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.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Some asymptotic limits of reaction–diffusion systems appearing in reversible chemistry

This paper concerns reaction–diffusion systems consisting of three or four equations, which come out of reversible chemistry. We introduce different scalings for those systems, which make sense in various situations (species with very different concentrations or very different diffusion rates, chemical reactions with very different rates, etc.). We show how recently introduced mathematical tools allow to prove that the formal asymptotics associated to those scalings indeed hold at the rigorous level.     

Qualitative properties of pulsating fronts for reaction–advection–diffusion equations in periodic excitable media

In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially.     

Solution with an inner transition layer of a two-dimensional boundary value reaction–diffusion–advection problem with discontinuous reaction and advection terms

Theoretical and Mathematical Physics - We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the...     

Isogeometric analysis in advection–diffusion problems: Tension splines approximation

We present a novel approach, within the new paradigm of isogeometric analysis introduced by Hughes et al. (2005) [6], to deal with advection dominated advection–diffusion problems. The key ingredient is the use of Galerkin approximating spaces of functions with high smoothness, as in IgA based on classical B-splines, but particularly well suited to describe sharp layers involving very strong gradients.     

Dynamical behaviors of a classical Lotka–Volterra competition–diffusion–advection system

This paper deals with a two-species competition model in a homogeneous advective environment, where two species are subjected to a net loss of individuals at the downstream end. Under the assumption that the advection and diffusion rates of two species are proportional, we give a basic classification on the global dynamics by employing the theory of monotone dynamical system. It turns out that bistability does not happen, but coexistence and competitive exclusion may occur. Furthermore, we present a complete classification on the global dynamics in terms of the growth rates of two species. However, once the above assumption does not hold, bistability may occur. In detail, there exists a tradeoff between growth rates of two species such that competition outcomes can shift between three possible scenarios, including competitive exclusion, bistability and coexistence. These results show that growth competence is important to determine dynamical behaviors.     

Asymptotic behavior of reaction–advection–diffusion population models with Allee effect

In this work, a qualitative analysis is carried out for reaction–advection–diffusion (RAD) systems modeling the interactions between two species with Allee effect. In particular, we study different scenarios: mutualism, competition, and a predator–prey relationship in order to investigate the survival or extinction of both populations. Global existence and uniqueness of positive solutions of the proposed RAD problems are demonstrated. Equilibrium states and asymptotic behavior of solutions are obtained using the monotone method and the upper and lower solutions technique. Numerical simulations by a Crank–Nicolson monotone iterative method of the different asymptotic solution dynamics are shown to illustrate our theoretical results.     

Bifurcation and stability of a two-species reaction–diffusion–advection competition model

This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions stemming from the two boundary steady states are obtained as well. In particular, we describe the feature of the coincidence of bifurcating coexistence steady-state solution branches. At the same time, the effect of advection on the stability of the bifurcating solution is also investigated, and our results suggest that the advection term may change the stability. Finally, we point out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov–Schmidt reduction, and bifurcation theory.     

Concentration profile of endemic equilibrium of a reaction–diffusion–advection SIS epidemic model

Cui and Lou (J Differ Equ 261:3305–3343, 2016) proposed a reaction–diffusion–advection SIS epidemic model in heterogeneous environments, and derived interesting results on the stability of the DFE (disease-free equilibrium) and the existence of EE (endemic equilibrium) under various conditions. In this paper, we are interested in the asymptotic profile of the EE (when it exists) in the three cases: (i) large advection; (ii) small diffusion of the susceptible population; (iii) small diffusion of the infected population. We prove that in case (i), the density of both the susceptible and infected populations concentrates only at the downstream behaving like a delta function; in case (ii), the density of the susceptible concentrates only at the downstream behaving like a delta function and the density of the infected vanishes on the entire habitat, and in case (iii), the density of the susceptible is positive while the density of the infected vanishes on the entire habitat. Our results show that in case (ii) and case (iii), the asymptotic profile is essentially different from that in the situation where no advection is present. As a consequence, we can conclude that the impact of advection on the spatial distribution of population densities is significant.     

Spline approximation of advection–diffusion problems using upwind type collocation nodes

A spline collocation method for linear advection–diffusion equations is proposed. The method is based on an operator-dependent collocation grid, and provides stabilized approximated solutions, with respect to the coefficient of the diffusive term, when problems are advection dominated.     

Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem in the case of a balanced nonlinearity

We study a singularly perturbed periodic problem for the parabolic reaction–advection–diffusion equation with small advection. We consider the case in which there exists an internal transition layer under the conditions of balanced nonlinearity. An asymptotic expansion of the solution is constructed. To substantiate this asymptotics, we use the asymptotic method of differential inequalities. The Lyapunov asymptotic stability of the periodic solution is analyzed.     

The spatial behavior of a competition–diffusion–advection system with strong competition

We study a competition–diffusion–advection system for two competitive species inhabiting a spatially heterogeneous environment. We show that they spatially segregate as the interspecific competition rate tends to infinity. Besides, by using a blow up method, we obtain the uniform Hölder bounds for solutions of the system.     

High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C¹$C^1$-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C¹$C^1$-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h⁴,k⁴)$O(h^4\text{,}{k}^4)$. Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C¹$C^1$-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.     

Interface procedures for finite difference approximations of the advection–diffusion equation

We investigate several existing interface procedures for finite difference methods applied to advection–diffusion problems. The accuracy, stiffness and reflecting properties of various interface procedures are investigated.The analysis and numerical experiments show that there are only minor differences between various methods once a proper parameter choice has been made.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Solution of the linear and nonlinear advection–diffusion problems on a sphere

Various linear advection–diffusion problems and nonlinear diffusion problems on a sphere are considered and solved using the direct, implicit and unconditionally stable finite-volume method of second-order approximation in space and time. In the absence of external forcing and dissipation, the method preserves the total mass of the substance and the norm of the solution. The component wise operator splitting allows us to develop the direct (noniterative) and fast numerical algorithm. The split problems in the longitudinal direction are solved using the Sherman-Morrison formula and Thomas algorithm. The direct solution of the split problems in the latitudinal direction requires the use of the bordering method for a block matrix, and the preliminary determination of the solution at the poles. The resulting systems with tridiagonal matrices are solved by the Thomas algorithm. The numerical experiments demonstrate that the method correctly describes the local advection–diffusion processes on the sphere, in particular, through the poles, and accurately simulate blow-up regimes (unlimited growing solutions) of nonlinear combustion, the propagation of nonlinear temperature and spiral waves, and solutions to Gray-Scott reaction–diffusion model.