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With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.

We study the large time behaviour of entropy solutions of the Cauchy problem for a possibly degenerate nonlinear diffusion equation with a nonlinear convection term. The initial function is assumed to have bounded total variation. We prove the convergence of the solution to the entropy solution of a Riemann problem for the corresponding first-order conservation law.

Spatially extended stationary and traveling states in the strongly nonlinear regime of convection in layers of binary fluid mixtures heated from below are described by a few-mode-model. It is derived from the proper hydrodynamic balance equations including experimentally relevant boundary conditions with a non-standard Galerkin approximation that uses numerically obtained, symmetry decomposed modes. Properties of the model are elucidated and compared with full numerical solutions of the field equations.

This paper focuses on the problem of the heat-induced moisture movement in a porous medium. In particular, the paper addresses the nonlinear convection-diffusion processes arising from such moisture transport under steady thermal gradients. The character of the governing differential equation is examined in relation to its solution structure. Conclusions with regard to the residual moisture distribution within an annular porous medium are established.

The scalar nonlinear convection-diffusion equationis considered, for given initial data and zero Dirichlet boundary conditions, in a smooth bounded domain Ω⊂ℝn. The homogeneous viscous Burgers' equation in one dimension is well-known to possess a unique, exponentially attracting equilibrium. These properties are shown to be preserved in the generalisation considered. Furthermore, the equilibrium is shown to be bounded in the maximum norm independently of the function a. The main methods used are maximum principles, and a variational method due to Stampacchia.

Nonlinear convection in a porous medium and rotating about vertical axis is studied in this paper. An upper bound to the heat flux is calculated by the method initiated first by Howard [6] for the case of infinite Prandtl number.ForTa≪0(1), the rotational effect is not significant. For0(1)≪Ta≪0(RlogR), the Nusselt number decreases with increasingTafor a given Rayleigh numberR. The flow has always a finite number of modes, but with increasingTain this region, the number of modes decreases. The functional dependence of the Nusselt number onRandTais found to have discontinuities as the number of modesN*reduces toN*−1. For0(RlogR)≪Ta≪0(R), the Nusselt number is proportional toRTa(logRTa). The stabilizing effect of rotation is so strong that the optimal solution has left with only one horizontal mode. ForTa=0(R), the Nusselt number becomes0(1)and the convection is inhibited entirely by rotation forTa>1π2R.

Undoped and strongly photoexcited semiconductor superlattices with field-dependent recombination behave as excitable or oscillatory media with spatially discrete nonlinear convection and diffusion. Infinitely long, dc-current-biased superlattices behaving as excitable media exhibit wave fronts with increasing or decreasing profiles, whose velocities can be calculated by means of asymptotic methods. These superlattices can also support pulses of the electric field. Pulses moving downstream with the flux of electrons can be constructed from their component wave fronts, whereas pulses advancing upstream do so slowly and experience saltatory motion: they change slowly in long intervals of time separated by fast transitions during which the pulses jump to the previous superlattice period. Photoexcited superlattices can also behave as oscillatory media and exhibit wave trains., Comment: 21 pages, 12 figures

The nonlinear effects on the convection-diffusion equation in the presence of absorption are addressed. The governing equations are derived and an exact similarity solution is found and discussed. The nonlinear effects on the concentration flux and concentration distributions for the case of an instantaneous point source solution are illustrated. The conditions for the existence of diffusive waves in the nonlinear convection-diffusion mechanism in the presence of absorption are also shown.

In this work stabilized higher-order finite element approximations of convection-diffusion-reactions models with nonlinear reaction mechanisms are studied. Streamline upwind Petrov–Galerkin (SUPG) stabilization together with anisotropic shock-capturing as an additional stabilization in crosswind-direction is used. The parameter design of the scheme is described precisely and error estimates are provided. Theoretical results are illustrated by numerical computations. The work extends former investigations for linear problems to more realistic nonlinear models.

