# A numerical scheme for solving a mathematical model derived from larvae- algae-mussel interactions

## Palavras-chave:

golden mussel, larvae-algae-mussel model, Navier-Stokes equations, numerical methods, nonlinear system of PDEs## Resumo

In this work we present a numerical formulation for solving a mathematical model, derived from larvae-

algae-mussel interactions in aquatic environments, proposed in [1]. The model is composed of three unsteady and

nonlinear advective-diffusive-reactive equations for species densities coupled with the Navier-Stokes equations to

simulate the velocity field of the water. We employ the operator splitting technique in the finite element method

context for solving the transport problem in two stages: first, given the velocity field, we solve the advective-

diffusive problem to obtain the densities of larvae, algae and mussels; then, we use this first step approximation as

initial condition for solving the system of ordinary differential equations for the reactions terms. In the first stage,

the nonlinear stabilized finite element method CAU and the two-step Backward Differentiation Formula of second

order are employed in the spatial and time discretizations. The nonlinear process is solved by a Picard fixed point

iteration. In the second stage, the system of ordinary differential equations for reactions is approximated by the

fourth-order Runge-Kutta scheme. The numerical formulation proposed is used to simulate the 3D dynamics of

species proliferation and quantify the golden mussel population in a stretch of the Pereira Barreto channel, located

in Brazil, with a focus on population control actions. The preliminary results as well as other considerations related

to the problem and the numerical model are discussed.