Landscape evolution models (LEMs) allow the study of earth surface responses to changing climatic and tectonic forcings. While much effort has been devoted to the development of LEMs that simulate a wide range of processes, the numerical accuracy of these models has received less attention. Most LEMs use first-order accurate numerical methods that suffer from substantial numerical diffusion. Numerical diffusion particularly affects the solution of the advection equation and thus the simulation of retreating landforms such as cliffs and river knickpoints. This has potential consequences for the integrated response of the simulated landscape. Here we test a higher-order flux-limiting finite volume method that is total variation diminishing (TVD-FVM) to solve the partial differential equations of river incision and tectonic displacement. We show that using the TVD-FVM to simulate river incision significantly influences the evolution of simulated landscapes and the spatial and temporal variability of catchment-wide erosion rates. Furthermore, a two-dimensional TVD-FVM accurately simulates the evolution of landscapes affected by lateral tectonic displacement, a process whose simulation was hitherto largely limited to LEMs with flexible spatial discretization. We implement the scheme in TTLEM (TopoToolbox Landscape Evolution Model), a spatially explicit, raster-based LEM for the study of fluvially eroding landscapes in TopoToolbox 2.

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.

In its simplest form, tectonic processes are represented by their kinematics
and the assumed vertical surface deformation field

Detachment-limited fluvial erosion (

River incision drives the development of erosional landscapes by setting the
base level for hillslope processes. Steepening of hillslope toes leads to
increased sediment fluxes from hillslopes to the river system. Hillslope
denudation (

In summary, TTLEM solves the following PDE, whereby an explicit distinction
is made between the fluvial and hillslope domain. The fluvial domain is
determined by cells having a contributing drainage area exceeding a critical
drainage area (

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).

TopoToolbox provides a function library for deriving the drainage network
and terrain attributes (Schwanghart and Scherler, 2014). The
calculation of flow-related terrain attributes, i.e., data derived from flow
directions, relies on a set of highly efficient algorithms that exploit the
directed and acyclic graph structure of the river flow network
(Phillips et al., 2015). Nodes of
the network are grid cells and edges represent the directed flow connections
between the cells in downstream direction. Topological sorting of this
network returns an ordered list of cells in which upstream cells appear
before their downstream neighbors. Based on this list, we calculate terrain
attributes such as upslope area with a linear scaling, thus enabling
efficient calculation (

DEMs of real landscapes frequently contain data artifacts that generate topographic sinks. Topographic sinks can also occur during simulations when diffusion on hillslopes creates “colluvial wedges” that dam sections of the river network. By adopting algorithms of flow network derivation from TopoToolbox, TTLEM makes use of an efficient and accurate technique for drainage enforcement to derive non-divergent (D8) flow networks (Schwanghart et al., 2013; Soille et al., 2003). Based on the thus-derived flow network, TTLEM uses downstream minima imposition (Soille et al., 2003) that ensures that downstream pixels in the network have lower or equal elevations than their upstream neighbors.

We implement a two-dimensional version of a flux-limiting total volume method to reduce
numerical diffusion when simulating tectonic displacements on a regular
grid. Equation (1) can be written as a scalar
conservation law:

The numerical TVD fluxes are calculated following Toro (2009). In the following, we illustrate how to derive
the flux over one out of the four cell boundaries:

TTLEM features a one-dimensional version of the flux-limiting TVD-FVM to solve for river
incision (Eq. 2), which is written as a scalar
conservation law:

In addition, we implement a first-order implicit FDM for the solution of the
SPL detailed in Braun and Willett (2013). The
method provides stable solutions regardless of the time step length, a
property desired when simulating landscape evolution over long timescales
and large spatial domains. Explicit schemes of river incision (both FDM and
TVD-FDM), in turn, require time steps that satisfy the Courant–Friedrich–Lewy condition (CFL):

Ideally, numerical methods are benchmarked against analytical solutions. Albeit analytical solutions are available for specific initial and boundary conditions only, they are accurate and grid-resolution independent, contrary to numerical solutions where model parameter values might depend on the grid resolution (Pelletier, 2010). We implemented an analytical solution for the SPL as an independent benchmark to compare the performance of the different numerical schemes of river incision under conditions where an analytical solution is available.

Model parameters used for the TTLEM simulations.

First, we created an artificial DEM with topography in steady state between
uplift and erosion (see Table 1). From this DEM, we extracted the drainage
network and corresponding river elevations by selecting all cells exceeding
10

We applied the slope patch solution to the steady-state pre-uplift river
profiles using the simulated uplift rates as input. We also assessed the
accuracy of the numerical methods with the root mean squared error (RMSE):

We implemented linear hillslope diffusion using the implicit Crank–Nicolson scheme (Pelletier, 2008). The scheme is unconditionally stable at large time steps. A numerical solution of the nonlinear hillslope equation, however, is more demanding. The maximum time step length of an explicit FDM sharply decreases as slopes approach the threshold gradient. To overcome this restriction, Perron (2011) developed Q-imp, an implicit solver that allows the increase of time step lengths by several orders of magnitude. Conversely, the per-operation computational cost of this algorithm is higher in comparison to the explicit solution, and the overall performance of this method is better than alternative solutions (Perron, 2011). Q-imp efficiently calculates hillslope diffusion even for high-resolution simulations. However, rapid incision during one time step may generate slopes along rivers that are greater than the threshold slope, a situation that Q-imp cannot solve. An approach is thus needed that adjusts hillslopes to the threshold slope prior to calculating Q-imp.

