Nonsteady relaxation and critical exponents at the depinning transition

We study the nonsteady relaxation of a driven one-dimensional elastic interface at the depinning transition by extensive numerical simulations concurrently implemented on graphics processing units. We compute the time-dependent velocity and roughness as the interface relaxes from a flat initial configuration at the thermodynamic random-manifold critical force. Above a first, nonuniversal microscopic time regime, we find a nontrivial long crossover towards the nonsteady macroscopic critical regime. This ``mesoscopic'' time regime is robust under changes of the microscopic disorder, including its random-bond or random-field character, and can be fairly described as power-law corrections to the asymptotic scaling forms, yielding the true critical exponents. In order to avoid fitting effective exponents with a systematic bias we implement a practical criterion of consistency and perform large-scale (L2²⁵) simulations for the nonsteady dynamics of the continuum displacement quenched Edwards-Wilkinson equation, getting accurate and consistent depinning exponents for this class: β=0.245±0.006, z=1.433±0.007, ζ=1.250±0.005, and ν=1.333±0.007. Our study may explain numerical discrepancies (as large as $30%$ for the velocity exponent β) found in the literature. It might also be relevant for the analysis of experimental protocols with driven interfaces keeping a long-term memory of the initial condition.