ARES: An efficient approach to adaptive time integration for stiff Differential-Algebraic Equations

Abstract

In the broad context of solving systems of stiff Differential-Algebraic Equations (DAE), in-between a basic Euler implicit scheme with a fixed timestep and adaptive timestep and higher order approaches, we propose the Adaptive Relaxed Euler Scheme (ARES): an implicit Euler scheme with an adaptive timestep, in conjunction with a nonlinear solver using the Newton method. We stick to a 1st-order time scheme and the adaptive quality uses very few additional operations and is therefore much less costly and easier to implement, while remaining adaptive to the local stiffness of the system. The overall principle of ARES: allowing to reduce accuracy of a transient calculation in order to get faster to a steady state, proves to be especially relevant in the context of complex industrial reactive transport simulations, where only the steady state of the plant is of interest, while eluding often evaluation through a direct calculation. In cases where computational time is of the essence, our approach is demonstrated through practical examples to offer a simple and valid way to obtain steady-state solutions reliably and fast.