One-dimensional Reaction-diffusion Research Articles

Model of periodic filament formation in laterally coupled gas discharges

A one-dimensional reaction–diffusion model was used to study periodic filament formation in a laterally coupled gas discharge. Our results demonstrate three types of pattern formation (localized solitary filaments, small oscillation and intermediate), which deform into one another subject to a control parameter θ(φa/pd) and a boundary condition (where pd is the pressure–distance and φa is the applied voltage). The work shows the same qualitative dependence as the experimental work of Guikema et al (2000 Phys. Rev. Lett. 85 3817).

Complex dynamics of the formation of spatially localized standing structures in the vicinity of saddle-node bifurcations of waves in the reaction-diffusion model of blood clotting

Local activation in a one-dimensional three-component reaction-diffusion model of blood clotting may lead to a formation of spatially localized standing structures (peaks) via several complex scenarios. In the first scenario, two concentration pulses first propagate from the site of activation, then stop and transform into peaks [Zarnitsina et al., Chaos 11, 57 (2001)]. Here, we examine this scenario, and also describe a different scenario of peak formation. In this scenario, two trigger waves propagate initially in opposite directions away from the site of activation. Then they stop and change direction of propagation toward each other to the activation site, where they interact and form a peak. Both of these scenarios of stable peak formation are observed in the vicinity of saddle-node bifurcation and may be viewed as a memory of the extinct wave modes.

Running pulses of complex shape in a reaction-diffusion model.

In a one-dimensional reaction-diffusion model of an active medium, stable steady-state wave pulses of a new type are described. They are called multihumped because their waveforms contain several maxima of similar size. Presumably, the multihumped pulses arise via a bifurcation at which an unstable trigger wave disappears. The parameter governing this bifurcation is the diffusion coefficient for the model inhibitor. The model is analyzed by varying this parameter to determine the conditions for the emergence of multihumped pulses. The results of this analysis show how their waveform and dynamics of excitation depend on the inhibitor diffusion coefficient.

Perturbative calculation of one-point functions of one-dimensional single-species reaction-diffusion systems

Perturbations around autonomous one-dimensional single-species reaction-diffusion systems are investigated. It is shown that the parameter space corresponding to the autonomous systems is divided into two parts: In one part, the system is stable against the perturbations, in the sense that the largest relaxation time of the one-point functions changes continuously with perturbations. In the other part, however, the system is unstable against perturbations, so that any small perturbation drastically modifies the large-time behavior of the one-point functions.

Cubic autocatalysis with Michaelis–Menten kinetics: semi-analytical solutions for the reaction–diffusion cell

Cubic-autocatalysis with Michaelis–Menten decay is considered in a one-dimensional reaction–diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to sets of transcendental equations. Singularity theory is then used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found by a local stability analysis of the semi-analytical model. An example of a stable limit-cycle is also considered. Comparison with numerical solutions of the governing partial differential equations shows that the semi-analytical solutions are very accurate.

Porous catalyst intraparticle status of parallel, equilibrium-restrained reactions

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction ζ CH 4 and shift reaction ζ CO 2 are fairly in agreement with the experimental values. ζ CH 4 is large, while ζ CO 2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account.

Exploiting ISAT to solve the reaction–diffusion equation

We develop and demonstrate a computationally efficient numerical splitting technique for solving the reaction–diffusion equation. The scheme is based on the Strang splitting technique wherein the portions of the governing equations containing stiff chemical reaction terms are separated from those parts containing the less-stiff transport terms. As demonstrated, the scheme achieves second-order accuracy in space through the use of centred finite differences; second-order accuracy in time is achieved through Strang splitting. To improve greatly the computational efficiency, the pure reaction sub-steps use in situ adaptive tabulation (ISAT) to compute efficiently the reaction mapping while the pure diffusion sub-steps use an implicit Crank–Nicolson finite-difference method. The scheme is applied to an unsteady one-dimensional reaction–diffusion model equation with detailed chemical kinetics. For this test problem, we show spatial and temporal convergence results, the impacts of ISAT and ODE solver error tolerances, and demonstrate computational speed-ups achieved by using ISAT over direct integration.

