![Largest adaptive (solid line) and fixed (dashed line) time step for a stable simulation of Briggs et al. [29] benchmark.](https://www.researchgate.net/profile/Sasan-Tavakkol/publication/351141132/figure/fig4/AS:1024831663386624@1621350317020/Largest-adaptive-solid-line-and-fixed-dashed-line-time-step-for-a-stable-simulation_Q320.jpg)
![Experimental (--) and numerical (-) time series for Briggs et al. [4]](https://www.researchgate.net/profile/Sasan-Tavakkol/publication/351141132/figure/fig6/AS:1134365954318337@1647465327129/Experimental--and-numerical-time-series-for-Briggs-et-al-4_Q320.jpg)
![Numerical (solid line) and measured (x) maximum horizontal run-up in Briggs et al. [4] benchmark.](https://www.researchgate.net/profile/Sasan-Tavakkol/publication/351141132/figure/fig7/AS:1134365954314240@1647465327166/Numerical-solid-line-and-measured-x-maximum-horizontal-run-up-in-Briggs-et-al-4_Q320.jpg)
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Adaptive Third Order Adams-Bashforth Time Stepping
for Extended Boussinesq Equations
Sasan Tavakkol1,
*
, Sangyoung Son1, 2, Patrick Lynett1
1- Department of Civil and Environmental Engineering, University of Southern California, Los
Angeles, California, USA
2- School of Civil, Environmental and Architectural Engineering, Korea University, Republic of
Korea
ABSTRACT
We develop the third-order adaptive Adams-Bashforth time stepping and the second-order finite
difference equation for variable time steps. We incorporate these schemes in the Celeris Advent
software to discretize and solve the 2D extended Boussinesq equations. This software uses a
hybrid finite volume – finite difference scheme and leverages the GPU to solve the equations
faster than real-time while concurrently visualizing them. We simulate several benchmarks using
the adaptive time stepping scheme of Celeris Advent and demonstrate the capability of the
software in modeling wave-breaking, wave runup, irregular waves, and rip currents. The
adaptive scheme significantly improves the robustness of the model while providing faster
computational performance.
Keywords
adaptive time stepping; Adams-Bashforth; Celeris Advent; Boussinesq; wave modeling
1. Introduction
Numerical simulations are now essential tools in understanding any coastal phenomena ranging
from wave behavior in ports to designing recreational surfing activities. Among nearshore
models, the Boussinesq-type models have become the most popular approximations of the
Navier–Stokes equations for coastal engineering, thanks to their ability to represent the major
forces and interactions while requiring significantly less computational power compared to any
3D model. Nevertheless, these models are still computationally more expensive than their
counterpart, non-linear shallow-water equations. This higher computational demand limits their
application in low budget engineering projects. However, recent advances in computer hardware
and software have lowered the barrier to entry for using the Boussinesq-type models. Recently,
Tavakkol and Lynett [33] introduced a GPU accelerated software to solve the extended
Boussinesq equations [23], called Celeris Advent. This software effectively democratized the use
*
Corresponding author: tavakkol@usc.edu, currently at Google Research, New York, New York, USA
of Boussinesq-type models by letting the users run faster than real-time simulations on a
consumer-level laptop and within a user-friendly interactive environment. In the current study,
we explain and validate the adaptive time stepping scheme that we developed and incorporated
in Celeris Advent [32].
The past three decades have seen a significant effort from the coastal research community
towards developing Boussinesq-type models. Peregrine [25] derived the “standard” Boussinesq
equations by assuming that both nonlinearity and frequency dispersion are weak and are in the
same order of magnitude, therefore retaining only the lowest orders of nonlinearity and
frequency dispersion terms. Because of this assumption, the standard Boussinesq equations are
not applicable to very shallow or deep water. In shallow water, nonlinearity becomes more
important than frequency dispersion as the wave gets closer to the shore, violating the
assumption of same order nonlinear and dispersion effects. In the deep-water condition, the
frequency dispersion cannot be considered weak for any depth greater than one-fifth of the
equivalent deep-water wavelength, further limiting the application of the standard Boussinesq
equations.
The deep-water restriction of the standard Boussinesq equations is often too limiting for
engineering applications, especially where the incident wave energy spectrum consists of many
frequency components. Several modified forms of Boussinesq equations have been successfully
developed to extend their applications to deeper water depth (e.g., [7, 22, 24]). Among these
extended Boussinesq equations, those introduced by Madsen and Sørensen [23] and Nwogu [24]
are widely in use.
