Introduction
Motivation
Setting initial conditions for unsteady simulations of the Saint-Venant
equations (SVEs) across large river networks can be challenging. Every element
of the river network must be given initial values of flow and depth, and
these values should be consistent with the inflow boundary conditions
(e.g., from a land surface model) at the starting time to prevent
instabilities. This issue has not been previously addressed in the
literature, arguably because (i) adequate initial conditions are fairly
trivial for small SVE reaches, (ii) hydrological models with large river
networks often do not use the full SVE e.g., or
use it over a smaller set of reaches e.g.,, and
(iii) time-marching models only consider results after spin-up time is
complete (i.e., after the effects of the initial conditions have been washed
out of the system solution), which implies the initial conditions are
entirely irrelevant in analyzing the model results. However, for SVE
solutions of river networks, the initial conditions can dramatically affect
both spin-up time and convergence. Indeed, in our experience, näive
initial conditions can cause the spin-up time to be longer than the time-marching
period of interest. In extreme cases, this can result in divergence and
complete failure of an SVE solver. Consistency between the initial conditions
and the boundary conditions appears to be necessary for short spin-up times,
and thus is the focus of this study. Note that the need for consistency in an
SVE model initialization is due to the coupling of nonlinearity and the water
surface slope in the momentum equation, so common reduced-physics models (not
discussed herein) may not be as sensitive to consistent initial conditions.
Saint-Venant equation modeling arguably dates from Preissmann's seminal work
, followed by decades of
advances in techniques and applications . These
models focused on hydraulics of short river reaches or main stem rivers that
are easy to initialize for flow and depth. It is only recently that the
solvers for large river networks have become practical
, and it is with large networks that
initial conditions are problematic. Indeed, initial conditions and associated
spin-up problems have been recently acknowledged and investigated for
hydrological models e.g., but without
consideration of a separate river network model. Work by
and show that hydrological
model spin-up computational times could be significant and were dominated by
the selected initial hydrological conditions.
Our experience with Saint-Venant river network modeling is that simple
approaches to initial conditions often cause localized numerical
instabilities, slow convergence of the time-marching numerical solution, and
long model spin-up times. Herein, we investigate the initial condition
problem for a Saint-Venant river network model for a given set of inflows
from a hydrological model.
Synthetic vs. observed initial conditions
Model spin-up time is completed when the effects of initial conditions cannot
be observed in the model results. Thus, by definition, the initial conditions
cannot affect the unsteady solution beyond spin-up time. It follows that
initial conditions are irrelevant to the quality of the time-marching
simulation and are only important in how they affect the spin-up duration. We
can imagine a “perfect” set of initial conditions with zero spin-up time,
which would require initial flows and depths consistent with (i) the actual
unsteady behavior prior to the model start time and (ii) the model boundary
conditions; the latter includes both the bathymetric model for the river
channels and the coupled hydrological model providing runoff and baseflows.
Such perfect initial conditions are practically unattainable due to the
sparsity of synoptic flow/depth data as well as unavoidable uncertainty and
errors in both bathymetric and hydrological models. Another way to think of
this is that perfect initial conditions also require perfect boundary
conditions (perfect bathymetry and hydrology), or else some spin-up time is
required to wash out inconsistencies. In general, the spin-up time will be
affected by how far the initial conditions are from the theoretical perfect
conditions.
The key point is that the exact observed river initial conditions (if such
were available throughout a network) will not eliminate or necessarily
reduce spin-up time if the observed data are inconsistent with the model
boundary conditions. Similarly, interpolations of sparse synoptic data will
not be a priori consistent with the boundary conditions and thus cannot
eliminate spin-up time. Inconsistency is a critical concept: the mismatches
between the initial conditions and the boundary conditions can lead to
unrealistic destabilizing impulses in time marching the SVE solution. Such
impulses can require extensive spin-up time to damp their effects. An extreme
example is a high runoff rate into an almost dry stream that can cause a
Gibbs phenomenon at a wave front and negative values for the computed
cross-sectional area . Although several studies show such numerical
discontinuities can be resolved , the high computational cost of damping or resolving is
a burden that seems unnecessary during spin-up
since it cannot affect the time-marching results.
We argue that the primary goal of initial conditions is providing consistency
with the boundary conditions to allow smooth, convergent spin-up of an
unsteady solver. This task can be accomplished with synthetic initial
conditions that are independent of observations. The only practical
discriminators between using observed and synthetic initial conditions are
(i) the effort required to prepare the initial condition data and (ii) the
length of spin-up time.
As demonstrated herein, consistent synthetic initial conditions can be
readily generated for even large complex networks – a task that is daunting
for interpolation/extrapolation of sparse synoptic observations. Indeed,
developing consistent synthetic initial conditions only requires the channel
geometry and hydrological model that are used for the SVE time marching,
supplemented by a steady-state SVE solver (Sect. ).
In contrast, interpolation/extrapolation of synoptic data requires analysis
of the data locations and model geometry, which is likely to require
customization for each river network.
Unfortunately, we cannot definitively prove our second discriminator; the
wide range of possible hydrological models, channel geometry models, and
synoptic observation systems makes it impossible to prove that initial
conditions based upon observations will always require longer spin-up times.
However, no one has proposed an approach for interpolation/extrapolation of
observed synoptic data that provides simultaneous consistency with a
hydrological model, a river geometry model, and the SVE. It follows that we
are on firm ground in stating that any existing approach using observed data
as SVE initial conditions will result in inconsistencies. Finally, as our
work demonstrates that inconsistent initial conditions cause long spin-up
times in SVE models of sufficiently large river networks, it can be argued
that observed data should be deprecated as initial conditions until a
consistent approach is demonstrated and the spin-up can be shown to be
shorter than provided by synthetic initial conditions.
Initial condition approaches
Approaches for specifying initial conditions for the SVE can be grouped into
three main categories: (i) a “synoptic start” applying an
interpolated/extrapolated set of sparse observational data, (ii) a “cold
start” with initial flow rates and flow depths prescribed either as zero
e.g., or from some analytical values, e.g., mean
annual flows and depths, and (iii) a “steady-state” start, which we
describe herein. The metric for evaluating initial conditions is not how
well they reflect available real-world observations but how effective they
are in efficiently providing a consistent set of initial conditions.
Based on our discussion above, the first approach (synoptic start) is
unlikely to be efficient for SVE initial conditions in a large river network
due to inconsistencies between observations and model boundary conditions as
well as inconsistencies caused by interpolating/extrapolating sparse
observations throughout a network. There are no proven approaches to
analyzing consistency and melding observations to hydrological model runoff,
so the river network model spin-up will be subject to random inconsistencies
and instabilities that can delay or prevent convergence.
The second approach, a cold start, provides innumerable possible ways to
create initial conditions. For example, mean annual flows and depths
(e.g., from the NHDPlus data in the US) can provide a smooth and consistent
set of flows and elevations throughout a network. Although such cold start
initial conditions can be internally consistent, they may be far from the
flows/depths implied by the initial hydrological forcing. For example, a
river network model that is started with mean annual values would be
substantially in error if the initial hydrological inflows were from the
monsoon season. As a result of inconsistencies between the selected cold
start values and the hydrological inflows, a cold start can require extensive
spin-up time to dilute or wash out the error. Indeed, the spin-up time
dominated the computational time for the large SVE networks that we previously
modeled in when we used a cold start with mean
annual values. It might be possible to design a cold start approach that is
consistent with the hydrological inflows; however, we suspect that any such
approach is likely to be merely a variant of the steady-state approach
discussed herein.
Herein, we investigate the third approach, steady-state initial conditions, as
a preferred method for initializing a large river network model. With this
idea, a set of consistent initial conditions is one that satisfies both the
t=0 hydrological forcing and the steady-state Saint-Venant equations at
t=0. That is, we know that we cannot match the unsteady SVE at t=0 as
we cannot perfectly know the time-varying nonlinear effects before the model
starts (unless, of course, we run a model for the prior period, which would
be simply time shifting the initial condition problem). The steady-state
approach has the advantage of providing flows and depths that are consistent
across the entire network with all the boundary conditions (inflows and
channel geometry) as well as consistent with the nonlinear governing
equations. These consistencies eliminate destabilizing impulses otherwise
caused by mismatches between the flow/depth in a river reach and the runoff,
so subsequent time marching of the unsteady solution is smooth. Furthermore,
the steady-state solution is the closest available proxy to the unknown
unsteady solution at t=0, so this approach should minimize the spin-up time
required to reach an unsteady time march that is independent of the initial
conditions.
Overview
Herein, we present an efficient approach to establishing a set of steady-state
conditions that provides a consistent and smooth starting point for
time marching an unsteady Saint-Venant simulation. A full model
initialization problem has two parts: (i) determining a set of flows and water
surface elevations that are consistent steady solutions of the SVE for
starting an unsteady solver and (ii) determining the spin-up time needed to
ensure errors in the initial conditions are washed out of the unsteady
solution. The second problem is highly dependent on the network
characteristics and the particular flow and boundary conditions during
spin-up, so for brevity, this work deals quantitatively with solving the
first problem and then illustrates the effects on the second problem.
Methods
Saint-Venant equations
The Saint-Venant equations for temporal (t) evolution of flow and water
surface elevation along one spatial dimension (x) following a river channel
are generally derived using the hydrostatic and Boussinesq approximations
applied to the incompressible Navier–Stokes and continuity equations
. Cross-section averaging to obtain the 1-D equations
is considered reasonable where cross-sectional gradients are smaller than
along-channel gradients. However, the equations are widely used even where
such assumptions are questionable (e.g., near a bridge with multiple immersed
piers), with the effects of significant cross-section gradients or
non-hydrostatic behavior being represented as empirical energy losses. A
number of conservative and non-conservative equation forms have been used,
with different advantages and disadvantages .
Herein, we follow in using cross-sectional area
(A) and flow rate (Q) as principle solution variables of the numerical
system and the local water depth (h) and friction slope (Sf) as
a secondary variables (i.e., variables that depend on A and Q through
auxiliary relationships). The equation set can be written as
∂A∂t+∂Q∂x=ql∂Q∂t+∂∂xQ2A+gA∂h∂x=gA(S0-Sf),
where boundary conditions are the local channel bottom slope (S0) and the
local lateral net inflow (ql), the latter representing both
inflows from the landscape and outflows to groundwater. Auxiliary equations
for h=h(A) are derived from river cross-section data. The Chezy–Manning
equation can be used to provide the friction slope as
ASf=ñ2Q2F,
where ñ is the standard Manning's n roughness coefficient and F
is a convenient equivalent friction geometry ,
which subsumes the conventional hydraulic radius (Rh) using a
definition of
F=1ARh4/3=P4A71/3,
where P=P(A) is the wetted perimeter and Rh=AP-1. Note
that Eq. () fixes a typographical error in Eq. (10)
of and Eq. (3.55) of .
Required boundary conditions for the unsteady Saint-Venant solution are
ql(t) for each stream segment, Qbc(t) at the furthest
upstream node (headwater) in river branches with a Strahler order of 1, and
h with an h(A) relationship at the downstream boundary (assumed
subcritical). The time-marching unsteady solution requires initial conditions
for (Q,A), which can also be given as (Q,h) with A=A(h). Implementation
details of the unsteady solver used herein can be found in
and .
Pseudo time-marching approach
The most obvious approach for finding steady-state initial conditions is to
time march an unsteady solver until a steady state is achieved. That is, we
apply the unsteady solver with time-invariant boundary conditions of
ql(t)=ql(0) and Qbc(t)=Qbc(0) for t0≤t<0 where t0 is our pseudo time start and t=0 is the time for which
we want a set of initial conditions. We call this the pseudo time-marching
method (PTM). The initial condition for PTM is a set of Q(t0) and
A(t0) for each stream segment (e.g., some cold start method as described
above). At first glance, the logic here might seem circular: we are trying to
solve for initial condition set Q(0),A(0) of the unsteady
model and PTM requires specifying Q(t0),A(t0). This begs
the question as to why PTM should be used rather than simply applying a cold
start of the unsteady solver with Q(0)=Q(t0) and A(0)=A(t0). The answer
is that the key difference between the PTM using Q(t0) and A(t0) and a
cold start of the unsteady solver with the same values is that the former has
time-invariant boundary conditions while the latter's are time varying. Thus,
an unsteady solver with time-varying boundary conditions is trying to take an
inconsistent starting condition and converge it to a moving target. In
contrast, the PTM takes the inconsistent starting conditions and attempts to
converge them to a time-invariant target, which is more likely to be
successful. However, the PTM does not a priori ensure consistency between
the Q(t0) and A(t0) starting conditions and the t=0 boundary
conditions. It follows that PTM performance can be subject to the same type
of problems as a cold start depending on the choice of Q(t0) and A(t0)
and the skill of the modeler in their selection. In Sect. ,
we show that PTM typically has problems for large river systems with complex
geometry because the complexity of selecting a reasonable set of
Q(t0),A(t0) to ensure convergence.
The PTM is outlined as Algorithm . A user-selected
parameter (ϵ) is used as a threshold tolerance value for declaring
convergence to the steady state. A typical choice of the tolerance ϵ
is the square root of the computer hardware tolerance. For example, on a
64-bit Intel architecture, the hardware tolerance for a double precision
floating point floating number is 2.2204×10-16, which means a
good choice of ϵ is 1.4901×10-8. As a practical matter,
ϵ of 10-6 or even 10-4 is likely to be sufficient for
initial conditions; that is, as further spin-up time is still required to
dilute initial condition errors, the convergence needs only to be sufficient
for consistency across the network. The method can use a time-step size that
is either constant or varying, with an automatic reduction in step size when
convergence is not achieved in a given time step .
To avoid infinite runtimes for non-convergent behavior (e.g., due to
instabilities developed with inconsistent starting conditions), the solution
is terminated (failure to converge) in Algorithm
after the user-selected Nmax iterations. The starting conditions
for Q(t0),A(t0) are discussed in Appendix A.
Pseudo time-marching method
procedure PSEUDOTIMEMARCHING (Aini,
Qini, ϵ, Nmax)
{Aini,Qini: initial guesses of A
and Q; ϵ: tolerance; Nmax: maximal iteration number}
A←Aini
Q←Qini
i←0
t0←0
for i=1 to Nmax do
Solve SVE at time point ti using unsteady method
Compute error: e←Qt-Qt-1+At-At-1
if e<ϵ then
return Success
end if
ti+1←ti+Δti
i←i+1
end for
return Failure
end procedure
Steady-solution method
The PTM approach (above) results in a steady solution of the unsteady
Saint-Venant equations that satisfies both momentum and continuity for
time-invariant ql(0) and Qbc(0) boundary conditions in the
unsteady solver. However, we can achieve a similar effect more directly by
writing a steady-state version of the Saint-Venant equations as
∂Q∂x=ql∂∂xQ2A+gA∂h∂x=gA(S0-Sf).
A key point, implied by Eq. (), is that the spatial
gradient of steady-state Q over a stream segment is entirely due to the
lateral inflow (ql) without any influence of A. It follows that
for steady ql and Qbc boundary conditions, the flow in the
ith river segment (Qi) that has Strahler order Si must be
the sum of all the Qj for all the j connected reaches of Strahler order
Sj<Si. That is, the steady flow at any point is
simply the sum of all the connected upstream t=0 boundary conditions. The
corresponding A (and hence depth h) can then be computed with a numerical
partial differential equation (PDE) solution of Eq. () for known Q values. Note that
for large river networks, the natural downstream boundary condition is
subcritical, which requires specification of h and the corresponding A as
the starting point. We call this a “steady-solution method” (SSM). To look
at this from another viewpoint, if A is uniform, then
Eq. () devolves the fundamental equation of gradually
varying flow, dE/dx=S0-Sf, where E is the
specific energy. Thus, the SSM corresponds to using the steady-state flow
based on all boundary conditions and solving for surface elevations with a
gradually varying flow solution for non-uniform cross sections.
To efficiently compute the conservative initial Q throughout the river
network, it is useful to apply graph theory as discussed in
. A river network can be classified as a “direct
acyclic graph” (DAG) as a river may split upstream or downstream at a
junction, but the flow cannot loop back to a starting point. The connectivity
of a DAG can be efficiently computed by applying existing graph methods, such
as depth-first search (DFS) or breadth-first search (BFS), which provide
simple and efficient approaches to computing Q(0) for each stream segment
over an entire network. Note that these methods were designed and named by
computer scientists, so “depth” in DFS and “breadth” in BFS do not refer
to hydraulics or river geometry but instead are jargon referring to the
graph network characteristics. For simplicity in the present work, we confine
ourselves to the subset of DAG systems that are simply connected trees,
i.e., where there is never more than a single downstream reach from any
junction (as shown in Fig. ) so that there are no
uncertainties in flow directions or magnitudes. Extending the method to
geometry with multiple downstream reaches (e.g., braiding, canals, deltas)
requires additional rules for downstream splitting of flows that are beyond
the scope of the present work.
Propagation of flow rate Q at a junction.
A simple DFS traversal for Q is shown in
Algorithm . From each headwater node (Qj), the
inflow boundary condition is propagated downstream by adding the value to the
downstream node and including any lateral ql from the upstream
reach (stored in Qk). For river networks, the DFS traversal is highly
efficient and requires negligible computational time for river networks of
105 computational nodes e.g.,. Based on our
experience, the DFS computational costs should be essentially trivial for
even continental-scale systems of 107 nodes.
DFS traversal for Q
procedure QTRAVERSAL
for all i do {initialization}
Qi←0
end for
for each headwater node j with BC Qj(t) do
Qj←Qj(t=0)
k← downstream node of node j
while k is not empty do
Qk←Qk+Qj(t=0)
k←
downstream node of node k
end while
end for
return
end procedure
After the steady Qi for each stream segment is computed,
Eq. () can be solved for the corresponding Ai. We
discretize this equation with the Preissmann scheme, similar to the approach
used for the unsteady Saint-Venant equation in .
The value and derivative for any term are approximated as
f(x,t)≃12(fj+1+fj)∂∂xf(x,t)≃1Δx(fj+1-fj),
where subscripts indicate a node in the discrete system. Using j+1/2 to
represent geometric data that are logically between nodes (i.e., roughness
ñ and S0), Eq. () becomes
2ΔxQj+12Aj+1-Qj2Aj+gΔxAj+1+Ajhj+1-hj-gAj+1+AjS0(j+1/2)+gñj+1/22Qj+12Fj+1+Qj2Fj=0.
These nonlinear equations are similar to the unsteady discrete equations,
except that Q for each computational node is known from the DFS traversal.
Newton's method is used to solve this system for A without linearization,
similar to the approach in . The SSM requires a
starting guess for A to solve the steady-state problem. Herein, we use a
bisection method with the Chezy–Manning equation for normal depth conditions
(discussed in Appendix A). The overall algorithm for SSM is illustrated in
Algorithm .
Steady-solution method
procedure STEADYSOULTION
Call QTraversal()
for all all node j in network do {Initial guess of A}
Call bisection
routine BiSection(Qj)
end for
Solve steady version of dynamic eqn
in Eq. ()
return
end procedure
Computational Tests
Overview
The performance of PTM and SSM are examined with a series of test cases
ranging from simple uniform cross sections over short river reaches to
15 000 km of a real river network. To demonstrate the robustness and
performance of the SSM, we conduct tests from three perspectives: (i) effects
of different cross-section geometries; (ii) scalability with an increasing
number of computational nodes; and (iii) real-world river networks. Two
different computers are used: the cross-section and scalability tests are run
on a computer with 2.00 GHz Intel Xeon D-1540 CPUs and 64 GB of RAM, while
the large network tests are run on a computer with 2.52 GHz Intel i7-870
CPUs and 8 GB of RAM. In both cases, Ubuntu Linux is the operating system
and GNU C++ compiler is used.
(a) Waller Creek and catchment in Austin (Texas, USA). (b) Surveyed
cross sections of main channel for Waller Creek (Texas). Only 149 of 327
cross sections are shown for clarity. Elevations are relative to mean sea
level (data courtesy of the City of Austin).
Effects of cross-section geometry
Test cases for cross-section geometry effects were conducted for synthetic
geometry of simple river reaches without tributaries. Cases included
rectangular, parabolic, trapezoidal, and non-uniform cross sections, with a
range of channel lengths, widths, and computational nodes, as provided in
Table .
Cross-section geometry test cases. WB and
Ssw represent bottom width and sidewall slope, respectively; f
represents the focal length of parabolic shape.
Channel
Number of
Test
length
computational
Cross-section
Cross-section
case
(km)
nodes
shape type
shape detail
Case 1
3.1
78
Uniform rectangular
WB=20 m
Case 2
0.2
6
Uniform trapezoidal
WB=1 m; Ssw=0.5
Case 3
0.3
6
Uniform trapezoidal
WB=0.1 m; Ssw=1.5
Case 4
5.6
71
Uniform trapezoidal
WB=10 m; Ssw=0.5
Case 5
10
167
Uniform quasi-parabolic
f=37.8
Case 6
10
1664
Surveyed bathymetry
Unsymmetrical cross section
Case 7
122
31
Surveyed bathymetry
Unsymmetrical cross section
Scalability
To demonstrate the scalability as the number of computational nodes
increases, we use the geometry and flow conditions of Case 4 in
Table and generate synthetic test cases with increasing
numbers of nodes from a few hundred to over a million in the set: { 560,
2800, 5600, 11 200, 22 400, 44 800, 89 600, 179 200, 358 400, 716 800, 1 433 600 }.
Large river networks
To examine the robustness of PTM and SSM for more realistic conditions over
both small and large scales, we use a section of Waller Creek (Texas, USA) as
well as the entire watershed of the San Antonio and Guadalupe river basins
(Texas, USA). The former is a small urban watershed for which dense
cross-section survey data are available, whereas the latter is a large river
basin that has been previously modeled with the RAPID Muskingum routing model
and the SPRNT Saint-Venant model
.
The Waller Creek study includes two stream reaches and the catchment area
illustrated in Fig. . The total stream length is
11.6 km, which drains an area of 14.3 km2. The layout of Waller Creek is
shown in Fig. a, and parts of the bathymetry
surveyed data from City of Austin are shown in
Fig. b for clarity. Two different model
geometries were considered, which are designated as WCA and WCB. For WCA, the
stream is discretized by 373 computational nodes based on separation of the
surveyed cross sections. WCA neglects the minor tributary of Waller Creek and
includes the full complexity of the surveyed cross sections shown in
Fig. b. In contrast, WCB includes both
tributaries but uses wider computational node separation with only 30 of the
373 surveyed cross sections.
San Antonio and
Guadalupe river network from an NHDPlus V2 flowline.
To test the initial condition approach for a large river network, we use the
San Antonio and Guadalupe river basins (Fig. ), which have
a combined total stream length of 12 728 km (excluding some minor
first-order segments). The model herein uses 63 777 computational nodes,
59 594 segments, and 2643 junctions but is otherwise similar to the model
setup with 1.3×105 nodes used in .
Although the unsteady SPRNT model is typically run by coupling with a
land surface model for headwater and lateral inflows, for the present
steady-state tests we used a synthetic inflow data set for the headwater
inflows. The synthetic flow at each headwater stream was computed based on a
downstream peak flow rate distributed uniformly across all the headwater
reaches. We used the peak flow rate recorded on the main stem of Guadalupe
River at Victoria (Texas) on 19 January 2010 by USGS gauge 08176500. As this
gauge does not include the San Antonio River flows, we divided the peak flow
rate (453 m3 s-1) by the total number of headwater streams in the
Guadalupe River (815) to get a single inflow value that was applied to each
headwater reach (0.55 m3 s-1). The same flow rate was used for the
725 headwater reaches of the San Antonio River network. This approach ensures
that there is flow in every branch in the river network.
