Landscape evolution models (LEMs) simulate how the earth surface evolves in response to different driving forces, including tectonics, climatic variability and human activity. LEMs are integrative because they amalgamate empirical data and conceptual models into a set of mathematical equations that can be used to reconstruct or predict terrestrial landscape evolution and corresponding sediment fluxes (Glotzbach, 2015; Howard, 1994). Studies that address how climate variability and land use changes will affect landscapes in the long term increasingly rely on LEMs (Gasparini and Whipple, 2014).

Landscape evolution is not always smooth and gradual. Instead, sudden tectonic displacements along tectonic faults can create distinct landforms with sharp geometries (Whittaker et al., 2007). These topographic discontinuities do not necessarily smooth out over time but may persist over long timescales in transient landscapes (Mudd, 2016; Vanacker et al., 2015). For example, faults may spawn knickpoints along river profiles. These knickpoints will propagate upstream as rapids or water falls (Hoke et al., 2007), thereby maintaining their geometry through time (Campforts and Govers, 2015). After an uplift pulse, the river will only regain a steady state when knickpoints finally arrive in the uppermost river reaches. Transiency is not limited to individual rivers but also affects entire orogens such as the Southern Alps of New Zealand where the landscape may never reach a condition of steady state due to the permanent asymmetry in vertical uplift, climatically driven denudation and horizontal tectonic advection (Herman and Braun, 2006).

Transient “shocks” and topographic discontinuities are inherently difficult to model accurately. Most of the widely applied LEMs use first-order accurate explicit or implicit finite difference methods to solve the partial differential equations (PDEs) that are used to simulate river incision (Valters, 2016). These schemes suffer from numerical diffusion (Campforts and Govers, 2015; Royden and Perron, 2013). Numerical diffusion will inevitably lead to the gradual disappearance of knickpoints and will result in ever-smoother shapes. It has already been shown that numerical smearing decreases the accuracy of modeled longitudinal river profiles (Campforts and Govers, 2015). Here, we hypothesize that it is also relevant for the simulation of hillslope processes: hillslopes respond to river incision and inaccuracies in river incision modeling will thus propagate to the hillslope domain. Whether and to what extent this occurs is still unexplored.

Tectonic displacement is similar to river knickpoint propagation; in both cases, sharp landscape forms are laterally moving. Numerical diffusion may therefore significantly alter landscape features when tectonic shortening or extension is simulated using first-order accurate methods. In principle, flexible gridding overcomes this problem through dynamically adapting the density of nodes on the modeling domain to the local rate of topographic change. However, models using flexible gridding have other constraints. They are more difficult to implement and impose the structure of the numerical grid on the natural drainage network since rivers must follow the grid structure. Furthermore, the output of flexible grid models is not directly compatible with most software that is available for topographic analysis.

Here we present TTLEM (TopoToolbox Landscape Evolution Model), a spatially explicit raster-based LEM, which is based on the object-oriented function library TopoToolbox 2 (Schwanghart and Scherler, 2014). Contrary to previously published LEMs, we solve the stream power river incision model using a flux-limiting finite volume method (FVM), which is total variation diminishing (TVD), in order to avoid numerical diffusion. Our numerical scheme expands on previous work (Campforts and Govers, 2015) by extending the mathematical formulation of the TVD method from one-dimensional to entire river networks. Moreover, we develop a two-dimensional TVD-FVM scheme to simulate horizontal tectonic displacement on regular grids, which enables simulation of three-dimensional variations in tectonic deformation. The objective of this paper is to evaluate TTLEM and assess the performance of the numerical methods for a variety of real and simulated topographic and tectonic situations.

We solve Eq. (6) using a set of numerical schemes that we implement in the software TTLEM (see also Fig. A1 in the Appendix). TTLEM is written in the MATLAB programming language and in C-code where this significantly improves performance (e.g., for the nonlinear hillslope diffusion algorithm of Perron, 2011). Integrating TTLEM into TopoToolbox (Schwanghart and Kuhn, 2010; Schwanghart and Scherler, 2014) provides access to efficient algorithms of digital elevation model (DEM) analysis, as well as numerous routines for visualizing and analyzing modeling outputs. In the following sections, we discuss the numerical schemes of TTLEM to solve the PDEs described in the previous section. The section numbers correspond to the processes indicated in the model flowchart in the Appendix (Fig. A1).

