numerical methods for nonlinear hillslope transport...

Numerical methods for nonlinear hillslope transport laws

J. Taylor Perron1

Received 19 June 2010; revised 7 January 2011; accepted 11 March 2011; published 17 June 2011.

[1] The numerical methods used to solve nonlinear sediment transport equations often setvery restrictive limits on the stability and accuracy of landscape evolution models. This isespecially true for hillslope transport laws in which sediment flux increases nonlinearly asthe surface slope approaches a limiting value. Explicit‐time finite difference methodsapplied to such laws are subject to fundamental limits on numerical stability that requiretime steps much shorter than the timescales over which landscapes evolve, creating aheavy computational burden. I present an implicit method for nonlinear hillslope transportthat builds on a previously proposed approach to modeling alluvial sediment transport andimproves stability and accuracy by avoiding the direct calculation of sediment flux.This method can be adapted to any transport law in which the expression for sediment fluxis differentiable. Comparisons of numerical solutions with analytic solutions in one andtwo dimensions show that the implicit method retains the accuracy of a standard explicitmethod while permitting time steps several orders of magnitude longer than the maximumstable time step for the explicit method. The ability to take long time steps affords asubstantial savings in overall computation time, despite the implicit method’s higherper‐iteration computational cost. Implicit models for hillslope evolution also offer adistinct advantage when modeling the response of hillslopes to incising channels.

Citation: Perron, J. T. (2011), Numerical methods for nonlinear hillslope transport laws, J. Geophys. Res., 116, F02021,doi:10.1029/2010JF001801.

1. Introduction

[2] The growing availability of high‐resolution topographicdata makes it desirable to model landscapes at a comparablyfine spatial scale, so that the topographic data can be com-pared with model predictions [e.g., Perron et al., 2009] orused as initial conditions [Roering, 2008]. But resolving fine‐scale features such as individual hillslopes often introducessediment transport laws with strong nonlinearities, com-pounding the already difficult problem of solving equationsfor fluvial erosion and sediment transport on large grids. Inthis paper, I examine different strategies for computing finitedifference solutions to nonlinear hillslope transport equa-tions. I briefly review the forms of linear and nonlinearhillslope transport laws, and discuss standard explicit solu-tion methods and their limitations. I then review an implicitapproach to nonlinear alluvial sediment transport proposedby Fagherazzi et al. [2002], and show how it can be adaptedto nonlinear hillslope transport. This approach also has lim-itations, however, including the need to calculate sedimentflux in discrete directions despite the fact that the quantity ofinterest is the flux divergence.[3] To overcome these restrictions, I introduce an implicit

method that borrows from the strategy of Fagherazzi et al.

[2002], but does not calculate sediment flux directly. I mea-sure the absolute error in equilibrium solutions by comparingall three methods with analytic solutions in one and twodimensions, and I assess the rate of error growth in transientsolutions as a function of time step length. The improvedmethod offers superior numerical stability to the other twomethods, permitting longer time steps, while retainingessentially the same accuracy as the explicit method. Thisstability advantage far outweighs the higher per‐iterationcomputational cost of the implicit method, and the timesavings become more substantial as the spatial grid resolu-tion becomes finer. I conclude by offering a practicalexample in which I use the implicit method to model theevolution of a surface derived from airborne laser altimetry.This example shows how an implicit model of hillslopeevolution overcomes a significant challenge in landscapeevolution models that couple hillslopes experiencing non-linear soil transport to rapidly incising channels.

2. Nonlinear Hillslope Transport Laws

[4] Most hillslope evolution models are based on a massconservation equation for an elevation surface z(x, y),

�s@z

@tþr � q

� �¼ �rU ; ð1Þ

where t is time, rs and rr are the bulk densities of sedimentand rock, q is the volume flux of transportable sediment perunit width of the land surface, and U is the rate of change of

1Department of Earth, Atmospheric and Planetary Sciences,Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.

Copyright 2011 by the American Geophysical Union.0148‐0227/11/2010JF001801

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, F02021, doi:10.1029/2010JF001801, 2011

F02021 1 of 13

http://dx.doi.org/10.1029/2010JF001801

bedrock elevation. This version of the continuity equationassumes that soil production keeps pace with surface ero-sion, such that no bedrock outcrops occur, and neglectsdissolution and mass transport in solution. The other mainingredient of hillslope evolution models is a set of transportlaws that express the sediment flux in terms of z, its spatialderivatives, and other parameters [e.g., Kirkby, 1971; Dietrichet al., 2003].[5] Many previous studies have treated hillslope soil

transport as a linear process, with a flux proportional to, andopposite in direction from, the topographic gradient [Culling,1960, 1963, 1965],

q ¼ �Drz; ð2Þ

where D is constant in space and time. There is evidence forthe form of equation (2) from field measurements of hillslopemorphology [Nash, 1980; Hanks et al., 1984; Rosenbloomand Anderson, 1994] and cosmogenic nuclide mass balance[Monaghan et al., 1992; McKean et al., 1993; Small et al.,1999], and values of D have been measured in a variety ofenvironments [Nash, 1980; Hanks et al., 1984; Reneau et al.,1989;McKean et al., 1993;Rosenbloom and Anderson, 1994;Enzel et al., 1996;Fernandes andDietrich, 1997; Small et al.,1999]; but there is also evidence that equation (2) does notapply in all cases. Several authors have proposed that, as thelandscape steepens, soil flux increases as a nonlinear functionof the surface gradient [e.g., Scheidegger, 1961;DePloey andSavat, 1968; Andrews and Hanks, 1985; Pierce and Colman,1986;Andrews andBucknam, 1987;Anderson, 1994;Howard,1994; Roering et al., 1999; Gabet, 2000]. The most com-monly used expression for this nonlinear flux, which wasderived from a force‐balance analysis on hillslope sedimentbyAndrews and Bucknam [1987] andRoering et al. [1999], is

