Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts

The paper presents an adapted numerical integration for advection–reaction–diffusion problems. The numerical scheme, exploiting the a-priori knowledge of the qualitative behaviour of the solution, gains advantages in terms of efficiency and accuracy with respect to classic schemes already known in literature. The adaptation is here carried out through the so-called trigonometrical fitting technique for the discretization in space, giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms. Due to this mixed nature of the vector field, an Implicit–Explicit (IMEX) method is here employed for the integration in time, based on the first order forward–backward Euler method. The coefficients of the method here introduced rely on unknown parameters which have to be properly estimated. In this work, such an estimate is performed by minimizing the leading term of the local truncation error. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and some numerical experiments.