FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation

Input : Terrain cells $\backslash mathcal\{T\}$ with elevation $z$, per cell coefficients $\backslash alpha$ and $\backslash beta$

Output: Eroded terrain $z\text{'}$

1 $z\text{'}\; \backslash leftarrow$ copy of $z$
2 for $i\; \backslash leftarrow\; 1$ to $\backslash log\_2(|\backslash mathcal\{T\}|)$ do
3   foreach cell $c\; \backslash in\; \backslash mathcal\{T\}$ in parallel do
4     $z\text{'}[c]\; \backslash leftarrow\; z\text{'}[c]\; +\; \backslash alpha[c]z\text{'}[\backslash text\{recipient\; of\; \}\; c]$
5     $\backslash alpha[c]\; \backslash leftarrow\; \backslash alpha[c]\backslash alpha[\backslash text\{recipient\; of\; \}\; c]$
6     recipient of $c\; \backslash leftarrow$ recipient of recipient of $c$
7   end
8 end

Fluvial erosion is not the only erosive process responsible for shaping terrain. Hillslope processes model the gradual accumulation of solid material at the base of mountains and hills [BS97] and is usually expressed as a diffusion equation. We follow Tzathas et al. [TGSC24] and approximate it by including additional terms in the Stream Power Equation, changing $kQ^m$ to $kQ^m\; +\; k\_t\; +\; k\_h\; A^\{-h\}$, where $k\_t$ and $h\; =\; 0.6$ are hillslope erosion parameters, $A$ is the drainage area (obtained via flow routing with precipitation set uniformly to $p\; =\; \backslash Delta\; x^2$). The parameter $k\_t$ accounts for slope-dependent effects (landslides, debris-flow), regrouped in computer graphics under the catch-all term thermal erosion [MKM89].

Sediment deposition. Finally, flow routing can more generally propagate any type of transportable material. A case in point is sediment deposition, which acts as a complement to hydraulic erosion. Incorporating sediment deposition requires an additional term in the Stream Power Law [YBG⁺19]:

$$\backslash frac\{\backslash partial\; z\}\{\backslash partial\; t\}\; =\; -kQ^m\; \backslash frac\{\backslash partial\; z\}\{\backslash partial\; x\}\; +\; k\_d\; \backslash frac\{Q\_s\}\{Q\},\; \backslash quad\; (5)$$

where $k\_d$ is the sediment deposition coefficient and $Q\_s$ is sediment flux, obtained by accumulating the negative elevation balance $-\backslash Delta\; x^2\; \backslash frac\{\backslash partial\; z\}\{\backslash partial\; t\}$ downstream with flow routing. While Yuan et

al. [YBG⁺19] suggest a fully implicit solution that requires iterating between Equation 5 and the accumulation of $Q\_s$, we propose a semi-implicit variant in which $Q\_s$ is computed from the elevation balance of the previous time-step. While our solution introduces a time dependency into mass conservation, it does not require iterating, which leads to a faster algorithm with no visual difference.