q ¼ �Krz

1� jrzj=Scð Þ2 ; ð3Þ

where K is a constant with the same units as D, and Sc is acritical surface slope (throughout the paper, “slope” refers tothe magnitude of the land surface gradient). Equation (3)implies that the relationship between flux and slope is nearlylinear when ∣rz∣ � Sc, but that q increases rapidly as theslope approaches Sc. It has been proposed that the apparent

nonlinearity in the relationship between flux and slope is aconsequence of nonlocal dependence of the linear flux lawintroduced by surface gradients that vary over the distanceof a typical transport event [Schumer et al., 2009; Foufoula‐Georgiou et al., 2010; Tucker and Bradley, 2010]. Whetherit is a description of process mechanics or a parameterizationof nonlocal effects, the applicability of the nonlinear law hasbeen demonstrated by field studies [Andrews and Bucknam,1987; Roering et al., 1999], laboratory experiments [Roeringet al., 2001a], and numerical experiments [Howard, 1994;Tucker and Bras, 1998], and a number of studies haveexplored the implications of the difference between the linearand nonlinear laws for hillslope form [e.g., Anderson, 1994;Roering et al., 1999, 2001a, 2001b, 2007]. The main differ-ence in equilibrium form is that hillslopes onwhich soil flux isnonlinear have profiles that are concave‐down near thetopographic divide, similar to the profile generated by a linearflux (Figure 1a), but become increasingly straight furtherfrom the divide, with slopes approaching Sc (Figure 1b).

3. Numerical Methods

[6] The different transport laws also have implications forthe numerical methods used to solve the model equations.For the linear model, substituting equation (2) into equation (1)yields

@z

@t¼ �r

�sU þ Dr2z; ð4Þ

which indicates that hillslope erosion can be treated as a lineardiffusive process, with D the diffusivity. As I explain insection 3.2, this allows the use of implicit numerical methodsthat offer unconditional stability without compromising accu-racy. For the nonlinear model, however, the expression for theeffective diffusivity, K/(1 − (∣rz∣/Sc)2), depends on spatialderivatives of z, and can therefore vary in space and time.This prevents the straightforward formulation of an implicitmethod, and so previous efforts to solve the nonlinear diffu-sion equation for hillslopes have used explicit methods[Andrews and Hanks, 1985; Andrews and Bucknam, 1987;Howard, 1994; Roering et al., 1999], which have significantlimitations. I discuss the most common explicit methods andtheir limitations in section 3.1 and propose alternatives insection 3.2.

Figure 1. Perspective views derived from laser altimetry maps of (a) Gabilan Mesa, Salinas Valley,California, and (b) Zabriskie Point, Death Valley, California. Grid spacing is 1 m, and tick spacing is100 m. Data are from the National Center for Airborne Laser Mapping (NCALM, www.ncalm.org).

PERRON: NUMERICAL METHODS FOR NONLINEAR HILLSLOPE TRANSPORT F02021F02021

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3.1. Explicit Methods

[7] Substituting equation (3) into equation (1), using theEuclidean norm for ∣rz∣, applying the divergence operator,and solving for the time derivative yields [Roering et al.,1999]

@z

@t¼ �r

�sU þ K

r2z

1� jrzj=Scð Þ2

26664

þ 2

@z

@x

� �2@2z

@x2þ @z

@y

� �2 @2

@y2þ 2

@z

@x

@z

@y

@2z

@x@y

� �

S2c 1� jrzj=Scð Þ2� �2

37775; ð5Þ

[8] The most straightforward finite difference approach tosolving an initial value problem based on equation (5) is theforward‐time, centered‐space (FTCS) method, which usesforward differencing in time and centered difference approx-imations for the spatial derivatives:

znþ1i; j � zni; j

Dt¼ �r

�sU þ K

znxx þ znyy

1� ðznxÞ2 þ ðznyÞ2� �

=S2c

264

þ2ðznxÞ2znxx þ ðznyÞ2znyy þ 2ðznxznyznxyÞS2c 1� ðznxÞ2 þ ðznyÞ2

� �=S2c

� �2375; ð6Þ

where the superscripts indicate the time level, with n thepresent time, Dt is the time step, and the second‐orderspatial difference approximations are

zx ¼ zi; jþ1 � zi; j�1

2Dx� @z

@xð7aÞ

zy ¼ ziþ1; j � zi�1; j

2Dy� @z

@yð7bÞ

zxx ¼ zi; jþ1 � 2zi; j þ zi; j�1

Dxð Þ2 � @2z

@x2ð7cÞ

zyy ¼ ziþ1; j � 2zi; j þ zi�1;j

Dyð Þ2 � @2z

@y2ð7dÞ

zxy ¼ ziþ1; jþ1 � ziþ1; j�1 � zi�1; jþ1 þ zi�1; j�1

4DxDy� @2z

@x@y; ð7eÞ

where subscripts i and j are the spatial indices in the y andx directions, respectively, and Dy and Dx are the grid spa-

cings. Note that r2z ≈ zxx + zyy and ∣rz∣ ≈ffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2x þ z2y

q.