As is often the case in large river networks, comprehensive cross-section
geometry data were not available for the San Antonio and Guadalupe rivers.
Indeed, noted data availability and our ability to
effectively use synthetic geometry as one of seven fundamental challenges to
continental river dynamics modeling.
Because the geometry affects both PTM and SSM solutions, we tested four
different estimation approaches for synthesizing geometry (Cases A, B,
C, and D). Case A uses synthetic trapezoidal cross sections using the
approach applied in based on .
In this method, trapezoidal widths (W) are computed from mean annual flows
(Qm) from the NHDPlus data set as W=αQm0.5 with α=1.5.
For the side slope of the trapezoidal cross section, an identical sidewall slope
(45∘) is used throughout the river network. Case B channels were
similar to Case A but included some minor changes to Manning's n, inflow
boundary conditions, and channel bottom slopes in reaches where instabilities
occurred, which was necessary to provide convergence for the PTM (see
Sect. ). Case C channels were based on work of
, who used USGS streamflow measurements in the San
Antonio and Guadalupe river network along with the at-a-station hydraulic
geometry approach to find the best trapezoidal
cross-section approximation for the drainage area. Using this approach, the
bed width (b0) is an exponential function of cumulative drainage area
(AD) as
b0=γADλ,
where b0 is meters, AD is km2, and the coefficients are
γ=12.59 and λ=0.382. The approach
provides reasonable values for trapezoidal channel sidewall slopes over most
of the basin but fails in many of the first-order streams with small
drainage areas (<25 km2) where the computed sidewall slopes are near
zero. For simplicity in the present test cases, a uniform value of
45∘ is used for the sidewall slopes
throughout the river network. Case D uses channel bathymetry data generated
from , which use a height above nearest drainage (HAND) analysis
applied to the National Elevation Dataset (NED) to
provide an automated approach for estimating trapezoid-based composite
cross sections.
Results
Comparison metrics
The overall algorithm efficiency is evaluated by the number of Newton
iterations required for convergence to steady state. The number of Newton
iterations reflects the difficulty in converging the nonlinear solution and
is proportional to the simulation runtime. As this metric is independent of
computer architecture, it provides a universal measure of algorithm
performance. For SSM, we use the number of iterations to converge the area
(A) solution of Eq. (), which is the dominant
computational cost (i.e., the non-iterative graph-traversal solution for Q
is negligible in comparison). For PTM, we use the cumulative sum of Newton
iterations for the (Q,A) solution over all pseudo time steps. Where
converged solutions of PTM and SSM both exist, comparisons (not shown)
indicate the resulting (Q,A) steady-state results are identical within the
convergence tolerance (ϵ=10-6).
Effects of cross-section geometry
Table provides a comparison of Newton iterations for the test
cases of Table for single reaches with different channel
cross sections. The SSM converges quickly across all cases, whereas the
performance of the PTM is always substantially slower than that of the SSM. The
performance of the PTM appears somewhat erratic, which is likely because the
overall number of pseudo time steps depends on how far the starting guess is
from the converged answer and the size of the time step used in the PTM
pseudo time march.
Newton iterations required to achieve
convergence for benchmark geometry test cases. The converged results are identical for both methods.
Relative
Test
PTM
SSM
speed-up
case
iterations
iterations
of SSM
Case 1
327
6
54×
Case 2
73
4
18×
Case 3
136
8
17×
Case 4
773
9
85×
Case 5
8634
76
113×
Case 6
13 765
4
3441×
Case 7
91 234
30
3041×
By comparing the geometric data from Table with the results
in Table , it can be seen that the largest discrepancies between
PTM and SSM performance (Cases 6, 7) are with non-uniform cross sections. In
both of these, the SSM performs O(103) times better, compared to O(10) to
O(102) improvements for simple geometry cases. This result is consistent
with the idea that the performance of PTM depends on how close the starting
guess for Q,A is to the steady-state solution. With
non-uniform cross-section geometry, the starting guess is generally quite
far from the steady-state condition, as it is difficult to a priori estimate
gradients of the water surface that match the nonlinear acceleration
associated with cross-section variability. In contrast, the benchmark tests
with simple cross-section geometry (Cases 1–5) show more modest speed-up by
SSM, which is consistent with the steady-state solution for PTM with simple
geometry being closer to the starting guess. For short reaches with simple
geometry and only a few computational nodes (Cases 2, 3), the speed-up by SSM
is essentially irrelevant.
Scalability
Computing initial conditions using models with varying numbers of
computational nodes for Case 4 in Table provides the
speed-up results shown in Fig. . These tests use simple
trapezoidal cross sections and, consistent with the results above, the
speed-up advantage of the SSM is relatively modest with less than 103
nodes. However, beyond this point, the effective speed-up with SSM is quite
dramatic. It appears that the SSM method becomes more effective than PTM both
with increasing complexity of the cross-sectional geometry and the increasing
number of computational nodes.
