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- 2006
33. An operator splitting scheme for the stream-function formulation of unsteady Navier–Stokes equations
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X.-H. Tang and Christo I. Christov
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Laplace transform ,business.industry ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Computational Mechanics ,Finite difference method ,Computational fluid dynamics ,Computer Science Applications ,Nonlinear system ,Mechanics of Materials ,Stream function ,Boundary value problem ,Navier–Stokes equations ,business ,Numerical stability ,Mathematics - Abstract
A fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher-order ultra-parabolic equation. The convergence with respect to the fictitious time (we call the latter 'internal iterations') allows one to obtain fully implicit nonlinear scheme in full time steps for the physical-time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second-order of approximation. For the solution of the internal iteration, a fractional-step scheme is proposed based on the splitting of the combination of the Laplace, bi-harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second-order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically.
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- 2006
34. A Fully Coupled Solver for Incompressible Navier–Stokes Equations using Operator Splitting
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Christo I. Christov, Tchavdar T. Marinov, and Rossitza S. Marinova
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Mechanical Engineering ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Computational Mechanics ,Energy Engineering and Power Technology ,Aerospace Engineering ,Non-dimensionalization and scaling of the Navier–Stokes equations ,Solver ,Condensed Matter Physics ,Physics::Fluid Dynamics ,Continuity equation ,Mechanics of Materials ,Pressure-correction method ,Hagen–Poiseuille flow from the Navier–Stokes equations ,Poisson's equation ,Reynolds-averaged Navier–Stokes equations ,Navier–Stokes equations ,Mathematics - Abstract
The steady incompressible Navier–Stokes equations are coupled by a Poisson equation for the pressure from which the continuity equation is subtracted. The equivalence to the original N–S problem is...
- Published
- 2003
35. Methods for the coupled Stokes-Darcy problem
- Author
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F. Feuillebois, S. Khabthani, L. Elasmi, A. Sellier, Michail D. Todorov, Christo I. Christov, Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur (LIMSI), Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université - UFR d'Ingénierie (UFR 919), Sorbonne Université (SU)-Sorbonne Université (SU)-Université Paris-Saclay-Université Paris-Sud - Paris 11 (UP11), Laboratoire d'Ingénierie Mathématique, École Polytechnique de Tunisie, rue El Khawarezmi, La Marsa, Tunisia, affiliation inconnue, Laboratoire d'hydrodynamique (LadHyX), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), and Michail D. Todorov and Christo I. Christov
- Subjects
Partial differential equation ,Mathematical analysis ,020206 networking & telecommunications ,02 engineering and technology ,Slip (materials science) ,Viscous liquid ,Integral equation ,Physics::Fluid Dynamics ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Particle size ,Boundary value problem ,Porous medium ,Asymptotic expansion ,Mathematics - Abstract
The motion of particles in a viscous fluid close to a porous membrane is modelled for the case when particles are large compared with the size of pores of the membrane. The hydrodynamic interactions of one particle with the membrane are detailed here. The model involves Stokes equations for the fluid motion around the particle together with Darcy equations for the flow in the porous membrane and Stokes equations for the flow on the other side of the membrane. Boundary conditions at the fluid‐membrane interface are the continuity of pressure and velocity in the normal direction and the Beavers and Joseph slip condition on the fluid side in the tangential directions. The no‐slip condition applies on the particle. This problem is solved here by two different methods.The first one is an extended boundary integral method (EBIM). A Green function is derived for the flow close to a porous membrane. This function is non‐symmetric, leading to difficulties hindering the application of the classical boundary integral method (BIM). Thus, an extended method is proposed, in which the unknown distribution of singularities on the particle surface is not the stress, like in the classical boundary integral method. Yet, the hydrodynamic force and torque on the particle are obtained by integrals of this distribution on the particle surface.The second method consists in searching the solution as an asymptotic expansion in term of a small parameter that is the ratio of the typical pore size to the particle size. The various boundary conditions are taken into account at successive orders: order (0) simply represents an impermeable wall without slip and order (1) an impermeable wall with a peculiar slip prescribed by order (0); at least the 3rd order is necessary to enforce all boundary conditions.The methods are applied numerically to a spherical particle and comparisons are made with earlier works in particular cases.The motion of particles in a viscous fluid close to a porous membrane is modelled for the case when particles are large compared with the size of pores of the membrane. The hydrodynamic interactions of one particle with the membrane are detailed here. The model involves Stokes equations for the fluid motion around the particle together with Darcy equations for the flow in the porous membrane and Stokes equations for the flow on the other side of the membrane. Boundary conditions at the fluid‐membrane interface are the continuity of pressure and velocity in the normal direction and the Beavers and Joseph slip condition on the fluid side in the tangential directions. The no‐slip condition applies on the particle. This problem is solved here by two different methods.The first one is an extended boundary integral method (EBIM). A Green function is derived for the flow close to a porous membrane. This function is non‐symmetric, leading to difficulties hindering the application of the classical boundary integra...
