Stability of solutions to extremum problems for the nonlinear convection–diffusion–reaction equation with the Dirichlet condition

The solvability of the boundary value and extremum problems for the convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of substances is proven. The role of the control in the extremum problem is played by the boundary function in the Dirichlet condition. For a particular reaction coefficient in the extremum problem, the optimality system and estimates of the local stability of its solution to small perturbations of the quality functional and one of specified functions is established.     

Stability of solutions of control problems for the convection–diffusion–reaction equation with a strong nonlinearity

We consider a boundary control problem for the stationary convection–diffusion–reaction equation in which the reaction constant depends on the concentration of matter in such a way that the equation has a fifth-order nonlinearity. We prove the solvability of the boundary value problem and an extremal problem, derive an optimality system, and analyze it to derive estimates for the local stability of the solution of the extremal problem under small perturbations of both the performance functional and one of the given functions.     

Analysis of a BDF–DGFE scheme for nonlinear convection–diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection–diffusion equation. We employ a combination of the discontinuous Galerkin finite element method for the space semi-discretization and the k-step backward difference formula for the time discretization. The diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the discrete L ^∞(L ²)-norm and the L ²(H ¹)-seminorm with respect to the mesh size h and time step τ for k = 2,3. Numerical examples verifying the theoretical results are presented. This work is a part of the research project MSM 0021620839 financed by the Ministry of Education of the Czech Republic and was partly supported by the Grant No. 316/2006/B-MAT/MFF of the Grant Agency of the Charles University Prague. The research of M. Vlasák was supported by the project LC06052 of the Ministry of Education of the Czech Republic (Jindřich Nečas Center for Mathematical Modelling).     

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”- and “DG”-norm formed by the “L ²(H ¹)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.     

On the solutions of time-fractional reaction–diffusion equations

In this paper, a new application of generalized differential transform method (GDTM) has been used for solving time-fractional reaction–diffusion equations. To illustrate the reliability of the method, some examples are provided.     

Approximate analytical solutions of reaction–diffusion equations with exponential source term: Homotopy perturbation method (HPM)

In this letter, the solutions of some nonlinear differential equations have been obtained by means of the homotopy perturbation method (HPM). Applications of the homotopy method to some nonlinear reaction–diffusion equations with exponential source term show rapid convergence of the sequence constructed by this method to the exact solutions.     

Maximum principle in linear finite element approximations of anisotropic diffusion–convection–reaction problems

A mesh condition is developed for linear finite element approximations of anisotropic diffusion–convection–reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition requires that the mesh be simplicial and \(\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)\) -nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix, where \(h\) denotes the mesh size and \(\varvec{b}\) and \(c\) are the coefficients of the convection and reaction terms. In two dimensions, the condition can be replaced by a weaker mesh condition (an \(\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)\) perturbation of a generalized Delaunay condition). These results include many existing mesh conditions as special cases. Numerical results are presented to verify the theoretical findings.     

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion are investigated. The direct problem consists in finding a solution of the corresponding boundary value problem for given data on the boundary of the domain of the independent variable. The peculiarity of the direct problem consists in the inhomogeneity and irregularity of mixed boundary data. Solvability and stability conditions are specified for the direct problem. The inverse boundary value problem consists in finding some traces of the solution of the corresponding boundary value problem for given standard and additional data on a certain part of the boundary of the domain of the independent variable. The peculiarity of the inverse problem consists in its ill-posedness. Regularizing methods and solution algorithms are developed for the inverse problem.     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Asymptotic profiles of solutions to convection–diffusion equations

The large time behavior of zero-mass solutions to the Cauchy problem for the convection–diffusion equation $\text{u}{}_{\mathrm{t}}\text{?u}{}_{\mathrm{xx}}\text{+(|u|}{}^{\mathrm{q}}\text{)}{}_{\mathrm{x}}\text{=0,}\text{u(x,0)=u}{}_0\text{(x)}$ is studied when q>1 and the initial datum u₀ belongs to $\text{L}{}^1\text{(}\text{R}\text{,(1+|x|)}\text{d}\text{x)}$ and satisfies $\text{∫}{}_{\mathrm{<mtext>R</mtext>}}\text{u}{}_0\text{(x)}\text{d}\text{x=0}$. We provide conditions on the size and shape of the initial datum u₀ as well as on the exponent q>1 such that the large time asymptotics of solutions is given either by the derivative of the Gauss–Weierstrass kernel, or by a self-similar solution of the equation, or by hyperbolic N-waves. To cite this article: S. Benachour et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Global existence and blowing-up of solutions to a class of coupled reaction–convection–diffusion systems

This paper concerns the global existence and blowing-up of solutions to the homogeneous Neumann problem of a coupled reaction–convection–diffusion system. The critical Fujita curve is determined and blowing-up theorem of Fujita type is established. An interesting phenomenon is that the critical Fujita curve even could be the infinite due to the convection.     

Traveling wave solutions of advection–diffusion equations with nonlinear diffusion

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c∈]_c?,+∞[$c\in ]c_{?},+\infty [$, where _c?>0$c_{?}>0$ is explicitly computed but may not be optimal. We also prove that a free boundary hypersurface separates a region where u=0$u=0$ and a region where u>0$u>0$, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u>0$u>0$, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.     

Bifurcation of positive solutions to scalar reaction–diffusion equations with nonlinear boundary condition

The bifurcation of non-trivial steady state solutions of a scalar reaction–diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.     

Hybridized schemes of the discontinuous Galerkin method for stationary convection–diffusion problems

For stationary linear convection–diffusion problems, we construct and study a new hybridized scheme of the discontinuous Galerkin method on the basis of an extended mixed statement of the problem. Discrete schemes can be used for the solution of equations degenerating in the leading part and are stated via approximations to the solution of the problem, its gradient, the flow, and the restriction of the solution to the boundaries of elements. For the spaces of finite elements, we represent minimal conditions responsible for the solvability, stability, accuracy, and superconvergence of the schemes. A new procedure for the post-processing of solutions of HDG-schemes is suggested.     

Alternating triangular schemes for convection–diffusion problems

Explicit–implicit approximations are used to approximate nonstationary convection–diffusion equations in time. In unconditionally stable two-level schemes, diffusion is taken from the upper time level, while convection, from the lower layer. In the case of three time levels, the resulting explicit–implicit schemes are second-order accurate in time. Explicit alternating triangular (asymmetric) schemes are used for parabolic problems with a self-adjoint elliptic operator. These schemes are unconditionally stable, but conditionally convergent. Three-level modifications of alternating triangular schemes with better approximating properties were proposed earlier. In this work, two- and three-level alternating triangular schemes for solving boundary value problems for nonstationary convection–diffusion equations are constructed. Numerical results are presented for a two-dimensional test problem on triangular meshes, such as Delaunay triangulations and Voronoi diagrams.     

Nitsche-mortaring for singularly perturbed convection–diffusion problems

In the present paper we analyse a finite element method for a singularly perturbed convection–diffusion problem with exponential boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented.