<p>Most of the recent studies modeling fluvial erosion in the context of tectonic geomorphology focus on the detachment-limited regime. One reason for this simplification is the direct relationship of the constitutive law used here – often called stream-power law – to empirical results on longitudinal river profiles. Another, not less important reason lies in the numerical effort that is much higher for transport-limited models than for detachment-limited models. This study proposes a simple formulation of transport-limited erosion that is as close to empirical results on river profiles as the stream-power law is. As a central point, a direct solver for the fully implicit scheme is presented. This solver requires no iteration for the linear version of the model, allows for arbitrarily large time increments, and is almost as efficient as the established implicit solver for transport-limited erosion. The numerical scheme can also be applied to linear models between the two extremes of detachment-limited and transport-limited erosion.</p>