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\begin{document}

\title{Implicit algorithm for threshold Stream Power Incision Model}



\author[1]{Jean Braun}%
\author[2]{Eric Deal}%
\affil[1]{GFZ-Potsdam}%
\affil[2]{ETH}%


\vspace{-1em}



  
  \date{June 10, 2024}


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\begin{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~(missing citation)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\textgreater{}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.%
\end{abstract}%



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