Speed-up multiplier of SSM compared to PTM for Case 4 as a function
of the number of computational nodes.
Waller Creek test cases
The results of initial condition convergence for two Waller Creek simulations
are shown in Table . The SSM method dramatically reduces the
total number of iterations to convergence, which is also reflected in
reducing the computer runtime by 99 and 92 % for WCA and WCB, respectively.
Although the absolute runtime for this small system is trivial for either PTM
or SSM, the disparity provides insight into the performance that is confirmed
with the more complicated river network (discussed below).
Total Newton's iterations required to achieve convergence of the
Waller Creek test case.
Pseudo time-marching
Steady-solution
Relative
PTM
SSM
Configuration
iterations (PTM)
iterations (SSM)
speed-up of SSM
runtime
runtime
WCA
2900
23
130×
1.570 s
0.011 s
WCB
890
13
70×
0.037 s
0.003 s
San Antonio and Guadalupe river basins
The results of the full river network computations are provided in
Table , which show the SSM was successful and used a
relatively small number of Newton iterations despite the complexity of the
system. Although results in Sect. and
Sect. indicate that PTM method is acceptable for small
systems, results for large-scale river network simulations are less
promising. In the San Antonio and Guadalupe river network configurations,
three out of four PTM solutions failed; that is, the method diverged from any
selected starting condition and finally caused convergence failure.
Configurations A, C, and D all showed evidence of numerical
instabilities leading to divergence. For example, configuration D in PTM
method failed at convergence after 497 iterations; the maximum L2 convergence
norm of the matrix reached up to 1.51e235, which is unquestionably
divergent. Our inability to converge PTM with any A, C, or D cases led
to development of the ad hoc B setup that provided the only PTM solution on
the full river network. To obtain the B configuration, we identified reaches
where instabilities developed and made minor ad hoc adjustments for Manning's
n, inflow boundary conditions, and channel bottom slopes until the model
showed reasonable convergence behavior (see below). Note that the modeler's
time to tune the system for the PTM method to successfully converge is not
included in the comparisons of Table .
The convergence behavior of the PTM for configuration B is shown in
Fig. . It can be seen that for several hundred time-marching
steps the solution was oscillating rather dramatically but
eventually settled down to a slow, smooth behavior. We believe this is
evidence of the PTM trying to overcome inconsistencies between the
Q(t0),A(t0) starting conditions and the boundary
conditions in the network. Note that PTM for B was not converged to the
same ϵ=10-6 tolerance used for SSM. Instead, the solution was
manually terminated after more than 9 h, when the convergence norm reached 1.6×10-4 and was sufficiently smooth so that it was clear that the
method would eventually converge.
Total Newton's iterations required to achieve convergence for
four configurations of the San Antonio and Guadalupe river network.
PTM Newton
SSM Newton
Relative
PTM
SSM
Configuration
iterations
iterations
speed-up of SSM
runtime*
runtime
A
convergence failure
61
–
–
3 s
B
192 527
51
>3775×
9 h 5 min 8 s
3 s
C
convergence failure
29
–
–
6 s
D
convergence failure
46
–
–
14 s
* The PTM
method was terminated after the L2 convergence norm reached 1.6×10-4, whereas the SSM was converged to the predefined tolerance of
10-6.
Convergence of the L2 norm between consecutive pseudo time-marching
solutions for the PTM with configuration B of the San Antonio and Guadalupe
river network. Note the above figure uses the number of time-marching steps as
compared to the larger number of Newton iterations provided in
Table .
Discussion
Effects on spin-up
As alluded to in the introduction, obtaining an effective model initial
condition is only one step in the initialization of an unsteady model. A
second step is understanding at what time the model results are independent
of any errors or inconsistencies in the initial conditions – i.e., the
spin-up time. Some model spin-up time is generally unavoidable as we never
have exactly the correct spatially distributed initial conditions that are
exactly consistent with spatially distributed boundary conditions. In effect,
eliminating spin-up time requires a set of initial conditions that are not
only consistent with the boundary conditions at t=0 but also consistent
with the boundary conditions for tm<t<0, where tm represents the
system “memory” (or the time interval to wash out a transient impulse).
As an illustration of the scale of the spin-up problem compared to the
initial condition problem, we have run the SPRNT unsteady SVE model
for the San Antonio and Guadalupe river network
using over 30 000 data points of unsteady lateral inflows for 14 days in January 2010.
