Abstract
We present an O(n) complexity and implicit algorithm for the
two-dimensional solution of the Stream Power Incision Model (SPIM)
enriched by a discharge threshold term and taking into account
variability in rainfall and thus discharge. The algorithm is based on
the formulation developed by
Deal et al (2018) and the
generalization of the FastScape algorithm \cite{BraunWillett2013}where the slope is approximated
by first-order accurate finite difference. We consider a variety of
discharge thresholds that vary in their dependence on channel slope. The
algorithm requires finding the root of a non-linear equation using a
Newton-Raphson iterative scheme. We show that the convergence of this
scheme is unconditional, except for a narrow range of model parameters
where the threshold increases with the slope and for low discharge
variability. We also show that the rate of convergence of the iterative
scheme is directly proportional to the slope exponent n in the SPIM.
We compare the algorithm to analytical solutions and to numerical
solutions obtained using a higher-order finite difference scheme. We
show that the accuracy of the FastScape algorithm and its generalization
presented here is comparable to other schemes for values of
n>1. We also confirm that the FastScape algorithm and
its generalization to variable discharge+threshold conditions does not
need to satisfy the CFL condition and provides an accurate solution
for both small and very long time steps. We finally use the new
algorithm to quantify how the existence of an erosional threshold
strongly affects the length of the post-orogenic decay of mountain
belts.