The weak nonlinearity restriction which limits the applicability of modified Boussinesq
equations in very shallow waters is removed by eliminating this assumption in fully non-linear
models (e.g., [17, 43]). FUNWAVE [9] and COULWAVE [19] are widely-used numerical
implementation of the fully non-linear Boussinesq equations. These models have proven
themselves successful in a wide variety of applications such as wave runup [21], wave-current
interaction [28], wave generation by underwater landslides [18], rip and longshore currents [5],
etc.
Fully non-linear models are known to better represent steep waves in shallow water and are
shown to agree better with controlled laboratory experiments as well as with analytical solutions;
however, their application in hindcasting or forecasting a real-world field site may not benefit
from their higher-order accuracy because of uncertainties in the field site conditions (e.g., in
boundary condition and bathymetry). Considering that these models are also computationally
more expensive than the weakly non-linear Boussinesq models, their application in real-world
scenarios might not be always justifiable. Bearing these facts in mind, we chose to solve the
extended Boussinesq equations introduced by Madsen and Sørensen [23] in Celeris Advent and
achieved sufficient accuracy with faster than real-time simulation speed.
Celeris Advent was originally developed to use a fixed third-order Adams-Bashforth time
stepping scheme as the predictor step and an optional fixed fourth-order Adams-Moulton time
stepping scheme as the corrector step. We later dropped the correction step as the predictor step
proved to be sufficiently accurate, provided that an adequately small time step is chosen. Celeris
Advent with the fixed time step is validated for wave and current simulation in [33, 36] by the
authors of the software as well as in [3, 26, 27] by other researchers. In this paper we describe
the development of our new numerical time stepping scheme which accepts variable time step
values and therefore allows the model to keep the maximum local CFL number constant by
adaptively calculating the required time step value over the time. We developed third-order
adaptive equations for the Adams-Bashforth time stepping scheme and incorporated them in
Celeris Advent [32]. The need for an adaptive time stepping formulation arose from our
observations of model instability in experiments with runup on steep surfaces. In these cases, the
flow velocity grows rapidly and the subsequent increase in the local Froude number leads to
instability. We validate the new adaptive scheme incorporated in Celeris Advent (v.1.3.4). The
open source code and its compiled version are available to download at www.celeria.org.
Shi et al. [29] introduced a high-order adaptive time stepping solver for Boussinesq-type
equations using Runge–Kutta time stepping. This scheme utilizes a fixed time step throughout
the multi-level time stepping phase and adjusts the time step only for the next time level
according to the CFL number. Therefore, this scheme is not theoretically fully adaptive. To the
best of our knowledge, we are the first to introduce a third order adaptive time stepping scheme
to solve Boussinesq equations [32]. We developed our third order adaptive scheme for Adams-
Bashforth time stepping in a general format such that it can be used to solve equations other than
the Boussinesq equations as well.
This paper is organized as follows. We describe the extended Boussinesq equations and a
specific rearrangement in Section 2. In Section 3, we briefly explain our hybrid finite volume –
finite difference scheme to solve the Boussinesq-type equations in space but explain in more
detail their solution on an adaptive time grid. We give a short explanation on the development of
Celeris Advent in Section 4, as the details are given previously in [32, 33]. In Section 5 we
demonstrate the capability of our model by applying it on four benchmarks including wave
breaking, wave runup, irregular waves, and rip currents. Conclusion, acknowledgments, and
references make up the last three sections of this paper.
2. Extended Boussinesq equations
Celeris solves the extended Boussinesq equations derived by Madsen and Sørensen [23]. These
equations for 2DH flow read as
(1)
where η is the free surface elevation measured from the still water surface elevation, h is the total
water depth, P and Q are the depth-integrated mass fluxes in x and y directions, respectively, g is
the gravitational acceleration coefficient, and f1 and f2 are the bottom friction terms. Subscripts x
and y denote spatial differentiation, with respect to the corresponding direction, and subscript t
denotes temporal differentiation. Finally, ψ1 and ψ2 are the modified dispersive terms defined as
(2)
(3)
where d is the still water depth and B is the calibration coefficient for dispersion properties of the
equations. We use B=1/15 as suggested originally in [22] and widely adopted thereafter.
