35 results on '"Christo I. Christov"'
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2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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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Numerical Analysis ,Natural convection ,General Computer Science ,Applied Mathematics ,Linear system ,Mathematical analysis ,Instability ,Theoretical Computer Science ,Wavelength ,Temperature gradient ,Nonlinear system ,Modeling and Simulation ,Stream function ,Boussinesq approximation (water waves) ,Mathematics - Abstract
A fully implicit finite difference scheme is developed for the stream function formulation of unsteady thermoconvective flows. An artificial time is added in the equation for the stream function rendering it into an ultra-parabolic type. For each time stage of the real time, a convergence is obtained with respect to artificial time (''internal iterations''). An implicit efficient operator-splitting time stepping is designed and proved to be absolutely stable. Employing a conservative central-difference approximation of the non-linear terms makes the scheme absolutely stable without using upwind differences. As a result, the scheme has no scheme viscosity and has virtually negligible phase error, which makes it a useful tool for investigating the intricate structure of the thermoconvective flow. The scheme is second-order approximation both in time and space. By means of the scheme developed, the convective flow in a vertical slot with differentially heated walls and vertical temperature gradient is studied for very large Rayleigh numbers. The model involves Boussinesq approximation and consists of a coupled system of a fourth-order in space equation for the stream function and a convection-diffusion equation for the temperature. The numerical results show that with increasing stratification parameter, the mode of the instability changes from traveling-wave to stationary-wave, which is consistent with the predictions of the linear theory of hydrodynamic instability. The role of the dimensionless wavelength is investigated and the issue of most dangerous wave is addressed numerically.
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- 2007
11. 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
12. 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
13. 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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Applied Mathematics ,Mathematical analysis ,Numerical technique ,Cnoidal wave ,Inverse problem ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Trivial solution ,Modeling and Simulation ,Boundary data ,Boussinesq approximation (water waves) ,Korteweg–de Vries equation ,Nonlinear Sciences::Pattern Formation and Solitons ,Engineering (miscellaneous) ,Bifurcation ,Mathematics - Abstract
A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg–de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth-order Boussinesq equation is considered for illustrations.
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- 2005
14. DISSIPATIVE QUASI-PARTICLES: THE GENERALIZED WAVE EQUATION APPROACH
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Christo I. Christov
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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.
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- 2002
15. An energy-consistent dispersive shallow-water model
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Christo I. Christov
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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.
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- 2001
16. Method of Variational Imbedding for the Inverse Problem of Boundary-Layer Thickness Identification
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Christo I. Christov and Tchavdar T. Marinov
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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
17. NUMERICAL STUDY OF PATTERNS AND THEIR EVOLUTION IN FINITE GEOMETRIES
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Christo I. Christov, José Pontes, and Manuel G. Velarde
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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.
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- 1996
18. On the Pseudolocalized Solutions in Multi-dimension of Boussinesq Equation
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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
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- 2012
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19. Pseudolocalized Three-Dimensional Solitary Waves as Quasi-Particles
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Christo I. Christov
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Hessian matrix ,General Physics and Astronomy ,Inverse ,FOS: Physical sciences ,Pattern Formation and Solitons (nlin.PS) ,01 natural sciences ,Square (algebra) ,010305 fluids & plasmas ,Dispersive partial differential equation ,symbols.namesake ,0103 physical sciences ,Field theory (psychology) ,010306 general physics ,Mathematical Physics ,Mathematics ,Applied Mathematics ,Mathematical analysis ,Equations of motion ,Mathematical Physics (math-ph) ,Nonlinear Sciences - Pattern Formation and Solitons ,Computational Mathematics ,Classical mechanics ,Square-integrable function ,Modeling and Simulation ,symbols ,Laplace operator - Abstract
