Implicit-explicit methods for reaction-diffusion problems in pattern formation

Reaction-diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems. To approximate the corresponding spatially discretized models, an explicit scheme can be used for the reaction term and an implicit scheme for the diffusion term. Such implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods in fluid flow problems.In this work, we analyze the performance of several of the best known linear multistep IMEX schemes for reaction-diffusion problems in pattern formation. For the linearized two chemical system, the growth properties exhibited by IMEX schemes are examined. Schemes which accurately represent the growth of the linearized problem for large time steps are identified. Numerical experiments show that first order accurate schemes, as well as schemes which produce only a weak decay of high frequency spatial error may yield plausible results which are nonetheless qualitatively incorrect. For such schemes, computations using refinements in the time step are likely to produce essentially the same (erroneous) results. Higher order schemes which produce a strong decay of high frequency errors are proposed instead.Our findings are demonstrated on several examples.The work of the author was partially supported by an NSERC Postgraduate Scholarship