Matrix-oriented discretization methods for reaction–diffusion PDEs: Comparisons and applications

Systems of reaction–diffusion partial differential equations (RD-PDEs) are widely applied for modeling life science and physico-chemical phenomena. In particular, the coupling between diffusion and nonlinear kinetics can lead to the so-called Turing instability, giving rise to a variety of spatial patterns (like labyrinths, spots, stripes, etc.) attained as steady state solutions for large time intervals. To capture the morphological peculiarities of the pattern itself, a very fine space discretization may be required, limiting the use of standard (vector-based) ODE solvers in time because of excessive computational costs. By exploiting the structure of the diffusion matrix, we show that matrix-based versions of time integrators, such as Implicit–Explicit (IMEX) and exponential schemes, allow for much finer problem discretizations. We illustrate our findings by numerically solving the Schnakenberg model, prototype of RD-PDE systems with Turing pattern solutions, and the DIB-morphochemical model describing metal growth during battery charging processes.