LAUSR

In this paper, we present a stable numerical scheme for solving two‐dimensional m$$ m $$ ‐component reaction–diffusion systems. The proposed approach utilizes the backward Euler method for temporal discretization and the hybridized discontinuous Galerkin (HDG) method for spatial discretization. We analyze the stability of the proposed HDG method for problems with Dirichlet and Neumann boundary conditions, demonstrating that this method is stable, in the sense of the energy method, under certain mild conditions on the stabilization parameters. Several numerical experiments are provided to validate the proposed scheme, with applications to two reaction–diffusion models: the Brusselator and glycolysis systems. Numerical results confirm that the proposed method achieves the expected optimal convergence rate for both the approximate solutions and their first derivatives. To exhibit relevant physical concepts, we demonstrate the convergence behavior of approximate solutions at the stable equilibrium points of the selected reaction–diffusion system with small diffusion coefficients. Furthermore, the nonconvergence behavior is given at unstable equilibrium points.