[9] An alternative explicit method can be arrived at bysubstituting equation (3) into equation (1) and writing theresult in terms of a nonlinear diffusivity, Dnl,

@z

@t¼ �r

�sU þr � Dnl zð Þrzð Þ; ð8Þ

where Dnl(z) = K/(1 − (∣rz∣/Sc)2). The divergence is thenapproximated by spatial differencing of the quantity Dnl(z)rz,


Dt¼ �r

�sU

þDnl zni; jþ1=2

� �zni; jþ1 � zni; j

� �Dxð Þ2

�Dnl zni; j�1=2

� �zni; j � zni; j�1

� �Dxð Þ2

þDnl zniþ1=2; j

� �zniþ1; j � zni; j

� �Dyð Þ2

�Dnl zni�1=2; j

� �zni; j � zni�1; j

� �Dyð Þ2 ; ð9Þ

and theDnl values at half steps are obtained by computingDnl

at the grid points,

Dnl zi;j� � ¼ K

1� z2x þ z2y

� �=S2c

; ð10Þ

and averaging the values at the two nearest grid points, forexample,

Dnl zni;jþ12

� �¼ 1

2Dnl zni;jþ1

� �þ Dnl zni;j

� �h i: ð11Þ

This method, which is essentially that used by Andrews andHanks [1985] to model nonlinear scarp diffusion, offersimproved stability relative to equation (6), but the gain issmall relative to that obtained with the implicit methodspresented in section 3.2. I therefore use equation (6), the morewidely used explicit method, for the comparisons in section 4.Each of these methods can be reduced to one independentdimension, such that z = z(x), by setting the y terms equal tozero.[10] Explicit methods such as equations (6) and (9) are

viable ways of solving the nonlinear diffusion equationforward in time, and they are relatively simple to program.But explicit methods suffer from the disadvantage that theyusually have very restrictive stability limits when applied todiffusion equations [Press et al., 1992]. The general rule isthat the maximum stable time step is of order (rx)2/D, thediffusion time across a cell. This is typically much shorterthan the time interval over which landforms evolve. For thecase of a linear hillslope diffusion equation with a soil dif-fusivity of 0.01 m2 yr−1 and Dx = 1 m, for example, thestability criterion requires Dt ] 100 years.[11] For a nonlinear law in which the diffusivity becomes

very large as slopes steepen, the stability limit becomes evenmore restrictive. To illustrate the magnitude of this effect,the nonlinear flux, equation (3), can be written as the sum ofa linear term and a nonlinear term,

q ¼ �Krz� Kjrzj=Scð Þ2

1� jrzj=Scð Þ2 !

rz; ð12Þ


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which, when substituted into equation (1), gives (in onedimension for simplicity)

@z

@t¼ �r

�sU þ K

@2z

@x2þ K

S

Sc

� �2 3� S=Scð Þ2

1� S=Scð Þ2� �2 @2z

@x2; ð13Þ

where S = ∂z/∂x. The stability of the trailing, nonlinear termin equation (13) is governed by the approximate criterionDt ] (Dx)2/Knl, where Knl is the prefactor on ∂2z/∂x2 in thatterm. If K = 0.01 m2 yr−1, Dx = 1 m, and S/Sc = 0.9, forexample, this requiresDt] 2 years. The closer S comes to Sc,the shorter the time step must be. The maximum stable timestep for a model using the nonlinear transport law cantherefore be limited to an impractically short interval by thesteepest locations on the model grid.

3.2. Implicit Methods

[12] An alternative approach, and one that is typicallyused to solve linear diffusion equations, is to use a differencingscheme that is at least partly implicit. If the spatial derivativesare evaluated at the unknown future time level n + 1, thedifferenced equation can often be written as a system oflinear equations that can be solved simultaneously for thefuture elevations of all the points on the grid. The mostcommon method used to solve the linear diffusion equation,for example, is the Crank‐Nicolson method, which is formedby taking the average of the explicit and implicit centereddifferencing schemes. The Crank‐Nicolson method, and itsextension to two dimensions via the alternating directionimplicit (ADI) method, have previously been applied to thelinear hillslope transport law [Perron et al., 2003, 2008;Pelletier, 2008]. Implicit methods are often unconditionallystable, and therefore allow time steps that are comparable tothe timescales of interest in the evolving system.[13] An implicit method cannot be formulated so easily for

the nonlinear hillslope transport law because the differencedequation (equation (6)) does not consist of a linear sum ofelevations, and therefore cannot be cast as a system of linearequations. This is a frequent occurrence in geomorphology,in which nonlinear flux laws are common, and alternativestrategies have been proposed, including spatially localizedapproximations of the governing equations [Willgoose et al.,1991] and approximations involving equilibrium solutions[Howard, 1994]. These strategies have limitations, however,and it is desirable to avoid sources of error beyond thoseintroduced by the underlying numerical approximation. Insection 3.2.1, I discuss a previously proposed implicit methodthat achieves much greater stability than explicit methods inexchange for a modest loss of accuracy, and illustrate how itcan be adapted to hillslope transport laws. In section 3.2.2,I build on the previous method to arrive at an improvedsolution technique that offers still greater stability whileretaining the accuracy of explicit methods.3.2.1. Discrete‐Flux Implicit Method[14] Equations for alluvial sediment transport pose a sim-

ilar challenge to equation (3), in that explicit numericalmethods are subject to stability constraints that requirevery short time steps [Willgoose et al., 1991; Howard, 1994;Willgoose, 2005], while the form of the transport law makesit difficult to formulate implicit methods. Fagherazzi et al.[2002] introduced an implicit method for alluvial sediment

transport that relaxes the constraint on time step duration.Their method can be adapted to any transport law in whichthe flux can be expressed as a function of z and its spatialderivatives.[15] Fagherazzi et al. [2002] approximate the divergence

term in equation (1) by summing all incoming and outgoingfluxes, Q, between point i, j and its neighbors,

