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

Systems of reaction-diffusion partial differential equations (RD-PDEs) are widely applied for modelling 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. We show that the structure of the diffusion matrix can be exploited so as to use matrix-based versions of time integrators, such as Implicit-Explicit (IMEX) and exponential schemes. This implementation entails the solution of a sequence of discrete matrix problems of significantly smaller dimensions than in the vector case, thus allowing for a much finer problem discretization. We illustrate our findings by numerically solving the Schnackenberg model, prototype of RD-PDE systems with Turing pattern solutions, and the DIB-morphochemical model describing metal growth during battery charging processes.