The modified dispersive terms (ψ1 and ψ2) approach to zero as d decreases to zero. This is a
favorable property because as a wave approaches the shore (i.e., the still water depth decreases),
it gets steeper, driving the waveheight (H) to still water depth ratio ε = H/d higher and the square
still water depth to wavelength (L) ratio μ2 = (d/L)2 lower. Note that ε and μ2 are representatives
of the nonlinear effects and the dispersive effects, respectively. This progressive change in the
values, as a wave gets closer to the shore, invalidate the underlying assumption of O(ε)=O(μ2) for
the derivation of Boussinesq equations and pushes the governing equations to the range where
NLSW equations suit better. For d = 0, the extended Boussinesq equations, Eq. (1), reduces to
the NLSW equations. In areas where the still water surface elevation is not defined, such as on
lands above the sea level, we set d = 0, so the solver automatically switches to NLSW equations.
The extended Boussinesq equations provide sufficiently accurate linear dispersion and shoaling
characteristics for values of kd < 3, where k is the wavenumber.
We rewrite Eq. (1) in a conservative form which is suitable to the applications of finite
volume method. Expressing the free surface elevation as η=h−d we have
(4)
We let b denote the bottom elevation from a fixed datum and ws, a constant number, denote
the still water elevation from this datum. Therefore, we have d=ws−b. Since ws is constant in
space and time, the derivative of the still water depth and the bottom elevation becomes equal,
but with a negative sign. Furthermore, the temporal derivative of the still water depth, dt,
becomes zero assuming a constant bottom elevation in time.
We make a variable change by introducing w=h+b, where w is the water surface elevation,
measured from the fixed datum. This variable change helps us employ well-balanced numerical
schemes for discretization of the advective terms, which is discussed in detail in [16]. Using the
new notations, we rewrite Eq. (1) as
(5)
where newly introduced variables are
In Eq. (5), U is the conservative variables vector, F(U) and G(U) are the advective flux vectors,
and S(U) is the source term which includes bottom slope, friction, and dispersive terms.
3. Numerical schemes
We use a hybrid finite volume – finite difference scheme on a uniform spatial Cartesian grid
which we introduced in [33] and refer to it as TL17. We developed TL17 following similar
works in [8] and [41]. In this scheme, the NLSW subset of the extended Boussinesq equations,
Eq. (1), is discretized using a second-order well-balanced positivity preserving central-upwind
scheme introduced by Kurganov and Petrova [16]. This scheme, known as KP07, is a finite
volume method (FVM) to solve the Saint-Venant system of shallow water equations. The
modified dispersive terms are discretized using the central finite difference method (FDM).
3.1 Spatial discretization
Following Wei and Kirby [42], Eq. (5) can be rearranged as
(6)
(7)
(8)
where
(9)
(10)
(11)
(12)
(13)
(14)
(15)
The left-hand side terms in Eq. (6)-(8) are discretized in time, [F*(Q)]t and [G*(P)]t are evaluated
by extrapolation in time, and the rest of the terms on the right hand side are known in the current
time step. This rearrangement allows us to rewrite Eq. (5) as ODE’s in time.
As mentioned before, we use KP07 to solve the NLSW subset of the extended Boussinesq
equations. We chose this scheme because it is well-balanced (i.e., preserves stationary steady
states) and guarantees the positivity of the computed fluid depth. Furthermore, it naturally
supports a dry state, with no need to keep track of the wet-dry front, and it can accommodate
discontinuous bottom topography. These qualities were required to develop Celeris Advent as an
interactive solver. TL17 uses KP07 as the FVM solver and adds the dispersive terms discretized
by central FDM to the source term in KP07. The spatial domain is discretized by rectangular
cells with fixed sizes of Δx and Δy. Each cell plays the role of a control volume for the FVM
discretization. Cell centers and their corresponding cell averaged values are used as the grid
points in FDM. The details of the KP07 and TL17 solver are explained in [16] and [33],
respectively.
3.2 Time Integration
3.2.1 Uniform Time stepping
Uniform time integration is performed by the third-order Adams-Bashforth scheme which reads
as
(16)
(17)
(18)
where the superscripts denote the step number in time, with n being the last step with known
values. This time stepping is explicit in time, meaning that all the variables on the right-hand side
of the equations are known. Since the variables at previous time steps are not defined in the very
first two time steps of the simulation (i.e., n=1 and n=2), a first order Euler time integration is
used to bootstrap the simulation until n=3.
