Keeping the edge: A numerical method that avoids knickpoint smearing when solving the stream power law

Abstract

The stream power equation is commonly used to model river incision into bedrock. Although specific conditions allow an analytical approach, finite difference methods (FDMs) are most frequently used to solve this equation. FDMs inevitably suffer from numerical smearing which may affect their suitability for transient river incision modeling. We propose the use of a finite volume method (FVM) which is total variation diminishing (TVD) to simulate river incision in a more accurate way. The TVD-FVM is designed to simulate sharp discontinuities, making it very suitable to simulate river incision pulses. We show that the TVD-FVM is much better capable of preserving propagating knickpoints than FDMs, using Niagara Falls as an example. Comparison of numerical results obtained using the TVD-FVM with analytical solutions shows a very good agreement. Furthermore, the uncertainty associated with parameter calibration is dramatically reduced when the TVD-FVM is applied. The high accuracy of the TVD-FDM allows correct simulation of transient incision waves as a consequence of older uplift pulses. This implies that the TVD-FVM is much more suitable than FDMs to reconstruct regional uplift histories from current river profile morphology and to simulate river incision processes in general.