Stable schemes for partial differential equations: the one-dimensional reaction–diffusion equation

Reaction–diffusion equations are fundamental in modelling several natural phenomena. The numerical schemes used to solve these equations often suffer from numerical stability problems. In this paper, a new type of algorithm to solve the diffusion equation in a stable and explicit manner is extended to the reaction–diffusion equation. The new scheme imposes a fixed value for the stability coefficient below the stability limit, and uses this information in order to determine a new grid. The values of the variables at this grid are then obtained by interpolation from the original grid. The scheme is applied to the linear single kinetic reaction–diffusion equation and to the classical Fisher equation. Different possibilities of extending the new scheme to the reaction–diffusion equations are discussed. It is shown that, for the linear case, including both terms (reaction and diffusion) in the computation of the new grid gives more accurate results and is more correct than just including the diffusion term. To solve the non-linear Fisher equation, a fractional-step method, where the reaction and diffusion terms are solved separately, is chosen. The new scheme provides realistic results when compared with analytic solutions.

Hysteresis in one-dimensional reaction-diffusion systems.

We introduce a simple nonequilibrium model for a driven diffusive system with nonconservative reaction kinetics in one dimension. The steady state exhibits a phase with broken ergodicity and hysteresis which has no analog in systems investigated previously. We identify the main dynamical mode, viz., the random motion of a shock in an effective potential, which provides a unified framework for understanding phase coexistence as well as ergodicity breaking. This picture also leads to the exact phase diagram of the system.

Singularly perturbed differential equations with discontinuous coefficients and concentrated factors

In this paper singularly perturbed one-dimensional reaction–diffusion equations with discontinuous coefficients and concentrated factors are examined. A new formulation which allows us to reduce the solving of general interface problems to the corresponding ones with only discontinuous coefficients is proposed. Then, a decomposition of the solution of the problem with only discontinuous coefficients into regular and singular parts (Shishkin decomposition) is derived. Asymptotic expansion up to fourth-order for the solution in the interfaces is also constructed.

Composition-Space Behavior of Diffusion-Reaction Systems

Species concentrations in homogeneous well-stirred reacting (WSR) systems with multiple time scales mostly access only a small portion (slow manifold) of the permitted composition space. This offers a mathematically sound basis for reducing complex kinetics. Computational investigation of such slow manifolds in inhomogeneous reaction systems undergoing diffusion in addition to chemical reaction is the subject of this paper. We perform simulations of a one-dimensional diffusion–reaction system from random non-premixed initial conditions to study the accessed composition space for a variety of diffusion to reaction time scale ratios. A detailed hydrogen–oxygen chemical kinetic set (6 species, 14 reactions) is used. Our results show that, to first order, the effect of random advection and diffusion is merely to increase the accessed state-space dimensionality over that of the WSR by the number of relevant mixture fractions. Circumstances under which the above simplification is not accurate are also identified.

Unstable trigger waves induce various intricate dynamic regimes in a reaction-diffusion system of blood clotting.

In this work we demonstrate that the unstable trigger waves, connecting stable and unstable spatially uniform steady states, can create intricate dynamic regimes in one-dimensional three-component reaction-diffusion model describing blood clotting. Among the most interesting regimes are the composite and replicating waves running at a constant velocity. The front part of the running composite wave remains constant, while its rear part oscillates in a complex manner. The rear part of the running replicating wave periodically gives rise to new daughter waves, which propagate in the direction opposite the parent wave. The domain of these intricate regimes in parameter space lies in the region of monostability near the region of bistability.

