Asymptotic solution for three-dimensional reaction-diffusion-advection equation with periodic boundary conditions

TL;DR

This paper develops an asymptotic method to approximate solutions of a three-dimensional reaction-diffusion-advection equation with periodic boundary conditions, effectively modeling moving fronts in complex systems like oil combustion.

Contribution

It introduces a novel asymptotic approach for singularly perturbed reaction-diffusion-advection problems in three dimensions, capturing internal layers and transition dynamics.

Findings

High accuracy of asymptotic solutions in numerical examples

Effective modeling of moving fronts in three-dimensional systems

Method applicable to various autowave and wave propagation phenomena

Abstract

In this study, we investigate the dynamics of moving fronts in three-dimensional spaces, which form as a result of in-situ combustion during oil production. This phenomenon is also observed in other contexts, such as various autowave models and the propagation of acoustic waves. Our analysis involves a singularly perturbed reaction-diffusion-advection type initial-boundary value problem of a general form. We employ methods from asymptotic theory to develop an approximate smooth solution with an internal layer. Using local coordinates, we focus on the transition layer, where the solution undergoes rapid changes. Once the location of the transition layer is established, we can describe the solution across the full domain of the problem. Numerical examples are provided, demonstrating the high accuracy of the asymptotic method in predicting the behaviors of moving fronts.