Highly efficient NURBS-based isogeometric analysis for coupled nonlinear diffusion–reaction equations with and without advection

Nonlinear diffusion–reaction systems model a multitude of physical phenomena. A common situation is biological development modeling where such systems have been widely used to study spatiotemporal phenomena in cell biology. Systems of coupled diffusion–reaction equations are usually subject to some complicated features directly related to their multiphysics nature. Moreover, the presence of advection is source of numerical instabilities, in general, and adds another challenge to these systems. In this study, we propose a NURBS-based isogeometric analysis (IgA) combined with a second-order Strang operator splitting to deal with the multiphysics nature of the problem. The advection part is treated in a semi-Lagrangian framework and the resulting diffusion–reaction equations are then solved using an efficient time-stepping algorithm based on operator splitting. The accuracy of the method is studied by means of a advection–diffusion–reaction system with analytical solution. To further examine the performance of the new method on geometries more general than rectangles (e.g., L-shaped domains and parts of annuli), the well-known Schnakenberg–Turing problem is considered with and without advection. Finally, a Gray–Scott system on a circular domain is also presented. The results obtained demonstrate the efficiency of our new algorithm to accurately reproduce the solution in the presence of complex patterns on more complicated geometries. Moreover, the new method clarifies the effect of geometry on Turing patterns.