Diffusion Models with Localized Reactions

6. An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains
by Pillay, M. Ward, A. Peirce, and T. Kolokolnikov
SIAM J. on Multiscale Modelling and Simulation, V8, No. 3, pp 803-835, 2010.


Abstract: The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded two-dimensional domain that contains N small non-overlapping absorbing windows on its boundary. The reciprocal of the MFPT of this narrow escape problem has wide applications in cellular biology where it may be used as an effective rst order rate constant to describe, for example, the nuclear export of messenger RNA molecules through nuclear pores. In the asymptotic limit where the absorbing patches have small measure, the method of matched asymptotic expansions is used to calculate the MFPT in an arbitrary two dimensional domain with smooth boundary. The theory is extended to treat the case where the boundary of the domain is piecewise smooth. The asymptotic results for the MFPT depend on the surface Neumann Green's function of the corresponding domain and its associated regular part. The known analytical formulae for the surface Neumann Green's function for the unit disk and the unit square provide explicit asymptotic approximations to the MFPT for these special domains. For an arbitrary two-dimensional domain with a smooth boundary, the asymptotic MFPT is evaluated by developing a novel boundary integral method to numerically calculate the required surface Neumann Green's function.
5. The blowup property of solutions to some Diffusion Equations with Localized Nonlinear Reactions
by . J. Chadam, A. Peirce, and H-M. Yin
 Journal of Mathematical Analysis and Applications, Vol. 169, No. 2, 313-328, 1992.

Abstract: In this paper we investigate the blowup property of solutions to the equation u_t = \Delta u+f(u(x0,t) where x0 is a fixed point in the
domain. We show that under certain conditions the solution blows up in finite time. Moreover, we prove that the set of all blowup points is the whole region. Furthermore, the growth rate of solutions near the blowup time is also derived. Finally, the results are generalized to the following nonlocal reaction-diffusion equation : u_t = \Delta u + \int_{\Omega}f(u)dx.
4. Convergence properties of a class of boundary element approximations to linear diffusion problems with localized nonlinear reactions
by A. Peirce, A. Askar and H. Rabitz
 Numerical Methods for Partial Differential Equations, Vol. 6, pp.75-108, 1990.

Abstract: We consider a boundary element (BE) Algorithm for solving linear diffusion desorption problems with localized nonlinear reactions. The proposed BE algorithm provides an elegant representation of the effect of localized nonlinear reactions, which enables the effects of arbitrarily oriented defect structures to be incorporated into BE models without having to perform severe mesh deformations. We propose a one-step recursion procedure to advance the BE solution of linear diffusion localized nonlinear reaction problems and investigate its convergence properties. The separation of the linear and nonlinear effects by the boundary integral formulation enables us to consider the convergence properties of approximations to the linear terms and nonlinear terms of the boundary integral equation separately. For the linear terms we investigate how the degree of piecewise polynomial collocation in space and the size of the spatial mesh relative to the time step affects the accumulation of errors in the one-step recursion scheme. We develop a novel convergence analysis that combines asymptotic methods with Lax's Equivalence Theorem. We identify a dimensionless meshing parameter \Theta whose magnitude governs the performance of the one-step BE schemes. In particular, we show that piecewise constant (PWC) and piecewise linear (PWL) BE schemes are conditionally convergent, have lower asymptotic bounds placed on the size of time steps, and which display excess numerical diffusion when small time steps are used. There is no asymptotic bound on how large the tic steps can be-this
allows the solution to be advanced in fewer, larger time steps. The piecewise quadratic (PWQ) BE scheme is shown to be unconditionally
convergent; there is no asymptotic restriction on the relative sizes of the time and spatial meshing and no numerical diffusion. We verify the theoretical convergence properties in numerical examples. This analysis provides useful information about the appropriate degree of spatial piecewise polynomial and the meshing strategy for a given problem. For the nonlinear terms we investigate the convergence of an explicit algorithm to advance the solution at an active site forward in time by means of Caratheodoiy iteration combined with piecewise linear interpolation. We consider a model problem comprising a singular nonlinear Volterra equation that represents the effect of the term in the BE formulation that is due to a single defect. We prove the convergence of the piecewise linear Caratheodory iteration algorithm to a solution of the model problem for as long as such a solution can be shown to exist. This analysis provides a theoretical justification for the use of piecewise linear Caratheodory iterates for advancing the effects of localized
reactions.
3. The effect of defect structures on chemically active surfaces: a continuum approach
by A. P. Peirce and H. Rabitz.
 Phys. Rev. B., Vol. 38, No. 3, pp. 1734-1753, 1988.