- Published
- 2010
36. DISSIPATIVE QUASI-PARTICLES: THE GENERALIZED WAVE EQUATION APPROACH
- Author
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Christo I. Christov
- Subjects
Physics ,Balance (metaphysics) ,Nonlinear system ,Conservation law ,Classical mechanics ,Applied Mathematics ,Modeling and Simulation ,Dissipative system ,Dissipation ,Wave equation ,Dispersion (water waves) ,Engineering (miscellaneous) ,Quasi particles - Abstract
Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.
- Published
- 2002
37. [Untitled]
- Author
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Christo I. Christov and M. A. Christou
- Subjects
Computational Mathematics ,symbols.namesake ,Fourier transform ,Rate of convergence ,Iterative method ,Mathematical analysis ,symbols ,Orthonormal basis ,Soliton ,Boussinesq approximation (water waves) ,Spectral method ,Galerkin method ,Mathematics - Abstract
Using a complete orthonormal system of functions in L 2(−∞ ,∞) a Fourier-Galerkin spectral technique is developed for computing of the localized solutions of equations with cubic nonlinearity. A formula expressing the triple product into series in the system is derived. Iterative algorithm implementing the spectral method is developed and tested on the soliton problem for the cubic Boussinesq equation. Solution is obtained and shown to compare quantitatively very well to the known analytical one. The issues of convergence rate and truncation error are discussed.
- Published
- 2002
38. Numerical scheme for Swift-Hohenberg equation with strict implementation of lyapunov functional
- Author
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Christo I. Christov and José Pontes
- Subjects
Lyapunov function ,Diffusion equation ,Differential equation ,Truncation error (numerical integration) ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Finite difference method ,Computer Science Applications ,symbols.namesake ,Modeling and Simulation ,Modelling and Simulation ,Functional equation ,symbols ,Lyapunov equation ,Lyapunov redesign ,Mathematics - Abstract
In this paper, we consider a nonlinear generalized diffusion equation called the Swift-Hohenberg equation (SH) for which a Lyapunov functional is known. We develop a computationally efficient second-order in time implicit difference scheme based on the operator-splitting method. Internal iterations are used to make the scheme both nonlinear and implicit. We prove that the scheme allows strict (independent of the truncation error) implementation of a discrete approximation of the Lyapunov functional. The new scheme is used to investigate the pattern formation from random initial conditions, and spatially chaotic states are found.
- Published
- 2002
- Full Text
- View/download PDF
39. An energy-consistent dispersive shallow-water model
- Author
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Christo I. Christov
- Subjects
Galilean invariance ,Computer simulation ,Applied Mathematics ,General Physics and Astronomy ,Invariant (physics) ,Galilean ,Physics::Fluid Dynamics ,Computational Mathematics ,Waves and shallow water ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Classical mechanics ,Inviscid flow ,Modeling and Simulation ,Free surface ,Boussinesq approximation (water waves) ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
The flow of inviscid liquid in a shallow layer with free surface is revisited in the framework of the Boussinesq approximation. The unnecessary approximations connected with the moving frame are removed and a Boussinesq model is derived which is Galilean invariant to the leading asymptotic order. The Hamiltonian structure of the new model is demonstrated. The conservation and/or balance laws for wave mass, energy and wave momentum (pseudo-momentum) are derived. A new localized solution is obtained analytically and compared to the classical Boussinesq sech. Numerical simulation of the collision of two solitary waves is conducted and the impact of Galilean invariance on phase shift is discussed.