A higher-order dispersive equation is introduced as a candidate for the governing equation of a field theory. A new class of solutions of the three-dimensional field equation are considered, which are not localized functions in the sense of the integrability of the square of the profile over an infinite domain. For this new class of solutions, the gradient and/or the Hessian/Laplacian are square integrable. In the linear limiting case, an analytical expression for the pseudolocalized solution is found and the method of variational approximation is applied to find the dynamics of the centers of the quasi-particles (QPs) corresponding to these solutions. A discrete Lagrangian can be derived due to the localization of the gradient and the Laplacian of the profile. The equations of motion of the QPs are derived from the discrete Lagrangian. The pseudomass ("wave mass") of a QP is defined as well as the potential of interaction. The most important trait of the new QPs is that at large distances, the force of attraction is proportional to the inverse square of the distance between the QPs. This can be considered analogous to the gravitational force in classical mechanics., Comment: 19 pages, 10 figures, elsarticle format; v2 includes revision in response to referees; accepted for publication in Wave Motion for the special issue on Mathematical Modeling and Physical Dynamics of Solitary Waves: From Continuum Mechanics to Field Theory
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- 2012
- Full Text
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20. Mathematical modeling of sterile insect technology for control of anopheles mosquito
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R. Anguelov, Y. Dumont, J. Lubuma, Michail D. Todorov, Christo I. Christov, Department of Mathematics and Applied Mathematics [Pretoria], University of Pretoria [South Africa], Botanique et Modélisation de l'Architecture des Plantes et des Végétations (UMR AMAP), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Institut National de la Recherche Agronomique (INRA)-Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut de Recherche pour le Développement (IRD [France-Sud]), SIT program (CRVOI, Réunion Island, France), French Ministry of Health, National Research Foundation of South Africa, Department of Mathematics and Applied Mathematics, and Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut National de la Recherche Agronomique (INRA)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])
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Mosquitocontrol ,Insect ,L73 - Maladies des animaux ,01 natural sciences ,010305 fluids & plasmas ,Dynamique des populations ,media_common ,Mathematics ,Expérimentation ,0303 health sciences ,education.field_of_study ,biology ,U10 - Informatique, mathématiques et statistiques ,Contrôle de maladies ,Sterile insect technology ,Compartmental modeling ,Anopheles ,Mosquito control ,Attraction ,Épidémiologie ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Biological system ,Modèle mathématique ,media_common.quotation_subject ,Population ,Dynamical system ,Stability (probability) ,03 medical and health sciences ,Solution of equations ,Environmental safety ,Control theory ,Modelling and Simulation ,Stability theory ,0103 physical sciences ,Monotone operators ,Applied mathematics ,Population animale ,0101 mathematics ,education ,030304 developmental biology ,Lutte anti-insecte ,Numerical analysis ,010102 general mathematics ,fungi ,Sterileinsecttechnology ,Stérilisation ,Modèle de simulation ,biology.organism_classification ,[INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation ,Monotone polygon ,[SDE.BE]Environmental Sciences/Biodiversity and Ecology - Abstract
ACL-12-43; International audience; Modelling the biomechanics of growing trees is a non-classical problem, as the usual framework of structural mechanics does not take into account the evolution of the domain geometry due to growth processes. Incremental approaches have been used in rod theory to bypass this problem and to model the addition of new material points on an existing deformed structure. However, these approaches are based on the explicit time numerical algorithm of an unknown continuous model, and thus, the accuracy of the numerical results obtained cannot be analysed. A new continuous space-time formulation has been recently proposed to model the biomechanical response of growing rods. The aim of this paper is to discretise the corresponding non-linear system of partial differential equations and the linearised system in order to compare the numerical results with analytical solutions of the linearised problem. The finite element method is implemented to compute the space boundary problem and different time integration schemes are considered to solve the associated initial value problem with a special attention to the forward Euler method which is the analogue of the previously used incremental approach. The numerical results point out that the accuracy of the time integration schemes strongly depends on the value of the parameters. The forward Euler method may present slow convergence property and errors with significant orders of magnitude. Nevertheless, attention must be paid to implicit methods since, for specific values of the parameters and large time steps, they may lead to spurious solutions that may come from numerical instabilities. Hence, the second order Heun's method is an interesting alternative even if it is more time consuming.