We investigate how numerical schemes implemented in TTLEM affect simulated landscape evolution. As we focus on evaluating the schemes' performance, all simulations have synthetically generated landscapes as initial surfaces. Hence, our simulations are uncalibrated and results remain untested against an actual landscape: however, the chosen parameter values are within the range of previous studies (e.g., Gasparini and Whipple, 2014; Whipple and Tucker, 1999). We distinguish between the effects on simulated river incision on the one hand and on simulated tectonic displacement on the other. To investigate the accuracy and implications of river incision methods, we compare the explicit TVD-FVM with the first-order implicit FDM and further differentiate between the implicit FDM where no limitation is set on the time step and the implicit FDM where the CFL criterion limits the time step length. To investigate the accuracy and implications of river incision methods we compare an explicit first-order GM with the two-dimensional TVD-FVM.

Our analysis of numerical solvers focuses on three interrelated issues: numerical accuracy, spatial resolution and computational efficiency. Adopting highly simplifying assumptions allow us to benchmark the solvers against analytical solutions. Our focus is on testing an implicit, first-order accurate FDM against TVD-FVM. The implicit FDM has several desirable properties. It is unconditionally stable and tolerates time step lengths exceeding those prescribed by the CFL criterion. LEMs are often run over time spans of millions of years and the CFL criterion is dictated by a few grid cells with high upslope areas. Adopting an implicit scheme is therefore potentially interesting since it allows the decrease of the computation time while enabling simulations at high spatial resolutions. Our results, however, show that this major advantage vanishes if the aim of a LEM simulation is to capture transiency correctly. For CFL > 1 the implicit FDM introduces significant numerical smearing, and for CFL ≫ 1, the approach tends to insert an artificial shock wave of uplift because gradual uplift is approximated by a step function if time steps are (very) large.

For time step lengths approaching those prescribed by the CFL criterion, we show that computational gains by implicit FDM are marginal compared to TVD-FVM. The TVD-FVM code can be vectorized, i.e., it exploits single-instruction multiple-data parallelism to save CPU time. The implicit FDM requires a lower number of numerical operations but all stream network nodes need to be treated sequentially. Simulations at higher spatial resolutions increase the numerical accuracy and may balance the low accuracy of the implicit, first-order accurate FDM. Our results indicate that there is indeed a strong gain in numerical accuracy for all methods (Figs. 4 and 5) with increasing spatial resolution. However, to achieve the same numerical accuracy as the TVD-FVM, the implicit method with a CFL < 1 constraint requires the use of spatial resolution that is about 3 times higher, resulting in a computation time that is ∼ 12 times higher (Fig. 5). In summary, while a first-order implicit scheme is stable and accurate for long-term steady-state solutions (Braun and Willett, 2013), it has severe shortcomings when simulating transient landscape evolution caused by knickpoint propagation in detachment-limited erosional basins. These shortcomings can, to a large extent, be avoided by using a TVD-FVM.

We also show that the impact of the numerical scheme used to simulate river incision is not limited to river profile development alone. Hillslopes adjust to local base level changes dictated by river incision. Hillslope denudation rates must therefore – at least partly – reflect the geometry and dynamics of a knickpoint and will respond differently to a diffuse signal that is the result of relatively slow, continuous uplift on the one hand and sharp discontinuity caused by a rapid base-level drop of major fault activity on the other hand. Our simulations show that, depending on the spatial and temporal resolution, catchment-wide erosion rates are more responsive to uplift when fluvial incision is calculated by TVD-FVM rather than by the first-order accurate implicit FDM. This is because first-order (explicit and implicit) FDMs fail to properly reproduce transient incision waves (Campforts and Govers, 2015) due to knickpoint smoothing. This also affects hillslope denudation since the drop in hillslope base level due to the passage of a knickpoint is smeared out in time when smoothing occurs. The response of catchment-wide erosion rates to uplift will therefore also be smoothed, resulting in significantly lower peak erosion rates. This effect will be most significant in upstream catchments that are far away from the base level since smoothing increases with time and knickpoint migration distance.

One might question the significance and necessity of numerical schemes that avoid diffusion of retreating knickpoints. Given the many assumptions and uncertainties that underlie many LEMs, numerical accuracy may seem like a problem of lesser importance. We argue that the simulations presented in this paper show that this is not the case and that it is indeed critical to simulate knickpoint retreat as accurately as possible. However, our analysis does not cover all situations wherein the accurate simulation of knickzones is important. Simulation of sharp knickpoints is also required in geomorphological and lithological settings where knickpoint retreat is caused by rock toppling, possibly triggered during extreme flood events (Baynes et al., 2015; Lamb et al., 2014; Mackey et al., 2014). Similarly, glacial incision often creates hanging valleys that are reshaped by migrating fluvial knickpoints after glacial retreat (Valla et al., 2010). In all of these cases simulation tools with a minimum of numerical diffusion are required to correctly quantify natural knickpoint diffusion and to study the underlying processes.