- Published
- 2001
40. Nonlinear dynamics of two-dimensional convection in a vertically stratified slot with and without gravity modulation
- Author
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Christo I. Christov and George M. Homsy
- Subjects
Physics ,Natural convection ,Mechanical Engineering ,Prandtl number ,Thermodynamics ,Mechanics ,Rayleigh number ,Condensed Matter Physics ,Instability ,Physics::Fluid Dynamics ,symbols.namesake ,Amplitude ,Mechanics of Materials ,symbols ,Streamlines, streaklines, and pathlines ,Jitter ,Dimensionless quantity - Abstract
The convective flow in a vertical slot with differentially heated walls and vertical temperature gradient is considered for very large Rayleigh numbers. Gravity is taken to be vertical and to consist of both a mean and a harmonic modulation (‘jitter’) at a given frequency and amplitude. The time-dependent Boussinesq equations governing the two-dimensional convection are solved numerically. To this end an economic operator-splitting scheme is devised combined with internal iterations within a given time step. The approximation of the nonlinear terms is conservative and no scheme viscosity is present in the approximation. The flow is investigated for a range of Prandtl numbers from Pr = 1000 when fluid inertia is insignificant and only thermal inertia plays a role to Pr = 0.73 when both are significant and of the same order. The flow is governed by several parameters. In the absence of jitter, these are the Prandtl number, Pr, the Rayleigh number, Ra, and the dimensionless critical stratification, τB. Simulations are reported for Pr = 103 and a range of τB and Ra, with emphasis on mode selection and finite-amplitude states. The presence of jitter adds two more parameters, i.e. the dimensionless jitter amplitude e and frequency ω, rendering the flow susceptible to new modes of parametric instability at a critical amplitude ec. Stability maps of ec vs. ω are given for a range of ω. Finally we investigate the response of the system to jitter near the neutral curves of the various instability modes.
- Published
- 2001
41. Gradient pattern analysis of Swift–Hohenberg dynamics: phase disorder characterization
- Author
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Erico L. Rempel, Rodrigues C.R. Neto, Christo I. Christov, José Pontes, Daniel Walgraef, Reinaldo R. Rosa, and Fernando M. Ramos
- Subjects
Statistics and Probability ,Combinatorics ,Amplitude ,Component (thermodynamics) ,Dynamics (mechanics) ,Phase (waves) ,Pattern formation ,Pattern analysis ,Statistical physics ,Condensed Matter Physics ,Characterization (materials science) ,Mathematics ,Numerical integration - Abstract
In this paper, we analyze the onset of phase-dominant dynamics in a uniformly forced system. The study is based on the numerical integration of the Swift–Hohenberg equation and adresses the characterization of phase disorder detected from gradient computational operators as complex entropic form (CEF). The transition from amplitude to phase dynamics is well characterized by means of the variance of the CEF phase component.
- Published
- 2000
42. Resonant thermocapillary and buoyant flows with finite frequency gravity modulation
- Author
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George M. Homsy, Christo I. Christov, and Vinod Suresh
- Subjects
Fluid Flow and Transfer Processes ,Physics ,Mechanical Engineering ,Prandtl number ,Computational Mechanics ,Mechanics ,Condensed Matter Physics ,Gravitational acceleration ,Physics::Fluid Dynamics ,symbols.namesake ,Temperature gradient ,Classical mechanics ,Mechanics of Materials ,Free surface ,symbols ,Relaxation (physics) ,Galerkin method ,Frequency modulation ,Dimensionless quantity - Abstract
Interaction between a base thermocapillary flow and a time-dependent buoyant force is studied for a slot geometry. A temperature gradient applied along a fluid-filled slot with thermocapillarity at a free surface produces a base parallel flow. The system is subjected to streamwise gravitational acceleration that varies harmonically in time. Grassia and Homsy [Phys. Fluids. 10, 1273 (1998)] have shown that in the limit of zero frequency modulation, coupling of the thermocapillary flow with long wave convective modes leads to singularities at critical points corresponding to the Rayleigh–Benard eigenvalues. In the case of small but finite frequency modulation studied here, inertial effects moderate the singularities which are replaced by a response that scales exponentially with the inverse of the dimensionless modulation frequency. An O(1) delay is observed in the onset of the resonant response even for small modulation frequencies. The response is also found to scale exponentially with the inverse Prandtl number for large Prandtl numbers and to be independent of Prandtl number for small Prandtl numbers. Relaxation oscillations are observed in certain parameter ranges as a result of the coupling between the fluid and thermal fields. A Galerkin approximation is used to reduce the problem to an equivalent dynamical system, the analysis of which gives analytical support to and insight into the numerical results.