The water surface elevation, wn+1, is directly calculated from Eq. (16). However, to calculate
the flux terms, P n+1 and Q n+1 the following set of implicit equations need to be solved:
(19)
(20)
where
(21)
The coefficient matrices in Eq. (19) and Eq. (20) are of tridiagonal form. We adopted the
Cyclic Reduction (CR) method to efficiently solve these set of equations on the GPU in Celeris
Advent.
3.2.2 Adaptive Time stepping
In the adaptive mode, the software keeps the maximum local CFL at a constant value, by using a
variable time step. We define the CFL number for Celeris Advent as
(22)
where c is the wave celerity in shallow water. The theoretical stability condition for KP07, and
thus for TL17, is CFL < 0.25 [16], however, in practice we often use a 0.5 safety factor and keep
CFL smaller than 0.125. In the adaptive time stepping, Celeris calculates the next time step size,
Δt, from Eq. (22) for a given constant CFL number and using the velocity and celerity values of
the current time step.
3.2.2.1 Third-order Adaptive Adams-Bashforth Equation
We aim to solve the following ODE:
(23)
where Xt denotes derivative of X with respect to t, and t0 denotes t = 0. Let Xi+1 denote our target
variable at the next time step, ti+1, and Xi denote the same variable in the current time step, ti. We
can make an approximation of f(t, X) by the third-degree polynomial, p(t), such that:
(24)
Employing the Lagrange formula for polynomial interpolation we have:
(25)
where Xtj = f(tj, Xj). Now we can write:
(26)
(27)
(28)
Using the sliding technique to substitute t with t+ti and introducing Δti=ti+1-ti, Δti-1=ti-ti-1, and Δti-
2=ti-1-ti-2 we can write:
(29)
Finally, after integration we have:
(30)
Eq. (30) is the third order adaptive Adams-Bashforth time integration equation. As a correctness
check, applying Δti = Δti-1 = Δti-2 = Δt in this equation yields to the same third order Adams-
Bashforth equation we use for uniform time stepping:
3.2.2.2 Variable-step Second Order Finite Difference Equations
To solve Eq. (7) and Eq. (8) using Eq. (30) we also need to derive the second order finite
difference discretization equation for variable time steps. Let’s approximate Y with , and its
derivative, , with . We use a polynomial approximation for such that it satisfies:
(31)
Let’s define:
(32)
The finite difference approximation of yields:
(33)
To meet the conditions in Eq. (31), we must have:
(34)
Inverting the matrix of coefficients in (34), we can find the value of c1:
(35)
For the special case where Δti-1 = Δti-2 = Δt
which is the well-known 2nd order backward difference.
Similarly, to derive , we can use the following polynomial approximation and solve it to
find
:
(36)
(37)
(38)
For the special case where Δti-1 = Δti-2 = Δt
which is the well-known 2nd order central difference.
To derive we can write:
(39)
(40)
(41)
For the special case where Δti-1 = Δti-2 = Δt
which is the well-known 2nd order forward difference.
3.2.2.3 Prediction Equations
Now, we have the tools to derive the prediction equations of w, U*, and V* with adaptive time
stepping. Deriving the equation for w is straightforward and yields:
(42)
For U* we have:
(43)
To calculate U* we must also derive the equations for F*t. We will use the second order finite
difference equations derived in the previous section:
(44)
(45)
(46)
Plugging Eqs. (44) to (46) in (43) gives the equation to calculate U* in the next step. The
equation for V* is derived similarly, and not represented here for the sake of brevity.
3.2.2.4 Implementation
We noticed that sudden increases in the time step sometimes lead to model instability. Therefore,
we define a custom version of the exponential moving average and use it to set the value of Δtn,
as follows:
(47)
where α is a pre-defined coefficient between 0 and 1. We found 0.01 < α < 0.5 to work well with
Celeris. We call Eq. (47) lazy exponential moving average as it lets the time step to drop
instantly, if required, but rise only gradually.
3.3 Boundary conditions
Two layers of ghost cells are considered at each boundary side and are used to implement the
boundary conditions. Five types of boundary condition are implemented in Celeris Advent: fully
reflective solid wall, sinewave maker [33], sponge layer [32], irregular wavemaker, and time
series [36]. These boundary conditions can be applied to any of the four boundaries of the field.
