Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Aleksei Liubavin, Mingkang Ni, Ye Zhang, Dmitrii Chaikovskii

TL;DR

This work develops an asymptotic framework for a three-dimensional reaction-diffusion-advection equation with periodic boundaries, addressing moving autowave fronts through an internal transition layer described by a surface $h(x,y,t)$. By separating outer (stationary) and inner (transition) regions, the authors construct a two-tier expansion with zero- and first-order inner problems in the stretched variable $\xi = r/\mu$, and derive a coupled set of conditions that determine both the outer fields and the front surface. A central result proves the existence of a solution $u(x,y,z,t,\mu)$ with $L$- and $M$-periodicity and a uniform asymptotic approximation $U_n$ that satisfies $|u-U_n| = O(\mu^{n+1})$, with $U_n$ built from the outer and inner contributions. A numerical example confirms the method by solving a concrete 3D RD-AD model, showing a moving transition layer near $h_0(x,y,t)$ and small relative errors against a benchmark solution, thereby illustrating practical applicability to front propagation problems in oil-recovery and related autowave 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.

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

TL;DR

This work develops an asymptotic framework for a three-dimensional reaction-diffusion-advection equation with periodic boundaries, addressing moving autowave fronts through an internal transition layer described by a surface . By separating outer (stationary) and inner (transition) regions, the authors construct a two-tier expansion with zero- and first-order inner problems in the stretched variable , and derive a coupled set of conditions that determine both the outer fields and the front surface. A central result proves the existence of a solution with - and -periodicity and a uniform asymptotic approximation that satisfies , with built from the outer and inner contributions. A numerical example confirms the method by solving a concrete 3D RD-AD model, showing a moving transition layer near and small relative errors against a benchmark solution, thereby illustrating practical applicability to front propagation problems in oil-recovery and related autowave 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.

Paper Structure

This paper contains 11 sections, 4 theorems, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Assumptions and local coordinates
  3. Asymptotic representations
  4. Outer functions
  5. Inner layer functions
  6. Zero-order inner layer functions
  7. First order and higher inner layer functions
  8. Asymptotic representation of the function $h$
  9. Main results
  10. A Numerical example
  11. Appendix: The well-posedness of conditions \ref{['C2']} and \ref{['C4']}.

Key Result

Theorem 4.1

If conditions C1, C2, C3, and C4 are met, then the system sec1.InitProb has a solution $u(x,y,z,t,\mu)$, which is $L$-periodic in the variable $x$ and $M$-periodic in the variable $y$. Additionally, this solution belongs to the class such that it lies between the upper and lower solutions for $t\in [0,T]$, and the function $U_n(x,y,z,t,\mu)$ is its uniform asymptotic approximation in the domain $

Figures (3)

  • Figure 1: Solution of \ref{['eq44']} (obtained numerically).
  • Figure 2: Comparison of the asymptotic solution \ref{['fig:asymptSOL']} and numerical solution \ref{['fig:numSOL']} at the moment $t_1=0.2$.
  • Figure 3: Comparison of the asymptotic solution \ref{['fig:asymptSOL2']} and numerical solution \ref{['fig:numSOL2']} at the moment $t_2=0.6$.

Theorems & Definitions (7)

  • Theorem 4.1
  • proof
  • Theorem 6.1
  • Theorem 6.2
  • proof
  • Theorem 6.3
  • proof