- Published
- 1999
43. Identification of heat-conduction coefficient via method of variational imbedding
- Author
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Tchavdar T. Marinov and Christo I. Christov
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Well-posed problem ,Elliptic curve ,Partial differential equation ,Modelling and Simulation ,Modeling and Simulation ,Mathematical analysis ,Heat equation ,Boundary value problem ,Calculus of variations ,Inverse problem ,Elliptic boundary value problem ,Computer Science Applications ,Mathematics - Abstract
The inverse parabolic problem of coefficient identification from over-posed data is embedded into a fourth order in space and second order in time elliptic boundary value problem. The latter is well posed for redundant data at boundaries. The equivalence of the two problems is demonstrated. A difference scheme of splitting-type is employed, and featuring examples are elaborated numerically.
- Published
- 1998
44. Method of Variational Imbedding for the Inverse Problem of Boundary-Layer Thickness Identification
- Author
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Christo I. Christov and Tchavdar T. Marinov
- Subjects
Physics::Fluid Dynamics ,Flow (mathematics) ,Differential equation ,Plane (geometry) ,Applied Mathematics ,Modeling and Simulation ,Mathematical analysis ,Boundary value problem ,Uniqueness ,Inverse problem ,Stagnation point ,Boundary layer thickness ,Mathematics - Abstract
The inverse problem of identification of boundary-layer thickness is replaced by the higher-order boundary value problem for the Euler–Lagrange equations for minimization of the quadratic functional of the original system (Method of Variational Imbedding – MVI). The imbedding problem is correct in the sense of Hadamard and consists of an explicit differential equation for the boundary-layer thickness. The existence and uniqueness of solution of the linearized imbedding problem is demonstrated and a difference scheme of splitting type is proposed for its numerical solution. The performance of the technique is demonstrated for three different boundary-layer problems: the Blasius problem, flow in the vicinity of plane stagnation point and the flow in the leading stagnation point on a circular cylinder. Comparisons with the self-similar solutions where available are quantitatively very good.
- Published
- 1997
45. Implicit time splitting for fourth-order parabolic equations
- Author
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José Pontes, Daniel Walgraef, Christo I. Christov, and Manuel G. Velarde
- Subjects
Convection ,Diffusion equation ,Mechanical Engineering ,Numerical analysis ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Computational Mechanics ,General Physics and Astronomy ,Parabolic partial differential equation ,Computer Science Applications ,Mechanics of Materials ,Initial value problem ,Boundary value problem ,Diffusion (business) ,Nonlinear Sciences::Pattern Formation and Solitons ,Bifurcation ,Mathematics - Abstract
A coordinate-splitting economic difference scheme is proposed for generalized parabolic equations (GPE) containing fourth-order diffusion operators and the algorithm for its implementation is developed. The performance of the scheme is demonstrated for different cases, e.g. for treating bifurcation phenomena. The technique is applied to the numerical solution of Swift-Hohenberg equation describing the Rayleigh-Benard convection and results are obtained for very large system sizes and for very long times on small computational platform.
- Published
- 1997
46. Investigation of the long-time evolution of localized solutions of a dispersive wave system
- Author
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M. D. Todorov and Christo I. Christov
- Subjects
Physics ,Nonlinear system ,Superposition principle ,Amplitude ,Classical mechanics ,Galilean invariance ,Time evolution ,Initial value problem ,Trailing edge ,Dispersion (water waves) - Abstract
We consider the long-time evolution of the solution of an energy-consistent system with dispersion and nonlinearity, which is the progenitor of the different Boussinesq equations. Unlike the classical Boussinesq models, the energy-consistent one possesses Galilean invariance. As initial condition we use the superposition of two analytical one-soliton solutions. We use a strongly dynamical implicit difference scheme with internal iterations, which allows us to follow the evolution of the solution at very long times. We focus on the behavior of traveling localized solutions developing from critical initial data. The main solitary waves appear virtually undeformed from the interaction, but additional oscillations are excited at the trailing edge of each one of them. We track their evolution for very long times when they tend to adopt a self-similar shape. We test a hypothesis about the dependence on time of the amplitude and the support of Airy-function shaped coherent structures. The investigation elucidates the mechanism of evolution of interacting solitary waves in the energy-consistent Boussinesq equation.
