Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction (Q1889898)

The authors are interested in reaction-diffusion problems of the type \((U^\varepsilon \in\mathbb{R}^n)\): \[ \partial_tU^\varepsilon_x \cdot\bigl(B (U^\varepsilon) \partial_xU^\varepsilon\bigr)=F^\varepsilon,\quad x\in \mathbb{R}^d,\;t\geq 0 \] where \(B(U^\varepsilon)\), called diffusion matrix, is a tensor of order \(d\times d\times n\). The solution of this dynamical system is denoted by \(U^\varepsilon=T^t_\varepsilon U_0\), where \(U_0\) is the initial condition and \(\varepsilon>0\) is a small parameter. Then the singular perturbation results are presented followed by various operator splittings. The theoretical results are illustrated numerically together with the influence of the discretization on the results.