This paper is concerned with error estimates in L2(H1)- and L∞(L2)-norm of the discontinuous Galerkin finite element method applied to the space semidiscretization of nonlinear nonstationary convection-diffusion problems. We discuss the discontinuos Galerkin method on shape regular meshes, which can be either conforming or nonconforming with hanging nodes. The main goal is to show that the results obtained under restrictive assumptions on the nonconformity of the meshes can be improved by using computational grids with less limiting properties.

In this paper the discontinuous Galerkin finite element method is used for the space-semidiscretization of a nonlinear nonstationary convection-diffusion problem in three dimensions. As in practical computations integrals appearing in the forms defining the approximate solution are evaluated with the use of quadrature formulae, the effect of numerical integration in the method is studied. An estimate of the error caused by the numerical integration is presented and it is shown which quadrature formulae guarantee preservation of the accuracy of the method with exact integration.

We investigate the origin of the experimentally observed multifractal scaling of vehicular traffic flows by studying a hydrodynamic model of traffic. We first extend and apply the formalism of generalized Hurst exponents H(q) to the case of random walkers that not only diffuse but rather also undergo nonlinear convection due to interactions with other walkers. We recover analytically, as expected, that H(q) equals 1/2 for a single random walker starting at the origin whose probability density function satisfies Burger's equation. Despite this result for a single walker, we find that for a collection of nonlinearly convecting diffusive particles, transient effects can give rise to multiscaling at given time scales for many initial conditions. In the Lighthill-Whitham-Richards hydrodynamic model of traffic, this multiscaling effect becomes more prominent for smaller diffusion constants and larger speed limits. We discuss the relevance of these findings for the realistic scenario of traffic that flows from small roads to large highways and vice versa, where transient effects can be expected to play a significant role.

In this paper we construct a characteristic difference method for two-dimensional nonlinear convection-diffusion problems. These problems arise in the modeling of fluid flow and transport in porous media, for example. The method is analysed mathematically and an error bound is derived. Convergence of the method can be achieved under a milder restriction in the temporal stepsize and spatial stepsize than those required by existing methods.

We investigate convection in a fluid channel uniformly heated from below and rotating about a vertical axis. When the width of the channel is moderate, convective instabilities are characterized by two three-dimensional traveling waves having the same frequency and wave number but traveling in opposite directions with different spatial structures. This Rapid Communication demonstrates that neither the progradely nor the retrogradely traveling wave is physically realizable in the vicinity of the instability threshold. The nature of convection is marked by nonlinear interactions of the two oppositely traveling three-dimensional waves which interfere strongly, leading to either vacillating or stationary convective flows.

International audience; no abstract

A distributed parameter model of a nonlinear two-dimensional convective system is formulated. Proper orthogonal decomposition is used to construct a reduced order state-space model of the system. Open-loop simulations of the full and reduced models are compared to demonstrate the validity of the reduced model. A linear quadratic regulator control problem is formulated for the system under boundary control. Control efiectiveness is demonstrated in reduced and full order simulations.

A Cauchy problem for a nonlinear convection-diffusion equation with periodic rapidly oscillating coefficients is studied. Under the assumption that the convection term is large, it is proved that the limit (homogenized) equation is a nonlinear diffusion equation which shows dispersion effects. The convergence of the homogenization procedure is justified by using a new version of a two-scale convergence technique adapted to rapidly moving coordinates.

This paper is concerned with the global stability of boundary layer solutions, together with the corresponding convergence rates, to the initial-boundary value problem for one-dimensional damped wave equation with a nonlinear convection term in the half space R+.

A large part of this book has been concerned with an up to date account of developments in the application of generalized energy methods in nonlinear convection problems. If any relevant literature has been omitted this is entirely unintentional. In this concluding chapter we give a brief indication of some areas where other generalized energy techniques are finding novel applications and we also briefly explore possible future uses of the energy method.

We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: satisfying a specific decay rate: with and . Here and . Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term defined on the half line .

