Runge-Kutta discontinuous Galerkin finite element schemes for nonlinear reaction diffusion equations

The reaction-diffusion equation plays an important role in dissipative dynamical systems for physical, chemical and biological phenomena. For example, in chemical physics, to describe concentration and temperature distributions, the heat and mass transfer are modeled by a nonlinear diffusion term, while the rate of heat and mass production are described by a nonlinear reaction term. In population dynamics, where the focus is on the evolution of a population density, diffusion terms correspond to random motions of individuals, and reaction terms describe their reproduction and interaction, as in a predator-prey model. Several authors pointed out that the population diffusion mechanisms are more realistically described by (degenerate) non-linear diffusion models like $\partial_t u-\Delta p(u)=r(u)\qquad \mbox{in}\Omega\times (0,+\infty)$ where $\Omega\subset\mathcal{R}^d,$ with $d=1,2$, is a bounded domain, $u$ is a density and $p(u), r(u)$ are the diffusivity and the reaction functions The previous equation is completed by suitable boundary conditions, usually homogeneous Neumann conditions. If $p(u)\simeq u^\gamma$, with $\gamma>0$, the equation is known as the porous media equation, which is degenerate in the sense that $p^\prime(0) = 0$. The solution can develop interfaces between the regions with zero and nonzero population densities, which represent population fronts. Reaction term is modeled by a generalized Kolmogorov-Fisher term, namely $r(u)=u(1-u^\beta)$. The discretization of this problem is challenging because the numerical scheme has to reproduce shock waves or fronts, and preserve stability and invariance properties. We present a new high-order family of numerical methods based on a discontinuous Galerkin space discretization and Runge-Kutta time stepping. These methods are capable to reproduce in the numerical solution some main properties of the analytical solution, like positivity, and to preserve stability and accuracy