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

TL;DR

This paper introduces a new second order explicit stabilized Runge-Kutta-Chebyshev method for advection-diffusion-reaction equations, offering improved stability and efficiency for high Peclet numbers through adaptive step size and parameter selection.

Contribution

The paper presents a novel explicit stabilized integrator based on Chebyshev polynomials, with an adaptive algorithm for automatic parameter tuning, outperforming existing methods.

Findings

Outperforms existing schemes at high Peclet numbers

Features an adaptive algorithm for step size and parameter selection

Demonstrates efficiency through numerical experiments

Abstract

A novel second order family of explicit stabilized Runge-Kutta-Chebyshev methods for advection-diffusion-reaction equations is introduced. The new methods outperform existing schemes for relatively high Peclet number due to their 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. An adaptive algorithm to implement the new scheme is proposed. This algorithm 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 algorithm are presented.