The effect of horizontal magnetic field on the onset of three-dimensional convection in a horizontal fluid layer is studied. It is found that the two-dimensional solutions are unstable to three-dimensional disturbances. A detailed bifurcation study is reported.

Formulations which complement the streamline-upwind/Petrov-Galerkin procedure are presented. These formulations minimize the oscillations about sharp internal and boundary layers in convection-dominated and reaction-dominated flows. The proposed methods are tested on various single- and multi-component transport problems.

A theoretical study is made of the effect of geometrical imperfections on the formation of weakly nonlinear convection in a shallow two-dimensional cavity uniformly heated from below. If the sidewalls of the cavity are not quite vertical then convective rolls first appear near these walls and subsequently spread inwards to the centre of the cavity as the Rayleigh number is increased. If the horizontal surfaces are not quite parallel then the major effect is a lateral modulation of the rolls due to a combination of the misalignment of the horizontal surfaces and the presence of the sidewalls. As an interesting special case, the solution for a cavity of uniformly sloping base is presented and asymptotic methods are found to provide a remarkably accurate prediction for the critical Rayleigh number as a function of angle of inclination.

The study reported has considered several alternatives to the widely used upwind-differencing scheme with the aim of identifying which (if any) could be regarded as the most suitable for inclusion in a general-purpose solving procedure for elliptic flows. The representations adopted are the ‘skewupwind’, the ‘quadratic-upwind’ and ‘locally-analytic’ treatments. All these discretizations gave more accurate results than standard upwind differencing though as implemented the ‘locally analytic’ scheme was not always stable. Overall, the ‘quadratic-upwind’ approximation emerged as the most satisfactory of the discretizations investigated.

Nonhysteretic infiltration in nonswelling soil is modelled by a nonlinear convection- diffusion equation. A parameter estimation scheme for the spatially varying soil water diffusivity is developed and its convergence is proved. A numerical simulation is performed to show the theoretical results.

We here investigate an existence and uniqueness of the nontrivial, nonnegative solution of a nonlinear ordinary differential equation: ( f m ) ″ + β r f ′ + α f + σ ( f q ) ′ = 0 satisfying a specific decay rate: lim r → ∞ r α / β f ( r ) = 0 with α = 1 / ( 2 q − m − 1 ) and β = ( q − m ) / ( 2 q − m − 1 ) . Here m > 1 , m q m + 1 and σ = 1 or −1. Such a solution arises naturally when we study a very singular solution for a slow diffusion equation with nonlinear convection: u t = ( u m ) x x + ( u q ) x defined either on the whole real line or on the half line ( − ∞ , 0 ] .

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We study the initial behavior of the fronts (for interfaces) generated by the solutions of the equation ut = (um)xx + b(uλ)x, where m, b, λ > 0 are real numbers. We prove a mass comparison principle that allows us to give necessary and sufficient conditions in order to have waiting time at the fronts. Different regions in the (λ, m) parameter space must be introduced leading to answer of very different nature.

Recent work (Hunter & Riahi 1975) on nonlinear convection in a rotating fluid is extended to a multi-modal regime. The schematic multi-boundary-layer method of Busse (1969) and the upper-bound technique of Howard (1963) are used to obtain upper bounds on the Nusselt number N . It is shown that there are infinitely many modes in the range $Ta \ll R^{\frac{3}{2}}$ , where Ta is the Taylor number and R is the Rayleigh number, and different types of mode optimize N in different regions of the parameter space ( R, Ta ). While the optimal N is independent of Ta for Ta [Lt ] R , it is found that it increases with Ta in $R \ll Ta \ll (R \log R)^{\frac{4}{3}}$ and decreases as Ta increases in $(R \log R)^{\frac{4}{3}} \ll Ta \ll R^{\frac{3}{2}}$ , and that the functional dependence of the optimal N on R and Ta is continuous (within a logarithmic term) throughout the region of R, Ta space.