These boundary condition data were generated from the North American Land Data
Assimilation System (NLDAS). The initial
conditions were generated using SSM, as described above. The initial
conditions were then perturbed by ±20% in every first-order reach,
which provides two slightly different initial condition data sets to compare
to the baseline. In Fig. , the time-marching results for
the perturbed and baseline initial condition cases reach the same state
throughout the network (0.001% threshold value) at 152 and 154 h of
simulation time, respectively. Thus, approximately 160 h represents a
conservative estimate of the expected time for errors in first-order streams
to be diluted in the higher-order (larger) river branches. Note that it only
takes 3.8 s of CPU time to compute initial conditions using SSM and an
additional 5 min of CPU time to compute the time marching during the
spin-up interval with the SPRNT unsteady model. This is 2 orders of
magnitude faster than the 9 h or more of CPU time required just to compute
initial conditions using PTM for the same system.
Spin-up for the San Antonio and Guadalupe river network with the SPRNT
unsteady SVE model initialized using the SSM approach. The positive and
negative 20 % perturbations are for the Q initial conditions in first-order
reaches.
Model performance
In general, the PTM performed poorly except on very simple systems. As the
river network complexity increases, the PTM changes from being somewhat slower
than SSM to being non-convergent. Indeed, the PTM has only one advantage over
the SSM in providing initial conditions to an unsteady SVE solver:
specifically, no new code is needed as PTM uses the same unsteady SVE code.
However, using PTM for large systems requires a frustrating trial and error
approach to tuning the system to obtain convergence. In contrast, the SSM
provides a rapid solution to the initial condition problem because Q is
computed from simple graph traversal (once through the network), and the
subsequent computation of A is “local” (in the sense there is no coupling
between distant computational nodes). Note that in contrast to PTM, the SSM
does not require the modeler to select a set of starting conditions; i.e., the
SSM starting condition (Q) is defined solely by the t=0 boundary
conditions. Thus, different modelers will produce exactly the same Q(0) and
A(0) with the same number of iterations when using the SSM on identical
geometry and boundary conditions. This same cannot be said for PTM as
modelers must select their Q(t0),A(t0) starting
condition and may need to resort to custom model tuning to obtain
convergence.
Herein, we only tested two methods for initial conditions, both based on
finding the steady-state Q,A that are consistent with the
boundary conditions. However, we can also argue that the cold start and
synoptic start (see Sect. ) would likely perform as bad
or worse than PTM. That the cold start would perform poorly follows from the
fact that it has the exact same problem as the PTM (converging over time from
inconsistent starting data) but increases the difficulty by trying to
converge to the unsteady boundary conditions. A cold start effectively turns
the initial condition problem into a spin-up problem. For a cold start model
performing similarly to the PTM for the San Antonio and Guadalupe river
network, we can expect spin-up to require more than 104 time steps of the
unsteady solver.
Although it is possible that a synoptic start could perform better than PTM
or a cold start, it seems likely that any approach to
interpolating/extrapolating sparse observational data across a larger river
network will necessarily result in inconsistencies between the initial
Q,A and the boundary conditions. If such inconsistencies
result in model instabilities (a difficult thing to predict), the overall
model spin-up time could be extensive. The key problem for the synoptic start
is that it requires judgment as to how to best interpolate/extrapolate
observational data for initial conditions, which is contrasted to the SSM
approach of simply using the actual Q(0) boundary conditions and the steady
solver without any further choices by the modeler.
Note that the poor performance of the PTM cannot be attributed to
inefficiencies in the unsteady solver. As discussed in
Sect. , the unsteady solver computed 150 h of unsteady
simulation from the SSM about 5 or 1800× faster than real
time. This implies that the unsteady solver computed 114 min of
real-world unsteady flows in the 3.8 s required for the SSM to compute the
steady-state initial condition. Thus, the SPRNT unsteady solver is quite
efficient – except when given an inconsistent set of initial and boundary
conditions as in the PTM solution.
Limitations
For simplicity, the SSM algorithms presented herein are for common river
networks with only a single reach downstream of each junction. Such tree
networks are a subset of the more general DAG river networks. Extension to
more complex DAG geometry requires the definition of splitting rules to
uniquely define partitioning of Q where the flow splits into multiple
downstream branches. Search algorithms for more complicated DAG forms are
available in the graph theory literature e.g., and
could be readily extended to the SSM.
Fundamental to the SSM approach is the presumption that there are no places
of net flow accumulation or loss throughout the network. All reservoirs and
hydraulic structures are treated as pass through so that all the upstream
flow is propagated through the downstream reaches. However, if there are
known locations of accumulation or loss, the river network could be divided
into subnetworks with separate SSM solutions. For example, upstream of a
reservoir an SSM solution can be used to obtain the inflow to the reservoir,
which can be used with operating information from the reservoir to provide
the correct outflow Q for use in the SSM solution for downstream reaches.
Hydraulic structures that affect h (or dh/dx) as a function of Q can be
readily included in the SSM. To do so merely requires changing the momentum
equation in a reach to model the physics of the structure (as is similarly
done in unsteady SVE models).