CDR-Solv: Solving the Convection-Diffusion-Reaction Equation with Algebraic Sub-Grid Scale Stabilization Using Python

The convection-diffusion-reaction (CDR) equation is a fundamental mathematical model for simulating the transport of pollutants. It is a crucial tool for addressing global environmental challenges. However, most existing computational solutions are proprietary and inaccessible, making the development of open-source educational platforms with advanced stabilization capabilities necessary. This study developed and validated a computational framework that solves CDR equations using algebraic sub-grid scale (ASGS) stabilization. The research addressed the fundamental challenge of spurious oscillations that emerge in standard Galerkin formulations when convective transport significantly exceeds diffusive processes. This is a prevalent issue in transport phenomena modeling. A novel, Python-based educational software platform called CDR-Solv was developed to demonstrate the effectiveness of ASGS stabilization across polynomial degrees ranging from linear to cubic approximations. Numerical experiments with minimal diffusion coefficients showed that numerical instabilities were successfully eliminated while maintaining solution accuracy across various source term configurations. The stabilization parameter, τK, was instrumental in achieving computational stability without compromising mathematical rigor. Comparative analysis revealed the superior performance of higher-order approximations in capturing boundary layer phenomena and sharp gradient regions. The primary contribution of this study is the development of an open-source educational platform that provides access to advanced stabilization techniques and algorithmic transparency. The CDR-Solv framework also allows for the systematic exploration of the effects of selecting different polynomial degrees on solution quality in transport-dominated regimes.