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- 2012
21. INELASTIC INTERACTION OF BOUSSINESQ SOLITONS
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Christo I. Christov and Manuel G. Velarde
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Surface (mathematics) ,Nonlinear system ,Amplitude ,Applied Mathematics ,Modeling and Simulation ,Mathematical analysis ,Phase (waves) ,Initial value problem ,Supersonic speed ,Wave equation ,Engineering (miscellaneous) ,Mathematics ,Sign (mathematics) - Abstract
Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the total pseudoenergy is conserved within 10−10. The latter allows us to claim that the pulses are solitons despite their “aging” (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.
- Published
- 1994
22. Robust Balanced Semi-coarsening AMLI Preconditioning of Biquadratic FEM Systems
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M. Lymbery, S. Margenov, Michail D. Todorov, and Christo I. Christov
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Elliptic operator ,Mathematical optimization ,Basis (linear algebra) ,Preconditioner ,Mathematik ,Piecewise ,Applied mathematics ,Constant (mathematics) ,Anisotropy ,Measure (mathematics) ,Finite element method ,Mathematics - Abstract
In the present study we demonstrate the construction of a robust multilevel preconditioner for biquadratic FE elliptic problems. In the general setting of an arbitrary elliptic operator it is well known that the standard hierarchical basis two‐level splittings for higher order FEM elliptic systems deteriorate with increasing the anisotropy ratio. An alternative approach resulting in a robust hierarchical two‐level splitting of the finite element space of continuos piecewise biquadratic functions involves the semi‐coarsening mesh procedure. This evokes us to analyze the behavior of the constant in the strengthened CBS inequality, which is a quality measure for hierarchical two‐level splittings of the FEM stiffness matrices, for the particular case of balanced semi‐coarsening mesh refinement. We present new theoretical estimates which further we support by numerically computed CBS constants over a rich set of parameters (coarsening factor and anisotropy ratio). An optimal order multilevel algorithm is constructed on the basis of the proven uniform estimates and the theory of the Algebraic MultiLevel Iteration (AMLI) methods. Its total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio
- Published
- 2011
23. On localized solutions of an equation governing Benard-Marangoni convection
- Author
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Christo I. Christov and Manuel G. Velarde
- Subjects
Condensed Matter::Soft Condensed Matter ,Physics::Fluid Dynamics ,Physics ,Convection ,Classical mechanics ,Marangoni effect ,Model equation ,Interfacial stress ,Combined forced and natural convection ,Modelling and Simulation ,Applied Mathematics ,Modeling and Simulation ,Nonlinear Sciences::Pattern Formation and Solitons - Abstract
Provided here is numerical evidence of localized solutions, solitary waves, in a model equation for Benard convection driven by interfacial stresses (Marangoni effect).
- Published
- 1993
24. Comment on 'On a class of exact solutions of the equations of motion of a second grade fluid' by C. Fetecau and J. Zierep (Acta Mech. 150, 135-138, 2001)
- Author
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Christo I. Christov and Ivan C. Christov
- Subjects
Class (set theory) ,Heaviside step function ,Mechanical Engineering ,Computation ,Fluid Dynamics (physics.flu-dyn) ,Computational Mechanics ,FOS: Physical sciences ,Equations of motion ,Derivative ,Physics - Fluid Dynamics ,Term (logic) ,symbols.namesake ,Exact solutions in general relativity ,symbols ,Applied mathematics ,Sine and cosine transforms ,Mathematics - Abstract
In a 2001 article, Fetecau and Zierep [1] considered Stokes' first problem for a second grade (SG) fluid, unaware that it had already been solved (correctly) by Puri [2] in 1984. These authors used the Fourier sine transform to obtain, what they believed to be, the exact solution of the resulting initial-boundary-value problem (IBVP). Unfortunately, due to their incorrect computation of the distributional derivative of the Heaviside function, an omission of a critical term in the subsidiary equation occurred, an elementary mistake (gracefully explained in [3]) apparently not uncommon in the literature. Therefore, the solution given in [1] is, generally speaking, incorrect., 3 pages, 1 figure (2 images); v2 corrects a few typos. The final publication is available at http://www.springerlink.com