We assume that hillslopes instantaneously adjust to oversteepening by
mobilizing the amount of material required to reduce the slope gradient to
the threshold value

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.

The impact of numerical diffusion on propagating river profile knickpoints is most obvious in situations where an analytical solution is available. The first simulation illustrates such a situation, with an artificial river profile characterized by a major knickzone between 8 and 12 km from the river head (Fig. 1). We assume that the drainage area is increasing in proportion to the square of the distance and uplift equals zero. For this simplified configuration, an analytical solution for the SPL relies on the method of characteristics (Luke, 1972). Notwithstanding the relatively high spatial resolution of 100 m, the first-order implicit FDM suffers from considerable numerical diffusion when river incision is calculated over a time span of 1 My (Fig. 1). The TVD-FVM systematically achieves a much higher accuracy over a wide range of spatial resolutions and parameter values (Campforts and Govers, 2015).

We assess the numerical accuracy of the entire drainage network with
spatially and temporally constant values for all model parameter values
(Table 1), assuming a fixed drainage network (see Sect. 3.3.2). We
first create a steady-state artificial landscape (Fig. 2) on a 50 km

Solution of the linear one-dimensional stream power law for a synthetic knickzone over a time span of 1 My. The analytical solution is obtained with the method of characteristics. The spatial resolution is 100 m. Table 1 lists other model parameter values.

A synthetic steady-state landscape produced as the testing
environment to verify and compare the different numerical schemes implemented
in TTLEM. Model runtime was 150 My, while uplift rate was assumed to be spatially
uniform over the area (block uplift) and fixed to 1 km My

Following steady state, we impose four consecutive sinusoidal uplift pulses
of equal magnitude on this artificial landscape over 1 My. Each uplift
pulse has a wavelength of 0.25 My and an amplitude of

Uplift imposed on the steady-state landscape shown in Fig. 2 to investigate the impact of different numerical schemes.

Figure 4 compares results obtained from the numerical methods and the analytical solution. The initial river profiles slightly differ depending on spatial resolution due to interpolation of the steady-state artificial landscape with a spatial resolution of 100 m. The results show that TVD-FVM and implicit numerical solutions converge at increasing spatial resolutions. Where the time step of the implicit scheme is unbounded by the CFL criterion, however, the solution deviates from those adhering to the CFL criterion. This illustrates that there is trade-off between numerical accuracy and numerical stability for an implicit scheme at long time steps. In addition, an implicit scheme at high spatial resolution and large time steps fails to converge to an analytical solution because uplift is modeled as a discrete stepwise function rather than a continuous function (e.g., the sinusoidal uplift history used here) that inserts artificial shocks in the solution.

The TVD-FVM is consistently more accurate than the implicit methods at all
spatial resolutions, although the implicit FDM (CFL < 1) approaches
the high accuracy of the TVD-FVM at very high resolutions (6.25 m) (Fig. 5a). At lower spatial resolutions (> 10 m) the numerical accuracy
of the TVD-FVM is significantly higher compared to the accuracy obtained
with the implicit methods at the cost of a slightly increased additional
computation time. To achieve the same numerical accuracy as the TVD-FVM at
500 m spatial resolution (RMSE

Comparison between different modeled resolutions for the river profile indicated in blue in Fig. 2. The green line is the analytical “true” solution, obtained with the slope patch method of Royden and Perron (2013). The full red line represents the first-order accurate implicit solution when the CFL < 1, and the dotted blue line represents the first-order accurate implicit solution when the time step is left free. The implicit solutions where CFL < 1 are simulated with a time step equal to the time step used for the TVD-FVM.

Temporal variation in simulated catchment-wide erosion rates using
different numerical methods to simulate river incision. The black lines
represent simulations where a flux-limiting TVD-FVM is used, the blue lines
represent the first-order accurate implicit FDM without constraints on the
time steps, and the red lines represent the first-order accurate FDM with an
inner time step calculated with the CFL criterion.

