Finite element modeling of nonlinear reaction–diffusion–advection systems of equations. Issue 11 (5th November 2018)

• Record Type: Journal Article
• Authors: Komala Sheshachala, Sanjay; Codina, Ramon
• Is Part Of: International journal of numerical methods for heat & fluid flow. Volume 28:Issue 11(2018)
• Page Start: 2688
• Page End: 2715
• Publication Date: 2018-11-05
• Subjects: Nonlinear reaction -- Predator–prey model -- Stabilized finite element methods; Heat -- Transmission -- Mathematics -- Periodicals; Fluid dynamics -- Mathematics -- Periodicals; 536.2
• DOI: 10.1108/HFF-02-2018-0077
• Languages: English
• ISSNs: 0961-5539

Abstract

Abstract : Purpose: This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed. Design/methodology/approach: A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation. Findings: The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model. Originality/value: A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.