- Published
- 2010
25. The Multi-index Mittag-Leffler Functions and Their Applications for Solving Fractional Order Problems in Applied Analysis
- Author
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V. S. Kiryakova, Yu. F. Luchko, Michail D. Todorov, and Christo I. Christov
- Subjects
Partial differential equation ,Integer ,Special functions ,Differential equation ,Orthogonal polynomials ,Calculus ,Applied mathematics ,Function (mathematics) ,Generalized hypergeometric function ,Mathematics ,Fractional calculus - Abstract
During the last few decades, differential equations and systems of fractional order (that is arbitrary one, not necessarily integer) begun to play an important role in modeling of various phenomena of physical, engineering, automatization, biological and biomedical, chemical, earth, economics, social relations, etc. nature. The so‐called Special Functions of Fractional Calculus (SF of FC) provide an important tool of Fractional Calculus (FC) and Applied Analysis (AA). In particular, they are often used to represent the solutions of fractional differential equations in explicit form. Among the most popular representatives of the SF of FC are: the Mittag‐Leffler (ML) function, the Wright generalized hypergeometric function pΨq, the more general Fox H‐function, and the Inayat‐HussainH‐function. The classical Special Functions (called also SF of Mathematical Physics), including the orthogonal polynomials, and the pFq‐hypergeometric functions fall in this scheme as examples of the simpler Meijer G‐function.In ...
- Published
- 2010
26. Parallel Numerical Simulations of Water Reservoirs
- Author
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Pedro Torres, Norberto Mangiavacchi, Michail D. Todorov, and Christo I. Christov
- Subjects
Mathematical optimization ,Discretization ,Water flow ,law ,Linear system ,Scalar (mathematics) ,Projection method ,Applied mathematics ,Navier–Stokes equations ,Finite element method ,LU decomposition ,law.invention ,Mathematics - Abstract
The study of the water flow and scalar transport in water reservoirs is important for the determination of the water quality during the initial stages of the reservoir filling and during the life of the reservoir. For this scope, a parallel 2D finite element code for solving the incompressible Navier‐Stokes equations coupled with scalar transport was implemented using the message‐passing programming model, in order to perform simulations of hidropower water reservoirs in a computer cluster environment. The spatial discretization is based on the MINI element that satisfies the Babuska‐Brezzi (BB) condition, which provides sufficient conditions for a stable mixed formulation. All the distributed data structures needed in the different stages of the code, such as preprocessing, solving and post processing, were implemented using the PETSc library. The resulting linear systems for the velocity and the pressure fields were solved using the projection method, implemented by an approximate block LU factorization...
- Published
- 2010
27. Biomathematics and Interval Analysis: A Prosperous Marriage
- Author
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S. M. Markov, Michail D. Todorov, and Christo I. Christov
- Subjects
Identification (information) ,Mathematical optimization ,Partial differential equation ,Bounded function ,Interval estimation ,Applied mathematics ,Point (geometry) ,Field (mathematics) ,Focus (optics) ,Mathematics ,Interval arithmetic - Abstract
In this survey paper we focus our attention on dynamical bio‐systems involving uncertainties and the use of interval methods for the modelling study of such systems. The kind of envisioned uncertain systems are those described by a dynamical model with parameters bounded in intervals. We point out to a fruitful symbiosis between dynamical modelling in biology and computational methods of interval analysis. Both fields are presently in the stage of rapid development and can benefit from each other. We point out on recent studies in the field of interval arithmetic from a new perspective—the midpoint‐radius arithmetic which explores the properties of error bounds and approximate numbers. The midpoint‐radius approach provides a bridge between interval methods and the “uncertain but bounded” approach used for model estimation and identification. We briefly discuss certain recently obtained algebraic properties of errors and approximate numbers.
