Redefined cubic B-splines collocation method for solving convection–diffusion equations

In this work, we discuss collocation method based on redefined cubic B-splines basis functions for solving convection–diffusion equation with Dirichlet’s type boundary conditions. Stability of this method has been discussed and shown that it is unconditionally stable. The developed method is tested on various problems and the numerical results are reported in tabular form. The computed results are compared wherever possible with those already available in literature. The method is shown to work for Péclet number ? 10. Easy and economical implementation process is the strength of it. This method can be easily extended to handle non-linear convection–diffusion partial differential equations.     

A study of isogeometric analysis for scalar convection–diffusion equations

Isogeometric analysis (IGA), in combination with the streamline upwind Petrov–Galerkin (SUPG) stabilization, is studied for the discretization of steady-state convection–diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solutions are much more accurate than those obtained with the finite element method, both in terms of the size of spurious oscillations and of the sharpness of layers.     

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).     

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.     

On numerical methods and error estimates for degenerate fractional convection–diffusion equations

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods—even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations.     

A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.     

A posteriori error estimates for subgrid viscosity stabilized approximations of convection–diffusion equations

We derive a posteriori error estimates for subgrid viscosity stabilized finite element approximations of convection–diffusion equations in the high Péclet number regime. Two estimators are analyzed: an asymptotically robust one and a fully robust one with respect to the Péclet number. Numerical results on test cases with boundary layers or internal layers show that the asymptotically robust estimator can be used to construct adaptive meshes.     

An error estimate for the finite difference approximation to degenerate convection–diffusion equations

We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection?Cdiffusion equations in one space dimension, and prove an L ¹ error estimate. Precisely, we show that the ${L^1_{\rm{loc}}}$ difference between the approximate solution and the unique entropy solution converges at a rate ${\mathcal{O}(\Delta x^{1/11})}$ , where ${\Delta x}$ is the spatial mesh size. If the diffusion is linear, we get the convergence rate ${\mathcal{O}(\Delta x^{1/2})}$ , the point being that the ${\mathcal{O}}$ is independent of the size of the diffusion.     

A systematic approximation for the equations governing convection–diffusion in a porous medium

In order to take into account thermal effects in flows through porous media, one makes ad hoc modifications to Darcy’s equation by appending a term that is similar to the one that is obtained in the Oberbeck–Boussinesq approximation for a fluid. In this short paper we outline a systematic procedure for obtaining an Oberbeck–Boussinesq type of approximation for the flow of a fluid through a porous medium. In addition to establishing the appropriate equation for a flow governed by Darcy’s equation, we proceed to obtain the approximations for flows governed by equations due to Forchheimer and Brinkman.     

Systems of reaction–convection–diffusion equations invariant under Galilean algebras

Systems, which are invariant under Galilean algebras (with and without mass operator), and their main extensions by operators of scale and projective transformations are selected from a class of systems of reaction–convection–diffusion equations for two-dimensional vector field with one space variable.     

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.     

Superconvergence of finite element method for singularly perturbed convection–diffusion equations in 1D

In this article, a singularly perturbed convection–diffusion equation is solved by a linear finite element method on a Shishkin mesh. By means of an analysis exploiting symmetries in the convective term of the bilinear form, a new superconvergence rate, which improves the existing result, is obtained.     

On one-dimensional forward–backward diffusion equations with linear convection and reaction

We study the initial-Neumann boundary value problem for a class of one-dimensional forward–backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to contain two forward-diffusion phases. We prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. Furthermore, we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant. The results are new even for the pure forward–backward diffusion problem without convection and reaction. Our primary approach relies on a combination of the convex integration and Baire's category methods for a related nonlocal differential inclusion.