- Published
- 2013
47. NUMERICAL STUDY OF PATTERNS AND THEIR EVOLUTION IN FINITE GEOMETRIES
- Author
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Christo I. Christov, José Pontes, and Manuel G. Velarde
- Subjects
Convection ,Buoyancy ,Biot number ,Applied Mathematics ,Mathematical analysis ,Pattern formation ,engineering.material ,Numerical integration ,Physics::Fluid Dynamics ,Modeling and Simulation ,Heat transfer ,engineering ,Boundary value problem ,Engineering (miscellaneous) ,Bifurcation ,Mathematics - Abstract
Pattern formation in a finite layer of fluid induced either by buoyancy or by a surface-tension gradient is considered. The fluid is confined between poor conducting horizontal boundaries, leading to patterns with a characteristic horizontal scale much larger than the fluid depth. The evolution of the system is studied by numerical integration of the (1+2)D equation introduced by Knobloch [1990]: [Formula: see text] Here µ is the scaled bifurcation parameter, κ=1, and a represents the effects of a heat transfer finite Biot number. The coefficients β, δ and γ do not vanish when the boundary conditions at top and bottom are not identical (β≠0, δ≠0) or when non-Boussinesq effects are taken into account (γ≠0). When the conductive state becomes unstable due to surface-tension inhomogeneities, it is shown that the system evolves towards a stationary pattern of hexagons with up or down flow depending on the relative value of the coefficients β and δ. In the case of buoyancy-driven convection (β=δ≠0), the system displays a tesselation of steady squares. Knobloch’s equation also describes time-dependent patterns at high thermal gradients, including spatio-temporal chaos, due to the non-variational character of the equation.
- Published
- 1996
48. Spatio-temporal chaos and intermittency in a 1-dimensional energy-conserving coupled map lattice
- Author
-
Grégoire Nicolis and Christo I. Christov
- Subjects
Statistics and Probability ,Mathematical analysis ,Vacuum state ,One-dimensional space ,Positive-definite matrix ,Type (model theory) ,Condensed Matter Physics ,Instability ,law.invention ,law ,Intermittency ,Energy (signal processing) ,Mathematics ,Coupled map lattice - Abstract
The Klein—Gordon equation with cubic nonlinearity (the φ 4 equation) is considered and an energy-conserving difference scheme is proposed for its solution. The scheme, extended to finite time increments and spacing, is then used to define a coupled map lattice for which an energy-like functional is conserved. The case of linear instability of the vacuum state is considered when this energy is not positive definite and found to lead, under certain additional conditions, to spatio-temporal chaos. The statistical properties of this type of solution such as probability densities and correlation functions are calculated. Strong intermittency, whereby the process wanders between two sub-manifolds, is found and studied in detail.
- Published
- 1996
49. Dissipative solitons
- Author
-
Christo I. Christov and Manuel G. Velarde
- Subjects
Physics ,Classical mechanics ,Dissipative system ,Statistical and Nonlinear Physics ,Condensed Matter Physics - Published
- 1995
50. On the Pseudolocalized Solutions in Multi-dimension of Boussinesq Equation
- Author
-
Christo I. Christov
- Subjects
Hessian matrix ,Numerical Analysis ,General Computer Science ,Applied Mathematics ,Mathematical analysis ,FOS: Physical sciences ,Pattern Formation and Solitons (nlin.PS) ,Differential operator ,01 natural sciences ,Nonlinear Sciences - Pattern Formation and Solitons ,Square (algebra) ,Domain (mathematical analysis) ,Theoretical Computer Science ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Quadratic equation ,Modeling and Simulation ,0103 physical sciences ,symbols ,Boundary value problem ,0101 mathematics ,010306 general physics ,Laplace operator ,Mathematics - Abstract
A new class of solutions of three-dimensional equations from the Boussinesq paradigm are considered. The corresponding profiles are not localized functions in the sense of the integrability of the square over an infinite domain. For the new type of solutions, the gradient and the Hessian/Laplacian are square integrable. In the linear limiting case, analytical expressions for the profiles of the pseudolocalized solutions are found. The nonlinear case is treated numerically with a special approximation of the differential operators with spherical symmetry that allows for automatic acknowledgement of the behavioral conditions at the origin of the coordinate system. The asymptotic boundary conditions stem from the $1/r$ behavior at infinity of the pseudolocalized profile. A special approximation is devised that allows us to obtain the proper behavior for much smaller computational box. The pseudolocalized solutions are obtained for both quadratic and cubic nonlinearity., Comment: 8 pages, 3 figures
- Published
- 2012
- Full Text
- View/download PDF
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