- Published
- 2010
28. ON THE PROBLEM OF HEAT CONDUCTION FOR RANDOM DISPERSIONS OF SPHERES ALLOWED TO OVERLAP
- Author
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Christo I. Christov and Konstantin Z. Markov
- Subjects
Matrix (mathematics) ,Field (physics) ,Series (mathematics) ,Truncation ,Applied Mathematics ,Modeling and Simulation ,Mathematical analysis ,Statistical theory ,Thermal conduction ,Transport phenomena ,Virial theorem ,Mathematics - Abstract
We consider a random two-phase medium which represents a matrix containing an array of allowed to overlap spherical inclusions with random radii. A statistical theory of transport phenomena in the medium, on the example of heat propagation, is constructed by means of the functional (Volterra-Wiener) series approach. The functional series for the temperature is rendered virial in the sense that its truncation after the p-tuple term yields results for all multipoint correlation functions of the temperature field that are asymptotically correct to the order np, where n is the mean number of spheres per unit volume. The case p=2 is considered in detail and the needed kernels of the factorial series are found to the order n2. In this way not only the effective conductivity, but also the full statistical solution, i.e., all needed correlation functions, can be expressed in a closed form, correct to the said order.
- Published
- 1992
29. Numerical Solvers for Generalized Algebraic Riccati Equations
- Author
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I. G. Ivanov, R. I. Rusinova, Michail D. Todorov, and Christo I. Christov
- Subjects
Preconditioner ,Iterative method ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Algebraic Riccati equation ,symbols.namesake ,Fixed-point iteration ,Power iteration ,Riccati equation ,symbols ,Applied mathematics ,Newton's method ,Eigendecomposition of a matrix ,Mathematics - Abstract
We consider a new type nonlinear matrix equation. We investigate the existence a positive definite solution and two iterative methods for computing this solution. The first method is the classical Newton procedure and the second is a new Stein iteration. In this paper it is proved that a new Stein iteration has convergence properties to those of the Newton iteration.
- Published
- 2009
30. On the Analysis of the Fedorenko Finite Superelement Method for Simulation of Processes with Small-Scale Singularities
- Author
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M. Galanin, S. Lazareva, Michail D. Todorov, and Christo I. Christov
- Subjects
Mathematical optimization ,Scale (ratio) ,Singularity theory ,Applied mathematics ,Gravitational singularity ,Superelement ,Finite element method ,Mathematics - Abstract
The paper shows the a‐priori error estimates for the Fedorenko finite superelement method in application to physical problems with small‐scale singularities.
- Published
- 2009
31. Stochastic functional expansion in elasticity of heterogeneous solids
- Author
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Konstantin Z. Markov and Christo I. Christov
- Subjects
Continuum mechanics ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Basis function ,Particle suspension ,Condensed Matter Physics ,Mechanics of Materials ,Modeling and Simulation ,Functional expansion ,Volume fraction ,General Materials Science ,SPHERES ,Elasticity (economics) ,Elastic modulus ,Mathematics - Abstract
The Volterra-Wiener functional expansion is employed to the analysis of statistic properties for random heterogeneous solids. For simplicity, the technique is displayed on an elastic suspension of spheres. The basis function in the expansion is chosen as that corresponding to the so-called “perfect disorder” of spheres (PDS), recently introduced by the authors. An infinite hierarchy of equations for the kernels in the expansion is derived whose truncating after the nth equation is shown to yield results for the averaged statistical characteristics which are valid to order c n f , where c f is the volume fraction of the spheres. The kernels for the first and the second approximations, n = 1, 2, are found and related to the displacement fields in an infinite elastic body containing, respectively, one and two spherical inhomogeneities. Within the frame of the so-called singular approximation the overall tensor of elastic moduli for a suspension of perfectly disordered spheres is shown to coincide to the order c 2 f with a formula, earlier obtained by means of the method of the effective field.