@z

@t� �r

�sU þ 1

DxDy

Xk

Qk : ð14Þ

where k denotes the eight neighbors in the three‐by‐threepoint neighborhood centered on i, j. Qk can be expressed as

Qk ¼ �kqk ; ð15Þ

where dk is the transport width between i, j and neighbor kthat converts the volume flux per unit width, qk, into avolume flux. For the nonlinear transport law in equation (3),qk can be approximated by

qk ¼ �KSk

1� Sk=Scð Þ2 ; ð16Þ

with

Sk ¼ zi; j � zkDk

;

and Dk, the distance from i, j to k, is Dx in the x direction,

Dy in the y direction, andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDxð Þ2þ Dyð Þ2

qon the diagonal.

dk in equation (16) can be approximated as Dy in the x direc-

tion, Dx in the y direction, and 2DxDy/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDxð Þ2þ Dyð Þ2

qon

the diagonal. Note that the sign of Sk in equation (16) accountsfor whether the flux is directed from i, j to k or vice versa.[16] Fagherazzi et al. [2002] formulated an implicit

method by approximating the right‐hand side of equation (14),denoted by f(z), with a first‐order Taylor series expansion attime n,

znþ1 � zn

Dt¼ f znþ1� � � f znð Þ þ df znð Þ

dznznþ1 � zn� �

: ð17Þ

When applied to point i, j, the trailing term in equation (17)is evaluated by summing the contributions of all the pointsin the nine‐point neighborhood centered on i, j,


Dt¼ f zni; j

� �þ@f zni; j

� �@zni; j


� �þXk

@f zni; j

� �@znk

� znþ1k � znk

� �: ð18Þ

By separating terms containing present and future elevations,they rewrote equation (18) as a set of linear equations inwhich the unknowns are the future elevations zn+1. Abbrevi-ating the partial derivatives by defining Fk

n = ∂f (zi, jn )/∂zkn, andnoting from examination of equations (14) and (16) that

Fi; j ¼ �Xk

Fk ; ð19Þ


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the separation of terms yields

1þDtXk

Fnk

!znþ1i; j �Dt

Xk

Fnk z

nþ1k

¼ 1þDtXk

Fnk

!zni; j þDt f zni; j

� ��Xk

Fnk z

nk

" #; ð20Þ

where the partial derivative Fk for the nonlinear hillslopetransport law is

Fk ¼ �kK

DxDyDk

1

1� Sk=Scð Þ2 þ2S2k

S2c 1� Sk=Scð Þ2� �2

264

375: ð21Þ

This system ofNxNy equations andNxNy unknowns (whereNx

andNy are the numbers of grid points in the x and y directions)can be solved in the usual way, by writing the unknown futureelevations and the right‐hand side of equation (20) as columnvectors of length NxNy, and constructing an NxNy × NxNy

matrix that operates on the vector of future elevations toproduce the left‐hand side of equation (20). This matrixoperator is sparse, containing at most nine elements per row(corresponding to the nine‐point neighborhood centered oni, j), which permits the use of efficient algorithms for findingthe solution [Saad, 2003; Davis, 2006].[17] A disadvantage of this approach is that the matrix

operator and right‐hand‐side vector must be recalculated ateach time step, whereas the operators in an implicit methodapplied to a linear equation with constant coefficients needonly be constructed once. However, the cost of constructingthe arrays is small relative to the cost of solving the system ofequations, and the net savings relative to an explicit methodare substantial (see section 4) [Fagherazzi et al., 2002].[18] My implementation of this method differs from that

of Fagherazzi et al. [2002] in several details. In addition toadapting the method to the nonlinear hillslope flux law, Iinclude the uplift term in equation (14), whereas they did notinclude any source terms. I explicitly include the transportwidth, d, whereas they used a transport law that accounts forfluvial channel width, and therefore gives volumetric fluxrather than flux per unit width. I calculate fluxes to and fromall neighboring points, consistent with transport by soil creep,whereas they calculated fluxes only along paths specified bya flow routing scheme (the D8 algorithm of O’Callaghanand Mark [1984]), consistent with fluvial transport. Here-after I refer to this method as Q8 to denote that it calculatesfluxes in the 8 discrete directions between a point and itsneighbors. The equations implementing Q8 can be reduced toone dimension by changingDxDy in equation (14) toDx, anddefining k to include the two neighboring points j − 1, j + 1.3.2.2. Improved Implicit Method[19] The Q8 method is a substantial improvement over

explicit methods because it permits much longer time steps.The main disadvantage is that the application of this methodin two‐dimensional models requires the calculation of sed-iment fluxes to and from a discrete set of neighboring gridpoints, rather than allowing the flux to vary continuously inall directions. As I show in section 4, this can make Q8 ordersofmagnitude less accurate than explicit methods, and can alsolimit the method’s stability.

