  Coupled nonlinear system of reaction-diffusion equations describing
multi-component (species) interactions with heterogeneous coefficients is
considered. Finite volume method based approximation for the space is used to
construct semi-discrete form for the computation of numerical solutions. Two
techniques for time approximations, namely, a fully implicit (FI) and a
semi-implicit (SI) schemes are examined. The fully implicit scheme is
constructed using Newton's method and leads to the coupled system of equations
on each nonlinear and time iterations which is computationally rather
expensive. In order to minimize the latter hurdle, an efficient and fast
multiscale solver is proposed for reaction-diffusion systems in heterogeneous
media. To construct fast solver, we apply a semi-implicit scheme that leads to
an uncoupled system for each individual component. To reduce the size of the
discrete system, we present a multiscale model reduction technique. Multiscale
solver is based on the uncoupled operator of the problem and constructed by the
use of Generalized Multiscale Finite Element Method (GMsFEM). In GMsFEM we use
a diffusion part of the operator and construct multiscale basis functions. We
collect multiscale basis functions to construct a projection/prolongation
matrix and generate reduced order model on the coarse grid for fast solution.
Moreover, the prolongation operator is used to reconstruct a fine-scale
solution and accurate approximation of the reaction part of the problem which
then leads to a very accurate and computationally effective multiscale solver.
We provide numerical results for two species competition test problems in
two-dimensional domain with heterogeneous inclusions. We investigate the
influence of number of the multiscale basis functions to the method accuracy
and ability to work with different values of the diffusion coefficients.