- Published
- 1985
32. Stochastic Functional Expansion for Random Media with Perfectly Disordered Constitution
- Author
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Konstantin Z. Markov and Christo I. Christov
- Subjects
Random field ,Hierarchy (mathematics) ,Applied Mathematics ,Mathematical analysis ,Functional equation ,Order (ring theory) ,Basis function ,Term (logic) ,Thermal conduction ,Virial theorem ,Mathematics - Abstract
The random field created by a system of random points which are centers of perfectly disordered equi-sized spheres of a finite radius is introduced and named the PDS-field. The notion of Perfect Disorder is defined statistically correctly taking the cumulants instead of moments to be $\delta $-functions. The Volterra-Wiener functional expansion with the PDS-field as a basis function is considered and a system of orthogonal Wiener functionals is explicitly constructed. The expansion is employed to the problem of specifying the overall conductivity for a random medium containing an array of perfectly disordered spherical inclusions. An infinite hierarchy of coupled equations for the kernels of the Wiener functionals is derived. It is shown that the Wiener expansion with respect to the PDS-field is also a virial one, i.e. the nth order term contributes quantities of order $c^n $, where c is the volume fraction of the inclusions. This allows the full stochastic solution to the problem of heat conduction throu...
- Published
- 1985
33. Orthogonal coordinate meshes with manageable Jacobian
- Author
-
Christo I. Christov
- Subjects
Computational Mathematics ,symbols.namesake ,Computer science ,Applied Mathematics ,Jacobian matrix and determinant ,symbols ,Polygon mesh ,Topology - Published
- 1982
34. Does the stationary viscous flow around a circular cylinder exist for large Reynolds numbers? A numerical solution via variational imbedding
- Author
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Tchavdar T. Marinov, Christo I. Christov, and Rossitza S. Marinova
- Subjects
Recurrence relation ,Variational imbedding ,Applied Mathematics ,Mathematical analysis ,Reynolds number ,Stokes flow ,Euler equations ,Viscous flow around circular cylinder ,Physics::Fluid Dynamics ,Computational Mathematics ,symbols.namesake ,Navier–Stokes equations ,Flow (mathematics) ,symbols ,Operator splitting ,Reynolds-averaged Navier–Stokes equations ,Numerical stability ,Mathematics - Abstract
We propose an approach to identifying the solutions of the steady incompressible Navier–Stokes equations for large Reynolds numbers. These cannot be obtained as initial-value problems for the unsteady system because of the instability of the latter. Our approach consists of replacing the original steady-state problem for the Navier–Stokes equations by a boundary-value problem for the Euler–Lagrange equations for minimization of the quadratic functional of the original equations. This technique is called Method of Variational Imbedding (MVI) and in this case it leads to a system of higher-order partial differential equations, which is solved by means of an operator-splitting method. As a featuring example we consider the classical flow around a circular cylinder which is known to lose stability as early as for Re=40. We find a stationary solution with recirculation zone for Reynolds numbers as large as Re=200. Thus, new information about the possible hybrid flow regimes is obtained.
- Full Text
- View/download PDF
35. Conservation properties of vectorial operator splitting
- Author
-
Tchavdar T. Marinov, Hideaki Aiso, Rossitza S. Marinova, Christo I. Christov, and Tadayasu Takahashi
- Subjects
Stability and convergence of difference schemes ,Conservation law ,Incompressible Navier–Stokes ,Applied Mathematics ,Operator (physics) ,Mathematical analysis ,Finite difference method ,Mathematics::Analysis of PDEs ,Differential operator ,Semi-elliptic operator ,Physics::Fluid Dynamics ,Computational Mathematics ,Incompressible flow ,Conservation properties ,Navier–Stokes equations ,Laplace operator ,Mathematics - Abstract
This work is concerned with the conservation properties of a new vectorial operator splitting scheme for solving the incompressible Navier–Stokes equations. It is proven that the difference approximation of the advection operator conserves square of velocity components and the kinetic energy as the differential operator does, while pressure term conserves only the kinetic energy. Some analytical requirements necessary to be satisfied of difference schemes for incompressible Navier–Stokes equations are formulated and discussed. The properties of the methods are illustrated with results from numerical computations for lid-driven cavity flow.
- Full Text
- View/download PDF
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