[20] If the expression for the flux is differentiable, it ispossible to avoid calculating the fluxes directly. Rather thansumming fluxes to obtain the rate of change of elevation(equation (14)), I evaluate the flux divergence analytically,and then perform the Taylor series expansion on the finitedifference approximation for the flux divergence, plus sourceterms. I refer to this method as Q‐imp because the fluxes areincluded implicitly in the expression for flux divergencerather than being evaluated directly. The implicit method isformulated as before,


Dt¼ f zni; j

� �þX‘

Xm

@f zni; j

� �@zn‘;m

znþ1‘;m � zn‘;m

� �; ð22Þ

but with f (z) defined as the right‐hand side of equation (6).The summation is performed over all nine points in thethree‐by‐three point neighborhood centered on i, j (‘ = i − 1,i, i + 1; m = j − 1, j, j + 1). The first‐order Taylor seriesapproximation is less accurate than iterative methods such asNewton’s method. An iterative method would, however,require a more complicated formulation and incur a highercomputational cost, and in practice the method in equation(22) yields sufficiently accurate results even for long timesteps (sections 4 and 5).[21] Abbreviating the partial derivatives as F‘,m, it is again

possible to write a system of linear equations in which theunknowns are the future elevations zn+1,

znþ1i; j �Dt

X‘

Xm

Fn‘;mz

nþ1‘;m ¼ zni; j þDt f zni; j

� ��X‘

Xm

Fn‘;mz

n‘;m

" #:

ð23ÞAs with the Q8 technique, the NxNy × NxNy matrix thatoperates on the vector of future elevations is sparse. Theabove equations that implement the Q‐imp method can bereduced to one dimension by setting y terms equal to zeroand performing the summation only over m. The expres-sions for the partial derivatives in one and two dimensionsare given in Appendix A.[22] Equation (23), like equation (20), is independent of the

form of the transport law. Provided the expression for sedi-ment flux is differentiable, the Q‐imp method can be adaptedto another transport law by substituting the appropriateexpressions for f andF‘,m. This is possible even if the equationfor surface erosion is part of a system of coupled equations.For example, if the transport coefficient is a function of soildepth, as studies have suggested for both the linear law[Ahnert, 1967;Furbish and Fagherazzi, 2001;Furbish, 2003;Heimsath et al., 2005;Mudd and Furbish, 2005;Furbish et al.,2009] and the nonlinear law [Roering, 2008], the soil thicknessh(x, y), supplied at each time step by a separate equation, wouldbe included in the differencing scheme and its partial deri-vatives. The coupling of equations, such as those describingthe evolution of surface topography and soil thickness, canimpose additional constraints on stability and accuracy, but asI show in sections 4 and 5, the stability limits of explicitmethods for hillslope evolution are likely to bemore restrictive.

4. Stability, Accuracy, and Efficiency

[23] I evaluated the stability and accuracy of the explicitFTCSmethod, the implicit Q8method, and the implicit Q‐imp


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method by comparing equilibrium numerical solutions inone and two dimensions to analytic solutions (Appendix B),and by determining how errors in transient solutions grow asDt increases. I evaluated the relative computational efficiencyof the methods by measuring the time required to perform thesame simulation on the same hardware over a range of spatialresolutions.[24] One‐dimensional solutions (Figure 2) were computed

for a 100 point array with Dx = 2 m. Initial profiles werehorizontal with an elevation of zero. Boundary points were setto have z = 0 for all t. Profiles at t = 250 kyr, which have peakelevations of approximately 50 m, were chosen as transientsolutions. Equilibrium solutions are the profiles reached aftera total time of 3 Myr, after which the elevations of all gridpoints had ceased to change with each iteration.[25] Two‐dimensional solutions (Figure 3) were computed

on a 100 × 100 point grid withDx,Dy = 2m. Initial elevationswere zero within a 99 m radius of the center of the grid, andequal to the analytic solution (equation (B7)) beyond thisradius. Points beyond this radius were set to have constantelevations for all t. As with the one‐dimensional runs, tran-sient solutions were chosen to be the grids at t = 250 kyrwith peak elevations of approximately 50 m, and iterationcontinued for a total time of 3 Myr to obtain equilibriumsolutions.[26] To evaluate the relative efficiencies of the Q‐imp and

FTCS methods, I performed two‐dimensional runs over1 Myr using the maximum stable time step for each method.The runs were performed on a single processor, and com-putation times were measured for each method for grid re-solutions ranging from 16.5 m to 0.25 m, which correspondto grid sizes of 13 × 13 points to 800 × 800 points for thescenario in Figure 3.[27] All runs used the same representative parameter values,

inferred for hillslopes in the Oregon Coast Range by Roeringet al. [1999]:K = 0.0035m2/yr, Sc = 1.25,U = 0.1 mm/yr, andrr/rs = 2. Thus, the accuracy, stability, and efficiency char-acteristics reported here are not absolute, but the relativecharacteristics of the methods are generalizable to otherparameter values.

4.1. Accuracy

[28] For the equilibrium solutions, error was measured asthe root‐mean‐square deviation of the model solution fromthe analytic solutions in Appendix B. Boundary points withfixed elevations were not included when calculating errors.For the transient solutions, the error for each method wasmeasured relative to the solution forDt = 1 year. The transienterrors reported here therefore reflect the rate at which errorsgrow with Dt, but not the absolute error relative to the truesolution.[29] At equilibrium, errors are independent ofDt (Figures 4a

and 4b) because the only errors are those due to the spatialdifferencing. This is illustrated by the fact that the FTCSand Q‐imp methods, which are based on the same spatialdifferencing scheme (equation (6)) have the same error in onedimension (Figure 4a), and nearly the same error in twodimensions (Figure 4b). In one dimension, equilibrium

Figure 2. One‐dimensional initial condition and solutions.Circles show the locations of grid points.

Figure 3. Two‐dimensional initial condition and solutions:(a) initial condition, (b) transient solution, and (c) equilib-rium solution. All grids are 100 × 100 points. Grid linesare drawn every 2 grid points for clarity.


