A fully adaptive explicit stabilized integrator for advection-diffusion-reaction problems

We introduce a novel second order family of explicit stabilized Runge-Kutta-Chebyshev methods for advection-diffusion-reaction equations which outperforms existing schemes for relatively high Peclet number due to its favorable stability properties and explicitly available coefficients. The construction of the new schemes is based on stabilization using second kind Chebyshev polynomials first used in the construction of the stochastic integrator SK-ROCK. We propose an adaptive algorithm to implement the new scheme that is able to automatically select the suitable step size, number of stages, and damping parameter at each integration step. Numerical experiments that illustrate the efficiency of the new methods are presented.