Abstract: A continuum approach is used to analyze the effect of defect structures on chemically active surfaces. The model comprises a linear diffusion equation with adsorption and desorption in which the defect structures are represented by nonlinear localized-reaction terms. The issue of multiple steady states and stability is treated, and a novel procedure is outlined that uses conformal mapping to derive stability criteria for these
localized-reaction diffusion equations. This conformal mapping procedure also provides insight into how the various physical processes affect the stability of the system. A class of reactive-trapping models is considered in which defects are assumed to act as sinks of material that ultimately desorbs as a chemical product. Other features included in the model are nonlinear enhanced reactivity with concentration, and saturation effects. The continuum assumption is tested by direct comparison with a discrete reactive-trapping model and found to be a remarkably good approximation, even when the number of inter-defect sites is as low as 20. We investigate the effect of relative defect locations on the balance between the desorptive processes that take place on the surface. The effect of defect locations on desorption is analyzed by considering symmetry-breaking perturbations to the defects in a periodic lattice. Two regimes of desorption are identified depending on the level of adsorption on the surface and the defect spacing, (i) Competitive: Defects that are moved closer by the perturbation compete for material, which reduces the trapping efficiency of the defect lattice and increases the bulk desorption rate; by considering the bulk desorption rate to be a function of the defect locations, we conclude that the situation of equally spaced defects is a local minimum of this function, (ii) Cooperative: Defects that are moved closer by perturbation in this regime act cooperatively to reduce the saturation level locally, which enhances the trapping efficiency of the defect lattice and reduces the bulk desorption rate. In this complex environment of competing physical effects it would be difficult to determine the dominant process without the analysis presented here. In order to determine whether these phenomena persist when the defects undergo finite random perturbations, we solve the continuum equations numerically using the boundary-element technique. The phenomena identified by the small perturbation case do persist when finite defect variations are considered.

2. Modeling the effect of changes in defect geometry on chemically active surfaces by the boundary element technique
by A. P. Peirce and H. Rabitz,
 Surface Science, Vol. 202, pp. 32-57, 1988.


Abstract: The boundary element (BE) technique is used to analyze the effect of defect structures upon desorption processes on two-dimensional chemically active surfaces. The standard BE algorithm for diffusion is modified to incorporate the effects of bulk desorption, and an explicit scheme is proposed for the treatment of the nonlinear equations associated with localized defect structures. The BE algorithm proposed here provides an elegant representation of the effects of localized non-linear reactions which allows arbitrarily oriented defect structures to be modelled without having to perform mesh deformation. A class of trapping reactions is assumed to occur along defect structures, and the effect of changes in defect geometry on the balance between the desorptive processes is explored. A number of interesting competitive/cooperative phenomena are observed to occur for the various shapes of defect geometry, including strong intrinsic competition in circular defect structures that form islands of nearly constant concentration, a redistribution of material along V-shaped defect structures in a way that reflects relative competitiveness of defects on opposite sides of the defect structure, and a reduction of competitiveness for defect distributions that are less regular in shape. The proposed BE algorithm is shown to provide a useful technique for modelling the effect of defect structures on chemically active surfaces.

1. An analysis of the effect of defect structures on catalytic surfaces by the boundary element technique
by A. P. Peirce and H. Rabitz,
 Surface Science, Vol. 202, pp. 1-31, 1988.

Abstract: The boundary element (BE) technique is used to analyze the effect of defects on one-dimensional chemically active surfaces. The standard BE algorithm for diffusion is modified to include the effects of bulk desorption by making use of an asymptotic expansion technique to evaluate influences near boundaries and defect sites. An explicit time evolution scheme is proposed to treat the non-linear equations associated with defect sites. The proposed BE algorithm is shown to provide an efficient and convergent algorithm for modelling localized non-linear behavior. Since it exploits the actual Green's function of the linear diffusion-desorption process that takes place on the surface, the BE algorithm is extremely stable.

The BE algorithm is applied to a number of interesting physical problems in which non-linear reactions occur at localized defects. The Lotka-Volterra system is considered in which the source, sink and predator-prey interaction terms are distributed at different defect sites in the domain and in which the defects are coupled by diffusion. This example provides a stringent test of the stability of the numerical algorithm. Marginal stability osculations are analyzed for the Prigogine-Lefever reaction that occurs on a lattice of defects. Dissipative effects are observed for large perturbations to the marginal stability state, and rapid spatial reorganization of uniformly distributed initial perturbations is seen to take place. In another series of examples the effect of defect locations on the balance between desorptive processes on chemically active surfaces is considered. The effect of dynamic pulsing at various time-scales is considered for a one species reactive trapping model. Similar competitive behavior between neighboring defects previously observed for static adsorption levels is shown to persist for dynamic loading of the surface. The analysis of a more complex three species reaction process also provides evidence of competitive behavior between neighboring defect sites. The proposed BE algorithm is shown to provide a useful technique for analyzing the effect of defect sites on chemically active surfaces.