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Figure 4. Errors in model solutions as a function of time step length for the three numerical methodsdiscussed in the text: forward‐time centered‐space (FTCS) and the two implicit methods Q8 and Q‐imp.(a) Equilibrium one‐dimensional solutions. (b) Equilibrium two‐dimensional solutions. (c) Transient solu-tions. Right‐hand vertical axes give the error as a fraction of the vertical relief. Shaded symbols show sta-bility limits. Stability limits for the Q‐imp method are not shown in Figure 4c because the maximum stabletime step is sufficiently long that runs could not be performed in an integer number of time steps. The changein slope at the smallest values ofDt in Figure 4c occurs because errors in transient solutions are expressedrelative to the solution for Dt = 1 year.


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solutions for all three methods are within ∼0.01% of theanalytic solution, though the Q8method is several times moreaccurate than the other two methods. In two dimensions, onthe other hand, the FTCS and Q‐imp methods yield similarlyhigh accuracy, but the error in the Q8 method is nearly threeorders of magnitude larger.[30] In both one and two dimensions, errors grow linearly

with Dt for all three methods (Figure 4c). The relative errorsin the two‐dimensional solutions are consistently higherthan the one‐dimensional errors by about a factor of 2.5. Buteven for time steps as long as 100 kyr (which are onlypossible with the Q‐imp method; see section 4.2), the rel-ative error in the two‐dimensional solution is less than ameter, which is less than 2% of the relief. Thus, the loss ofaccuracy associated with the use of a time step comparableto landscape evolution timescales is relatively small.

4.2. Stability

[31] As expected, both implicit methods are stable formuch longer time steps than the explicit method. For theparameters used here, both the one‐ and two‐dimensionalexplicit solutions became unstable for Dt > 50 years. TheQ8 method was stable for Dt ≤ 10 kyr in one dimension andDt ≤ 5 kyr in two dimensions, and the Q‐imp method wasstable for Dt ≤ 300 kyr in one dimension and Dt ≤ 100 kyrin two dimensions.[32] Neither implicit method is unconditionally stable

due to an inherent instability in the governing equation: if∣rz∣ > Sc, the Laplacian term in equation (5) becomesnegative, creating antidiffusion that amplifies small pertur-bations in elevation. Situations in which ∣rz∣ exceeds Sc canoccur in initial conditions (though this is not the case for theinitial conditions used here), or because of small errorsin the evolving solution at points where S is close to Sc.Because transient errors grow withDt (Figure 4c), situationswith ∣rz∣ > Sc become more likely when taking large timesteps. Instabilities in the implicit methods typically arose

near the boundaries, where slopes are steepest, whereas in-stabilities in the explicit method arose at various locations onthe grids. When the uplift source term is equal to zero andthe initial conditions contain no slopes steeper than Sc, theimplicit methods are unconditionally stable.[33] Despite the stability limit, the Q‐imp method allows

time steps long enough that the nonlinear hillslope transportlaw is unlikely to be the rate limiting component of a modelthat includes other nonlinear laws. Moreover, the growthrate of transient errors (Figure 4c) suggests that accuracycould become a concern for time steps longer than a fewhundred kyr.[34] Despite the lower accuracy and stability of the Q8

method, there may be situations in which this method ispreferable. For example, it would be appropriate to apply itto a transport law in which the expression for flux containsrules, such as a transport threshold, or is otherwise un-differentiable. Another situation in which the calculation offluxes in discrete directions would be preferred is if fluxesare constrained to follow particular flow paths, as in theapplication by Fagherazzi et al. [2002] to fluvial sedimenttransport. But if the transport law has a differentiable form,the Q‐imp method described here offers greater stability andaccuracy, and is clearly preferable on hillslopes, where fluxestypically vary continuously with direction.

4.3. Efficiency

[35] The per‐iteration computational cost is higher for theimplicit methods than for explicit methods, because bothimplicit methods require solving a large system of linearequations. But the larger time steps permitted by the implicitmethods outweigh the increased per‐iteration cost, and theoverall time savings increases rapidly as the grid resolutiongrows finer. Figure 5 shows the ratio of computation timesfor the Q‐imp method and the FTCS method as a function ofgrid size and resolution. This ratio exceeds one, indicatingthat the implicit method is more efficient, for all grid sizesconsidered here. The time advantage is ∼10X for grids of10 × 10 elements, and increases to ∼50X for grids of 100 ×100 elements. Between grid sizes of 100 × 100 and 300 × 300,the time advantage increases approximately as N3, such thatfor 300 × 300 grids, the code using the implicit methodexecutes roughly 1000 times faster than the FTCS code. Forlarger grids, the time advantage of the implicit method growsmore slowly (perhaps because of the memory overheadinvolved in constructing and solving a large, linear system ateach time step), reaching a ratio of 4200 for an 800 × 800 gridwithDx = 0.25 m. Although the exact time ratios in Figure 5are specific to this particular numerical experiment, otherexperiments not presented here confirm that the time savingsfor other parameter combinations show similar magnitudesand patterns of increase.

5. Coupling Hillslopes to Channel Incision

[36] The coupling of hillslope evolution to channel incisionis of central importance in landscape evolution, but couplinghillslopes that experience nonlinear transport to incisingchannels poses a challenge for numerical landscape evolutionmodels. Real hillslopes can in principle respond continuouslyto the lowering of channels, as steepening toe slopes produceincreases in sediment flux. But if channel incision and hill-

Figure 5. Ratio of computation times for the explicit FTCSmethod and the implicit Q‐imp method as a function of gridsize and resolution for the two‐dimensional scenario inFigure 3. The ratios were calculated from wall‐clock timesrather than CPU times.


