Front-like Solutions Research Articles

Micro-combustion modelling with RBF-FD: A high-order meshfree method for reactive flows in complex geometries

New micro-devices, such as unmanned aerial vehicles or micro-robots, have increased the demand of a new generation of small-scale combustion power system that go beyond the energy-density limitations of batteries or fuel cells. The characteristics short residence times and intense heat losses reduce the efficiency of combustion-based devices, a key factor that requires of an acute modelling effort to understand the competing physicochemical phenomena that hamper their efficient operation. With this objective in mind, this paper is devoted to the development of a high-order meshfree method to model combustion inside complex geometries using radial basis functions-generated finite differences (RBF-FD) based on polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly). In our model, the combustion chamber of a micro-rotary engine is simulated by a system of unsteady reaction-diffusion equations coupled with a steady flow passing a bidimensional stenotic channel of great slenderness. The conversion efficiency is characterized by identifying the different combustion regimes that emerged as a function of the ignition point. We show that PHS+poly based RBF-FD is able to achieve high-order algebraic convergence on scattered node distributions, enabling for node refinement in key regions of the fluid domain. This feature makes it specially well adapted to integrate problems in irregular geometries with front-like solutions, such as reactive fronts or shock waves. Several numerical tests are carried out to demonstrate the accuracy and effectiveness of our approach.

Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on ℝ

Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on ℝ

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.

Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity

The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates.

Transition fronts for inhomogeneous monostable reaction–diffusion equations via linearization at zero

We prove the existence of transition fronts for a large class of reaction–diffusion equations in one dimension, with inhomogeneous monostable reactions. We construct these as perturbations of corresponding front-like solutions to the linearization of the PDE at u = 0. While a close relationship between the solutions to the two PDEs is well known and has been exploited for KPP reactions (and our method is an extension of such ideas from Zlatoš A 2012 (J. Math. Pure Appl. 98 89–102)), to the best of our knowledge this is the first time such an approach has been used in the construction and study of fronts for non-KPP monostable reactions.

Bifurcation analysis of nonlinear reaction–diffusion problems using wavelet-based reduction techniques

Using a computational method for numerical homogenization, we perform the coarse-scale bifurcation analysis of nonlinear reaction–diffusion problems in both uniform and spatially varying media. The method is based on wavelet decomposition and projection of the differential equation on coarse scale wavelet spaces. The approach is capable of capturing turning points and pitchfork bifurcations of sharp, front-like solutions at the coarse level.

Pattern formation in homogeneous reactor models

AbstractThis work analyzes pattern formation mechanisms in the homogeneous model of a fixed catalytic bed for reactions with oscillatory kinetics. Two cases are analyzed: a nonadiabatic reactor with a continuous mass‐supply (either by a preceding reaction or via a membrane wall), and a simple adiabatic or cooled reactor. In the former case, the system may reach asymptotic space‐independent solutions, and when bistability of such solutions exists fronts may be established. Stationary or oscillatory front solutions, oscillatory states that sweep the whole surface, or excitation fronts may be realized then and the reactor behavior can be predicted from the sequence of phase planes spanned by the reactor. In an adiabatic reactor, fronts are formed only for sufficiently small Pe numbers, but these frontlike solutions do not separate different steady states. The patterns that can be realized in this case are quite similar to those in the previous case. The reactor behavior can be predicted by the sequence of phase planes spanned by the reactor, using an approximate finite difference presentation.

Modulated modulations approach to the loss of stability of periodic solutions for the degenerate Ginzburg-Landau equation

The degenerate Ginzburg-Landau equation gives a description of patterns which arise in the case of weakly unstable PDEs with a unbounded spatial direction when the Landau constant (characterizing the influence of the nonlinearity) is small. This equation possesses a family of periodic solutions, moreover there exists a band of stable periodic solutions among them. We give the full description of the possible behaviour of the system just outside this stable band. This is done through derivation of the so-called modulated modulation equations. We also study some solutions of these equations among which stationary periodic and heteroclinic solutions, moving solitons, cnoidal waves and front-like solutions are found.

APPLICATION OF AN OPERATOR-SPLITTING ALGORITHM FOR ADVECTION-DIFFUSION PROBLEMS

An operator-splitting method for solving convective heat and mass transfer equations is described. The method is shown to be equally accurate at low and high Pfelet numbers. At high Pfelet numbers the frontlike solutions are seen to be resolved without excessive damping or oscillations. This method has been tested against analytical solutions of model one-dimensional transient problems and a steady two-dimensional problem to investigate grid orientation effects.

Mathematical modeling of the excitation process in myocardial tissue: influence of fiber rotation on wavefront propagation and potential field

In our macroscopic model the heart tissue is represented as a bidomain coupling the intra- and extracellular media. Owing to the fiber structure of the myocardium, these media are anisotropic, and their conductivity tensors have a principal axis parallel to the local fiber direction. A reaction-diffusion system is derived that governs the distribution and evolution of the extracellular and transmembrane potentials during the depolarization phase of the heart beat. To investigate frontlike solutions, the system is rescaled and transformed into a system dependent on a small parameter. Subsequently a perturbation analysis is carried out that yields zero- and first-order approximations called eikonal equations. The effects of the transmural fiber rotation on wavefront propagation and the corresponding potential field, elicited by point stimulations, are investigated by means of numerical simulations.