We hypothesize that the diffusive nature of commonly applied first-order FDMs
is not restricted to the simulation of river longitudinal profiles but has
systematic consequences for other measures derived from LEM simulations.
Such measures include catchment-wide erosion rates that constitute the basis
for model–field data comparison and model parametrization (Gasparini and Whipple, 2014; Moon et
al., 2015). In order to investigate the sensitivity of LEM-derived
catchment-wide erosion rates to different numerical schemes of the river
incision model, we use the steady-state artificial landscape described in
the previous experiments (Sect. 4.1.2). The simulation runs over 5 My
with four consecutive uplift pulses of equal amplitude and a wavelength of
1.25 My with Dirichlet boundary conditions and a planform fixed drainage
network. We use two spatial resolutions (100 and 500 m) and three
different numerical methods (implicit FDM without time step limitation,
implicit FDM with time step limitation (CFL condition applied) and TVD-FVM)
to simulate river incision. The maximum length of

We compare differences in simulated erosion rates by randomly selecting
> 200 catchments with drainage areas ranging between 1 and 50 km

Spatial variation of differences between simulated erosion rates
calculated with a flux-limiting TVD-FVM for simulating river incision and a
first-order accurate implicit FDM. Here, we compare methods that are both run
with an inner time step constrained with the CFL criterion (see text).

We rank the catchments in increasing order of

For most catchments, we detect differences in catchment-wide erosion rates between the three numerical methods at a spatial resolution of 100 m. Generally, the amplitude of the response to a tectonic uplift pulse increases when using TVD-FVM: the use of a first-order implicit FDM without time step restriction results in a much smoother response in comparison to the TVD-FVM. The variations in response amplitude are significant: the majority of the catchments record amplitude reductions by more than 50 % when modeled with the implicit FDM without time step restriction. Time step restriction (and thereby sacrificing the main advantage of the implicit FDM) significantly reduces numerical diffusion so that most catchments display an erosional response comparable to that simulated by the TVD-FVM. However, this is only true for simulations with a 100 m spatial resolution. The advantage of a time-step-restricted implicit FDM over a nonrestricted implicit FDM disappears almost completely for a coarser grid resolution of 500 m.

Figure 7 shows that erosion rates diverge between the different methods with increasing distance to the outlet of the main river, while they are similar for larger catchments. A smaller effect of the numerical scheme on large catchment areas may partly arise from stronger averaging of local variations in catchment erosion rates. In addition, catchments at a large distance from the outlet – and thus likely with smaller catchment areas – will experience upstream migrating knickpoints only after several model time steps. If catchments are far from the fault zone, knickpoints will then be significantly smoothed by a first-order accurate implicit FDM, which will ultimately affect the response of the catchment. Again, spatial resolution matters: a larger grid size not only results in larger differences on average but also in larger differences between small and large catchments (Fig. 7).

The differences in catchment response relate to the differences in simulated erosion rates within the catchments. Figure 8 illustrates the spatial difference in erosion rates calculated with the two numerical methods during the final step of the model run (after 5 My). This figure shows that spatial differences are significant and form a systematic banded pattern related to the upslope migration of the erosion waves of the individual uplift pulses.

Spatial pattern of erosion rates during one model time step when
simulating landscape evolution with the flux-limiting TVD-FVM vs. the first-order accurate implicit FDM.

Impact of numerical schemes when simulating horizontal shortening on
a fixed grid. The simulations are performed at a spatial resolution of 50 m
and a CFL of 0.5.

We test the performance of the two-dimensional version of the flux-limiting TVD-FVM to
simulate tectonic displacement. A synthetic DEM forms the initial surface
for a simulation of a constant lateral tectonic displacement with neither
fluvial incision nor hillslope diffusion. Theoretically, this should result
in a laterally displaced landscape that, apart from this displacement,
remains unchanged in comparison to the initial state. We compare the flux-limiting TVD-FVM with a first-order accurate upwind GM
simulating a tectonic displacement in two directions (

In order to quantify the amount of numerical diffusion (

We find that numerical diffusivity of the GM exceeds commonly used values of hillslope diffusivities as soon as spatial resolution exceeds 90 m (Fig. 10a). The two-dimensional TVD-FVM decreases numerical diffusion by a factor of 5–60 compared to the GM (Fig. 10b). The accuracy increases for both schemes with increasing resolution and increasing CFL numbers. However, the gain in accuracy with increasing spatial resolution is higher for the TVD-FVM than for the GM. Our analysis shows that the explicit FDM performs best with a CFL criterion close to one where additional required iterations within a given time interval are at a minimum (Gulliver, 2007).

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

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

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

The model architecture of TTLEM is illustrated in Fig. A1.

We illustrate the impact of different hillslope process models on simulated
landscape evolution using a 30 m resolution DEM of the Big Tujunga region in
California as an example (Fig. A2). TTLEM allows the simulation of hillslope
processes assuming (non)-linear slope-dependent diffusion with the
consideration of a threshold hillslope. Figure A2 illustrates how different
hillslope process algorithms affect the evolution of hillslopes in the Big
Tujunga region, California (Fig. A2a). We assume no tectonic displacement and
use standard parameter values for river incision and hillslope diffusion
(Table 1) and a threshold slope (

Schematic representation of the TTLEM model flow. The numbered methods correspond with the paragraphs from Sect. 3 in the main text.

Hillslope response to river incision.

The authors declare that they have no conflict of interest.

This work was motivated by the meeting “Landscape evolution modeling –
bridging the gap between field evidence and numerical models” in Hannover,
21–23 October 2015, which was organized by the