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slope transport are treated in separate steps in a model, as isusually the case, the channel incision that occurs over a singletime step can oversteepen the toes of hillslopes beyond thecritical slope, Sc, in the nonlinear transport law (equation (3)).As noted in section 4.2, this creates a negative diffusivity, andthe solution becomes unstable. This boundary effect can limitthe maximum stable time step of explicit methods even moreseverely than the effects described in section 3.1: if the hill-

slope is nearly at Sc where it borders a channel (or anyboundary), and the channel lowers at a rate �, the time stepmust satisfy �Dt/Dx � Sc to avoid antidiffusion.[37] This challenge highlights another advantage of implicit

methods for modeling hillslope evolution. By incorporatingsource terms, which can include fluvial incision in addition touplift, into an implicit scheme, this stability limit can besurpassed. This is already apparent in the examples in section 4.

Figure 6. Modeled evolution of Dark Canyon, Eel River, California. (a) Shaded relief map of the initialsurface, based on laser altimetry from NCALM. (b) Shaded relief map of the surface after 25 kyr. (c) Plotof root‐mean‐square deviation of the final surface from the reference solution as a function of the timestep. The reference solution was chosen to be the solution using Dt = 1 year. Circles are for the explicitFTCS method, and squares for the implicit Q‐imp method.


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Consider the one‐dimensional case in Figure 2, which usesblock uplift of the interior points relative to fixed boundarypoints, a situation equivalent to a pair of incising channels atthe two boundaries. The implicit method’s maximum stabletime step of 300,000 years, if used in an explicit calculation,would produce an increase in the gradient at the boundaries of�r�sU Dt/Dx= 2 × 0.0001 m yr−1 × 300,000 years /2 m = 30,

which would far exceed Sc in a single time step even ifoperating on a horizontal surface. But the implicit method,which solves a system of simultaneous equations rather thantreating all points on the grid independently, allows theentire grid to adjust to the boundary lowering at once.[38] In a landscape evolution model that includes uplift

and fluvial incision in addition to hillslope transport, thecontinuity equation is usually written as

@z

@t¼

�r�sU �r � q on soil-mantled hillslopes

U � � in bedrock channels:

8<: ð24Þ

The implementation proceeds as follows. Calculate the ratesof uplift and fluvial incision in equation (24) at each gridpoint without modifying the elevations in the grid. Includethese rates as source and sink terms in the implicit scheme,in the same way that the source term �r

�sU was included in

equations (6) and (22). Assign boundary conditions suchthat hillslope processes do not modify the elevations atchannel points. Finally, update all elevations in the grid inone implicit step.[39] To illustrate this procedure, and to test the perfor-

mance of the method in a more realistic scenario than theideal hillslopes examined in section 4, I modeled the evo-lution of a steep landscape with an incising channel net-work, with an initial surface derived from airborne laseraltimetry. A 4 m DEM of a roughly 2 km × 1 km section ofDark Canyon, a tributary of the South Fork of the Eel Riverin California, was smoothed so that the surface gradient atall points was less than 1.5. A channel network was definedusing a steepest‐descent flow routing algorithm and a mini-mum drainage area of 3000 m2. The channel network wasassumed to incise at the same rate as rock uplift, such thatchannel elevations remained fixed. Both the explicit FTCSmethod and the Q‐imp method were used to advance thesolution for 25 kyr using time steps ranging from 1 year tothe maximum stable time step for each method. Parametersfor all runs were K = 0.005 m2/yr, Sc = 1.5, U = 0.1 mm/yr,and rr/rs = 2.[40] Figure 6 shows the initial and final surfaces and

compares the stability and convergence of the two methods.The maximum stable time step for the explicit method was10 years, and even this became unstable if the run continuedbeyond 28 kyr. These instabilities always arose where hill-slopes bordered channels, especially near channel junctionsor channel tips. The maximum stable time step for the Q‐impmethod was longer than 25 kyr. Errors grow approximatelylinearly withDt for both methods, and remain tolerably small(0.1% of the relief) even when the Q‐imp method is used tocalculate the solution in a single 25 kyr time step. Thiscomparison with an established explicit method shows that,in realistic scenarios, Q‐imp offers comparable accuracy and

superior stability. In addition, it shows how an implicit modelof hillslope evolution alleviates the problem of couplingsteep slopes shaped by nonlinear soil transport to rapidlyincising channels.

6. Conclusions

[41] I present an implicit method for computing finitedifference solutions to nonlinear hillslope evolution equa-tions. An implicit method previously proposed for alluvialsediment transport and adapted here to nonlinear hillslopetransport improves on the stability of explicit methods, butintroduces errors arising from the calculation of fluxes indiscrete directions. The new approach, called Q‐imp, drawson the strategy of the discrete‐flux method but avoids cal-culating fluxes directly. Comparisons with analytic solutionsin one and two dimensions show that the Q‐imp methodretains the accuracy of explicit methods while offering evenlarger stability gains than the previous implicit method. Theability to take much longer time steps far outweighs the addedper‐iteration computational cost of the implicit method,especially when hillslopes are coupled to incising channels.The Q‐imp method can be adapted to any transport law inwhich the expression for flux is differentiable, and should aidin modeling landscape evolution over large spatial domainswithout sacrificing spatial resolution.