First-order numerical methods also inadequately simulate lateral tectonic displacement on a regular grid. The amount of numerical diffusion that is introduced by these methods will, in many cases, exceed natural diffusion rates, thus making accurate simulation of hillslope development impossible. A two-dimensional variant of the TVD-FVM reduces the amount of numerical diffusion to values well below natural diffusivity values, an effect that is especially apparent at high spatial resolutions. The two-dimensional TVD-FVM thus allows the accurate modeling of this process, which significantly impacts the evolution of topography and river networks (Willett, 1999), using a fixed grid. This was hitherto only possible with flexible spatial discretization schemes.

Although most LEMs use first-order accurate discretization schemes (Valters, 2016), the problem of numerical diffusion has been discussed in the broader geophysical community (Durran, 2010; Gerya, 2010). An alternative family of shock-capturing Eulerian methods are MPDATA advection schemes (Jaruga et al., 2015). These schemes are based on a two-step approach in which the solution is first approximated with a first-order upwind numerical scheme and then corrected by adding an anti-diffusion term (Pelletier, 2008). However, contrary to the TVD-FVM, the standard MPDATA scheme (Smolarkiewicz, 1983) is not monotonicity preserving (i.e., it is not TVD). Instead, MPDATA introduces dispersive oscillations in the solution if combined with a source term (such as uplift) in the equation (Durran, 2010). Adding limiters to the solution of the anti-diffusive step (Smolarkiewicz and Grabowski, 1990) renders the MPDATA scheme oscillation free (Jaruga et al., 2015). However, by adding this additional correction, the method approaches the numerical nature of the TVD-FVM, which does not require further adjustments in any case.

Some of the weaknesses of the tested numerical solutions can be reduced by using LEMs that rely on irregular grid geometries. Irregular grids, for example, allow the simulation of tectonic shortening using a Lagrangian approach where grid nodes are advected with the tectonically imposed velocity field (e.g., Herman and Braun, 2006). In TTLEM the TVD-FVM solvers are implemented using a fixed grid, which has several advantages. First, input data such as topography, climate, lithology or tectonic displacement fields are typically available as raster datasets and thus require only minor modifications, whereas irregular grids require substantial preprocessing. Second, TTLEM output can instantly be analyzed and visualized using the TopoToolbox library (Schwanghart and Kuhn, 2010; Schwanghart and Scherler, 2014) or any other geographic information system. Thus, while irregular grid geometries and flexible grids may have some advantages over rectangular grids, TTLEM's implementation of numerically accurate algorithms strongly reduces the shortcomings of rectangular grids while facilitating straightforward processing of model input and output.

Despite the growing interest in the development and use of LEMs, accuracy assessment of the numerical methods has received little attention. First-order accurate FDMs are the most commonly applied numerical methods. However, they introduce numerical diffusion and artificially smooth discontinuities that are inherent in transient landscapes. To overcome this problem, we developed the TVD-FVM. The TVD-FVM solves river incision more accurately than the first-order accurate FDMs with significant influences on the geometry of modeled river profiles and implications for catchment-wide erosion rates. Errors due to numerical diffusion depend on the spatial and temporal resolution as well as on the position of the catchment in the landscape. In addition, we introduce a two-dimensional version of the TVD-FVM that allows the simulation of lateral tectonic displacement with low numerical diffusion on a fixed computational domain. Our new numerical techniques are implemented in the open-access raster-based Landscape Evolution Model (TTLEM) contained within TopoToolbox. Together with numerical implementations of common hillslope process models, TTLEM provides the community with a novel simulation tool for the accurate reconstruction, exploration and prediction of landscape evolution. In its current form, TTLEM is limited to uplifting, fluvially eroding landscapes. Further development will integrate other processes (e.g., glacial erosion) as well as the explicit routing of sediment through the landscape.

TTLEM 1.0 is part of TopoToolbox version 2.2. The source code and future updates are available in the GIT repository https://github.com/wschwanghart/topotoolbox. TTLEM is platform independent and requires MATLAB 2014b or higher and the Image Processing Toolbox. Documentation and user manuals for the most current release version of TopoToolbox and TTLEM can be found in the GIT repository in the help folders of the software. The user manual of TTLEM includes three tutorials that can be accessed from the command window in MATLAB. Example landscape evolution movies of different model configurations are presented online in Campforts et al. (2016b, c). The source code for the solution of the one-dimensional stream power law (SPLM) can be downloaded from the GIT repository https://github.com/BCampforts/SPLM. SPLM contains the solution for the one-dimensional river incision codes including four examples.