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<p>In this paper, a novel finite element numerical scheme is established to preserve positivity for a specific class of nonlinear reaction-diffusion-advection models. To facilitate the decoupling of nonlinear terms, we incorporate a discrete seminorm into the weak formulation of the model. The semi-discrete system established through this approach effectively preserve the positivity, boundedness, and asymptotic stability inherent in the nonlinear models. During the stability analysis, we propose a numerical stability threshold that maintain the qualitative characteristics of the exact stability threshold. Furthermore, we develop a fully discrete system by employing the backward Euler method for time integration. Subsequently, we conduct a thorough analysis of the fully discrete numerical solution, examining its positivity, convergence, and numerical stability. Finally, the validity of our conclusions is demonstrated through numerical experiments.</p>

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A positivity-preserving Galerkin finite element scheme for a nonlinear reaction-diffusion-advection Mussel-Algal model with Danckwerts boundary conditions

  • 期刊论文

  • 发表日期：

    2026年2月3日

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  • Liu Xing,
  • Wang Jie,
  • Lang Yanhua,
  • Yang Huizi,
  • Zhou Zhongxin

摘要

In this paper, a novel finite element numerical scheme is established to preserve positivity for a specific class of nonlinear reaction-diffusion-advection models. To facilitate the decoupling of nonlinear terms, we incorporate a discrete seminorm into the weak formulation of the model. The semi-discrete system established through this approach effectively preserve the positivity, boundedness, and asymptotic stability inherent in the nonlinear models. During the stability analysis, we propose a numerical stability threshold that maintain the qualitative characteristics of the exact stability threshold. Furthermore, we develop a fully discrete system by employing the backward Euler method for time integration. Subsequently, we conduct a thorough analysis of the fully discrete numerical solution, examining its positivity, convergence, and numerical stability. Finally, the validity of our conclusions is demonstrated through numerical experiments.