Appendix A: Partial Derivatives of FiniteDifference Approximations

[42] For the one‐dimensional case, the partial derivativesFm in equation (23) are

Fj ¼ �2 ac1þ 2 bc z2x

Dxð Þ2 ; ðA1aÞ

Fjþ1 ¼ ac1þ 2 bc z2x

Dxð Þ2

þ abc21þ 2 bc z2x� �

zx zxxDx

þ 2 abc21þ bc z2x� �

zx zxxDx

; ðA1bÞ

Fj�1 ¼ ac1þ 2 bc z2x

Dxð Þ2

� abc21þ 2 bc z2x� �

zx zxxDx

� 2 abc21þ bc z2x� �

zx zxxDx

; ðA1cÞ

where

a ¼ K

b ¼ 1

S2c

c ¼ 1

1� bz2x:


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[43] For the two‐dimensional case, the partial derivativesF‘,m in equation (23) are

Fi;j ¼ �2 ad1

Dxð Þ2 þ1

Dyð Þ2 !

� 4 abd2z2xDxð Þ2 þ

z2y

Dyð Þ2 !

; ðA2aÞ

Fi;j�1 ¼ ad

Dxð Þ2 � abd2zx zxx þ zyy� �

Dx

� 4 ab2d3z2xzxx þ z2yzyy þ 2 zx zy zxy� �

zx

Dx

� 2 abd2zx zxxDx

� z2xDxð Þ2 þ

zy zxyDx

!; ðA2bÞ

Fi;jþ1 ¼ ad

Dxð Þ2 þ abd2zx zxx þ zyy� �

Dx

þ 4 ab2d3z2xzxx þ z2yzyy þ 2 zx zy zxy� �

zx

Dx

þ 2 abd2zx zxxDx

þ z2xDxð Þ2 þ

zy zxyDx

!; ðA2cÞ

Fi�1;j ¼ ad

Dyð Þ2 � abd2zy zxx þ zyy� �

Dy

� 4 ab2d3z2xzxx þ z2yzyy þ 2 zx zy zxy� �

zy

Dy

� 2 abd2zy zyyDy

� z2y

Dyð Þ2 þzx zxyDy

!; ðA2dÞ

Fiþ1;j ¼ ad

Dyð Þ2 þ abd2zy zxx þ zyy� �

Dy

þ 4 ab2d3z2xzxx þ z2yzyy þ 2 zx zy zxy� �

zy

Dy

þ 2 abd2zy zyyDy

þ z2y

Dyð Þ2 þzx zxyDy

!; ðA2eÞ

Fiþ1;jþ1 ¼ Fi�1;j�1 ¼ �Fiþ1;j�1 ¼ �Fi�1;jþ1

¼ abd2zx zyDx Dy

; ðA2f Þ

where a and b are defined as above, and

d ¼ 1

1� b z2x þ z2y

� � :

[44] The approximations for the spatial derivatives of z inequations (A1) and (A2) are defined as in equation (7).

Appendix B: Analytic Solutions

[45] The equilibrium profile of a one‐dimensional hill-slope with a flux given by equation (3), which was derivedby Roering et al. [2007], is

z* ¼ 1� ln 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x*2

pþ ln 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x*2

p� �; ðB1Þ

where the dimensionless coordinates are defined as x* = x/Land z* = z/(LSc), and the length scale L is 1

2�s�rK Sc/U. The

coordinate system is defined so that the apex of the hillslopeis at x = 0, z = 0.[46] It is also useful to derive an equilibrium solution for an

axisymmetric hillslope for comparison with two‐dimensionalnumerical solutions. The continuity equation for an elevationfield in cylindrical coordinates, z(r, �), is

�s@z

@t¼ �r � �sqð Þ þ �rU ; ðB2Þ

where q = qr̂r + q��̂, and r̂ and �̂ are unit vectors in the r and� directions. I assume a topographic equilibrium (∂z/∂t = 0)and seek a solution for an axisymmetric hillslope with q� =0, so that equation (B2) becomes

@qr@r

¼ �r�s

U � qrr: ðB3Þ

Integrating and applying the boundary condition qr = 0 at theapex of the hillslope (r = 0) yields

qr rð Þ ¼ 1

2

�r�s

Ur: ðB4Þ

Using the transport law in equation (3) and again assumingq� = 0 gives

�KSr

1� Sr=Scð Þ2 ¼1

2

�r�s

Ur; ðB5Þ

where Sr ≡ ∂z/∂r, the radial component of the surface gra-dient. Solving equation (B5) for Sr yields an expression forslope as a function of radial distance,

Sr rð Þ ¼ @z

@r¼ Sc

2L

r�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2L

r

� �2s0

@1A; ðB6Þ

where L is the length scale defined above. Integratingequation (B6) with respect to r yields the radial elevationprofile of the hillslope,

z rð Þ ¼ 2LSc 1� ln 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r

2L

� �2rþ ln 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r

2L

� �2r !" #;

ðB7Þ

where the integration constant has been chosen such thatz = 0 at r = 0. Defining the dimensionless coordinates r* =


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r/L and z* = z/(LSc) reduces equation (B7) to a form similarto that of the one‐dimensional profile in equation (B1),

z* ¼ 2 1� ln 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r*

2

� �2s

þ ln 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r*

2

� �2s0

@1A

24

35:ðB8Þ

For a given value of L, the axisymmetric hillslope has lowerrelief than the one‐dimensional hillslope, as expected for thediverging soil flux.

[47] Acknowledgments. I thank Josh Roering and Brendan Meadefor helpful discussions. Reviews by Greg Tucker and three anonymous refer-ees led to several improvements in the manuscript. This study was supportedby NSF award EAR‐0951672.

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J. T. Perron, Department of Earth, Atmospheric and Planetary Sciences,Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge,MA 02139, USA. ([email protected])


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