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Convergence of approximate solutions constructed by the finite volume method for the moisture transport model in porous media

Akiko Morimura, Toyohiko Aiki

TL;DR

This work analyzes a Richards-type nonlinear parabolic equation in one spatial dimension, modeling moisture transport in porous media with a time-dependent source term $p(t,x)$. It develops a dual equation (adjoint) framework to establish uniqueness of weak solutions under mild integrability and structural assumptions, and shows that a finite-volume discretization converges to the weak solution. By mollifying $p$ and applying discrete Aubin–Lions compactness, it proves both existence and convergence of FV approximations, and secures convergence of the full sequence when $p\in L^4(0,T;H)$. Overall, the results validate the finite-volume method for time-dependent moisture transport models and provide a rigorous pathway to existence, uniqueness, and convergence for this class of nonlinear parabolic equations in porous media.

Abstract

We consider the initial-boundary value problem for a nonlinear parabolic equation in the one-dimensional interval. This problem is motivated by a mathematical model for moisture transport in porous media. We establish the uniqueness of weak solutions to the problem by using the dual equation method. Moreover, we prove the convergence of approximate solutions constructed with the finite volume method.

Convergence of approximate solutions constructed by the finite volume method for the moisture transport model in porous media

TL;DR

This work analyzes a Richards-type nonlinear parabolic equation in one spatial dimension, modeling moisture transport in porous media with a time-dependent source term . It develops a dual equation (adjoint) framework to establish uniqueness of weak solutions under mild integrability and structural assumptions, and shows that a finite-volume discretization converges to the weak solution. By mollifying and applying discrete Aubin–Lions compactness, it proves both existence and convergence of FV approximations, and secures convergence of the full sequence when . Overall, the results validate the finite-volume method for time-dependent moisture transport models and provide a rigorous pathway to existence, uniqueness, and convergence for this class of nonlinear parabolic equations in porous media.

Abstract

We consider the initial-boundary value problem for a nonlinear parabolic equation in the one-dimensional interval. This problem is motivated by a mathematical model for moisture transport in porous media. We establish the uniqueness of weak solutions to the problem by using the dual equation method. Moreover, we prove the convergence of approximate solutions constructed with the finite volume method.
Paper Structure (4 sections, 12 theorems, 96 equations)

This paper contains 4 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Notation and results
  3. Uniqueness of weak solutions
  4. Convergence of approximate solutions

Key Result

Proposition 2.1

Let $T>0$. Assume (A1) and (A2). If $p \in L^2(0,T; H)$ and $v_0 \in H$, then (P)($p, v_0$) has a weak solution.

Theorems & Definitions (24)

  • Definition 2.1
  • Proposition 2.1: Roubicek
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.1
  • ...and 14 more