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.

Abstract

In this article we present the approximation of the coupled model of the equations of motion of a fluid in shallow waters with the convection-diffusion-reaction (CDR) equation of pollutant transport. This approximation is carried out using high order finite elements and using stabilised variational sub-scale methods. We write the coupled system of equations, previously discretised in time and linearised, as a transient vector equation of CDR. The stabilised finite element methods used are the known ASGS and OSS sub-scale methods, the same ones that allow us to use the same interpolation for all unknowns, as well as to deal with dominant convection and reaction flows. We consider the possibility of non-linearity in both the convective and reaction terms. We will not consider the possible development of shocks in the solution. In order to examine the accuracy and robustness of the ASGS and OSS methods, we present four test cases: mesh convergence, transport of a pollutant in a square cavity, transport of a pollutant in the Gulf of Roses and at the river Guadalquivir mouth, and the predator-prey model, which can be written as a transient CDR vector equation with non-linearity in the reaction term.