A Mixed-Potential Model to Predict Fuel (Uranium Dioxide) Corrosion within a Failed Nuclear Waste Container

Abstract A mixed-potential model is described to predict the corrosion behavior of used nuclear fuel inside a steel-lined failed Canadian nuclear waste container under anticipated waste vault (repository) conditions. The model accounts for the effects of the alpha radiolysis of water, the precipitation of corrosion products on both the fuel and the carbon steel (CS), and redox reactions between species produced by either radiolysis or corrosion at the fuel surface and by corrosion on the CS liner. The model is based on a series of ten one-dimensional reaction-diffusion equations, each describing the mass-transport, precipitation/dissolution, adsorption/desorption, and redox processes of the ten chemical species included in the model. These equations are solved using finite-difference techniques. A three-layer spatial grid is used, with the two outer layers (of time-varying thickness) representing porous precipitated corrosion products on the uranium dioxide (UO2) and CS surfaces. The middle layer represen...

Static- and dynamical-phase transition in one-dimensional reaction-diffusion systems with boundaries

A general system of particles (of one or several species) on a one dimensional lattice with boundaries is considered. Two general behaviors of such systems are investigated. The stationary behavior of the system, and the dominant way of the relaxation of the system toward its stationary system. Bases on the first behavior, static phase transitions (discontinuous changes in the stationary profiles of the system) are studied. Based on the second behavior, dynamical phase transitions (discontinuous changes in the relaxation-times of the system) are studied. The investigation is specialized on systems in which the evolution equation of one-point functions are closed (the autonomous systems).

Reaction‐Diffusion Processes and Their Connection with Integrable Quantum Spin Chains

AbstractFor Abstract see ChemInform Abstract in Full Text.

Dynamical phase transition in the two-point functions of the autonomous one-dimensional single-species reaction-diffusion systems

The evolution of the two-point functions of autonomous one-dimensional single-species reaction-diffusion systems with nearest-neighbor interaction and translationally-invariant initial conditions is investigated. It is shown that the dynamical phase structure of such systems consists of five phases. As an example, a one-parameter family is introduced which can be in each of these phases.

Exact Finite Dimensional Feedback Control via Inertial Manifold Theory with Application to the Chafee–Infante Equation

A class of nonlinear parabolic partial differential equations is considered, and an exact finite dimensional feedback control law is designed in order to force the systems to behave in a prescribed way. The feedback law is obtained via inertial manifold theory by reducing the system to finite dimensions. The control achieved is exact, as opposed to approximate, as obtained in a previous work. The result is applied to the Chafee–Infante equation, a one-dimensional scalar reaction-diffusion equation, with distributed control.

Analysis of Immersed Interface Difference Schemes for Reaction-diffusion Problems with Singular Own Sources

AbstractThis paper analyzes immersed interface difference schemes for onedimensional reaction-diffusion equations with singular own linear and nonlinear sources. Error bounds in the infinity norm based on the maximum principle are derived. Sharper bounds and a more detailed structure of the error are obtained using the asymptotic error expansion analysis method. Numerical examples confirm the theoretical results.

A quasi-random walk method for one-dimensional reaction–diffusion equations

Probabilistic methods are presented to solve one-dimensional nonlinear reaction–diffusion equations. Computational particles are used to approximate the spatial derivative of the solution. The random walk principle is used to model the diffusion term. We investigate the effect of replacing pseudo-random numbers by quasi-random numbers in the random walk steps. This cannot be implemented in a straightforward fashion, because of correlations. If the particles are reordered according to their position at each time step, this has the effect of breaking correlations. For simple demonstration problems, the error is found to be significantly less when quasi-random sequences are used than when a standard random walk calculation is performed using pseudo-random points.

Pinning stationary planar fronts in diffusion-convection-reaction systems

This paper considers various strategies for controlling a stationary planar front solution, in a rectangular domain with a diffusion-reaction distributed system, by pinning the solution to one or few points and using actuators with the simplest possible spatial dependence. We review previous results obtained for one-dimensional diffusion-reaction (with or without convection) systems, for which we applied two approaches: an approximate model reduction to a form that follows the front position while approximating the front velocity, and linear stability analysis. We apply the same two approaches for the planar fronts. The approximate model reduction allows us to analyze qualitatively various control strategies and to predict the critical width below which the control mode of the one-dimensional system is sufficient. These results are corroborated by linear analysis of a truncated model with the spectral methods representation, using concepts of finite and infinite zeros of linear multidimensional systems.