Absorbing boundary conditions for diffusion equations

Summary. We construct and analyse a family of absorbing boundary conditions for diffusion equations with variable coefficients, curved artifical boundary, and arbitrary convection. It relies on the geometric identification of the Dirichlet to Neumann map and rational interpolation of in the complex plane. The boundary conditions are stable, accurate, and practical for computations. Received December 12, 1992 / Revised version received July 4, 1994     

Absorbing boundary conditions for the linearized shallow water equations

A mixed problem imitating the Cauchy problem for the linearized shallow water equations is considered. This problem is also a mixed problem with perfectly absorbing conditions (cp. [1], [3]). An exact formula for the conditions has been given.     

Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions

In this paper we prove the existence and uniqueness of a weak solution for a non-autonomous reaction-diffusion model with dynamical boundary conditions. After that, a continuous dependence result is established via an energy method, including in particular some compactness properties. Finally, the precedent results are used in order to ensure the existence of minimal pullback attractors in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition. The relation among these families is also discussed.     

Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions

This paper is concerned with some dynamical property of a reaction-diffusion equation with nonlocal boundary condition. Under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function, the asymptotic behavior of the time-dependent solution is characterized in relation to a finite or an infinite set of constant steady-state solutions. This characterization is determined solely by the initial function and it leads to the stability and instability of the various steady-state solutions. In the case of finite constant steady-state solutions, the time-dependent solution blows up in finite time when the initial function in greater than the largest constant solution. Also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain.     

On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions

We give conditions on the nonlinearities of a reaction-diffusion equation with nonlinear boundary conditions that guarantee that any solution starting at bounded initial data is bounded locally around a certain point of the boundary, uniformly for all positive time. The conditions imposed are of a local nature and need only to hold in a small neighborhood of the point .

     

Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions

We study the blow-up behavior for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. The main purpose is to understand how the reaction terms and the absorption terms affect the blow-up properties. We obtain a necessary and sufficient condition for blow-up, derive the upper bound and lower bound for the blow-up rate, and find the blow-up set under certain assumptions.     

Absorbing boundary conditions for electromagnetic wave propagation

In this paper, the theoretical perfectly absorbing boundary condition on the boundary of a half-space domain is developed for the Maxwell system by considering the system as a whole instead of considering each component of the electromagnetic fields individually. This boundary condition allows any wave motion generated within the domain to pass through the boundary of the domain without generating any reflections back into the interior. By approximating this theoretical boundary condition a class of local absorbing boundary conditions for the Maxwell system can be constructed. Well-posedness in the sense of Kreiss of the Maxwell system with each of these local absorbing boundary conditions is established, and the reflection coefficients are computed as a plane wave strikes the artificial boundary. Numerical experiments are also provided to show the performance of these local absorbing boundary conditions

     

Derivation of effective interface conditions for reaction-diffusion equations

We address the derivation of effective interface conditions for reaction-diffusion systems. The considered system is defined in a domain containing a thin layer that shrinks to the interface when its thickness ε tends to zero. The evolution of the system can be written in the form of an energy balance involving an energy and a dissipation functional. Using the Mosco convergence of the dual of the dissipation functional for ε → 0 it is possible to do a limit passage in the energy balance and obtain a limit system that describes the evolution on the interface. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-sided bounds for linked unknown nonlinear boundary conditions of reaction-diffusion

Two enzymes bound at opposite ends of a finite interval affect each other via activation and/or inhibition by their respective products. The local concentrations of the diffusing products, in the vicinity of the other enzyme, determines the rate of production by that enzyme of its product. A mathematical model (cf. Thames and Elster [J. Theor. Biol. 59 (1976), 415–427]) consists of linear diffusion equations coupled through unknown and nonlinear boundary conditions. When the (nonlinear) functions describing the boundary conditions have certain monotone properties it is shown that the boundary values can be found iteratively by means of convergent two sided bounds. Some results for reaction chains involving more that two enzymes are presented.     

Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data

Explicit criteria for the asymptotic stability (or instability) of bifurcating closed orbits are given for a class of abstract evolution equations.     

Open boundary conditions for hyperbolic equations

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10^?2 of the incident wave.     

Radiation boundary conditions for wave-like equations

In the numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary. In this study we construct a sequence of radiating boundary conditions for wave-like equations. We prove that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r^?m?1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature and utility of the boundary conditions.     

Long-time behavior of reaction-diffusion equations with dynamical boundary condition

In this paper, we study the long-time behavior of the reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order. Some asymptotic regularity of the solution has been proved. As an application of the asymptotic regularity results, we can not only obtain the existence of a global attractor A in (H¹(Ω)∩L^p(Ω))×L^q(Γ) immediately, but also can show further that A attracts every L²(Ω)×L²(Γ)-bounded subset with (H¹(Ω)∩L^p+δ(Ω))×L^q+κ(Γ)-norm for any δ,κ∈[0,∞).     

Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition

This paper deals with the blow-up of positive solutions for a nonlinear reaction-diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.     

Absorbing boundary conditions for the wave equation and parallel computing

Absorbing boundary conditions have been developed for various types of problems to truncate infinite domains in order to perform computations. But absorbing boundary conditions have a second, recent and important application: parallel computing. We show that absorbing boundary conditions are essential for a good performance of the Schwarz waveform relaxation algorithm applied to the wave equation. In turn this application gives the idea of introducing a layer close to the truncation boundary which leads to a new way of optimizing absorbing boundary conditions for truncating domains. We optimize the conditions in the case of straight boundaries and illustrate our analysis with numerical experiments both for truncating domains and the Schwarz waveform relaxation algorithm.

     

Integrable boundary conditions for many-component burgers equations

Infinite series of boundary conditions that are consistent with even-order higher symmetries and ensure the integrability of a Burgers type equation are constructed. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 888–901, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-16008 and by the International Science Foundation under grant Nos. MLY00 and RK2000.     

Transfer of boundary conditions for difference equations

It is well-known that the idea of transferring boundary conditions offers a universal and, in addition, elementary means how to investigate almost all methods for solving boundary value problems for ordinary differential equations. The aim of this paper is to show that the same approach works also for discrete problems, i.e., for difference equations. Moreover, it will be found out that some results of this kind may be obtained also for some particular two-dimensional problems.     

Absorbing boundary conditions and optimized Schwarz waveform relaxation

The numerical evaluation of the transforms in the title, and their inverses, is considered, using a variety of decomposition, truncation, and quadrature methods. Extensive numerical testing is provided and an application given to the numerical evaluation of the kernel of a Fredholm integral equation of interest in mixed boundary value problems on wedge-shaped domains. AMS subject classification (2000) 44A15, 65D30, 65R10     

Discontinuous reaction-diffusion equations under discontinuous and nonlocal flux conditions

In this paper, we consider a quasilinear parabolic equation with discontinuous source term in a bounded cylindrical domain under nonlocal and discontinuous flux conditions. Our main goal is to prove the existence of extremal solutions within a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, an abstract fixed-point result in partially ordered sets, compact embeddings, comparison, and truncation techniques.