3.4 Wave breaking
Wave breaking is not implemented in Celeris with a direct treatment. However, our experiments
[33, 36] show that the numerical dissipation of the scheme caused primarily by using the
minmod limiter imitates physical dissipation introduced by wave breaking. Kirby et al. [29] also
discuss that in models with shock-capturing schemes, the implementation of an explicit
formulation for breaking wave dissipation might be unnecessary. The MOST and GeoClaw
models, commonly used in tsunami studies, are other examples in which numerical dissipation
mimics wave breaking [2, 37, 39]. In the next sections we show that the wave breaking effect
caused by the use of the limiter is able to adequately mimic the physical phenomenon as shown
by comparisons of the numerical results with experimental measurements. As discussed before,
our solver automatically reduces to the NLSW equations to continue simulating the runup on the
beach.
3.5 Wet and dry cells
There is no definition for wet/dry cells in Celeris. All the cells are considered wet, though some
with water depth of zero or close to zero. This treatment is possible thanks to the finite volume
method and KP07 scheme and is favorable for our GPU implementation as it avoids branching in
the GPU computations. To determine the runup or inundated area, one needs to define a
minimum depth of inundation to distinguish wet and dry cells.
4. Software development
Development of Celeris consisted of two major steps. Firstly, a robust solution for the
mathematical model was developed such that it could be implemented on the GPU. The second
step was the implementation of a user-friendly software which can drive the model. To fulfill the
first step, we first implemented the derived equations and the mathematical model in MATLAB
to validate them and improve them where necessary. We call this MATLAB series of the model
Celeris Zero [32]. The purpose of developing Celeris Zero, was mostly validating the underlying
mathematical model of the software, and not its implementation on the GPU. Development of the
mathematical model and its implementation in Celeris Zero started in spring of 2014 and the first
version was running in early 2015.
After we became confident that our model was suitable for our goal of developing an
interactive and immersive coastal simulation software, we started the developments of the first
official series of Celeris, called Celeris Advent [32], in winter 2015. Our goal in development of
Celeris Advent was to provide a hassle-free software which can run on off-the-shelf Windows
machines with minimum preparation. Therefore, we selected Microsoft’s Direct3D library and
its HLSL shader language to harness the power of the GPU, rather than general purpose GPU
programming libraries such as CUDA, which requires some level of prior knowledge for
preparation of the system and runs only on devices with NVIDIA GPU’s. Celeris Advent is
implemented mostly in C++ and HLSL, and it is an open-source code developed and
redistributed under the terms of the GNU General Public License as published by the Free
Software Foundation.
We released Celeris Advent (v.1.0.0) to the public in December 2016 [33]. At the time of
writing this paper, Celeris Advent is downloaded over 2000 times by users from academia,
industry, and government spanning over 50 countries and its user manual has been translated to
Farsi, Spanish and Italian by independent users. Applications of Celeris Advent extend from
research on coastal phenomena to recreational surf forecasts. One of our applications at
University of Southern California, is a website which provides a five-day forecast of wave
conditions at several US coasts, available at http://coastal.usc.edu/waves/. The adaptive time
integration scheme is implemented in Celeris Advent (v.1.3.4) which is available to download at
www.celeria.org. We recently introduced a new series of the Celeris software, called Celeris
Base [34, 35], in which a modern game engine is used for implementation and immersive
visualization capabilities are added to the software. Celeris Base is more suitable to researchers
who would like to extend the capabilities of the software. We still recommend using Celeris
Advent for general purpose simulations.
4.1 GPU implementation
Celeris harnesses the power of the GPU to run the TL17 numerical model as well as to
concurrently visualize the results. We use shader programming for this purpose. The advantage
of shaders compared to general purpose GPU programming languages such as CUDA is their
portability between hardware and to some extent, between operating systems. Shaders are the
core of 3D graphics rendering and game development. They were originated with the purpose of
tweaking lighting effects in 3D rendering (hence the naming) but quickly became much more
powerful and today are widely used in fixed-function rendering pipelines in graphics API’s such
as OpenGL and Direct3D.
The disadvantage of using shaders to implement our numerical solver is the need to restate
the problem in terms of graphics primitives and setting up a dummy rendering pipeline. In
Celeris Advent, we use Direct3D and its shader programming language, HLSL, to solve the
governing equations on the GPU. This required setting up a dummy rendering pipeline to render
a quad with two triangles and six vertices. We then divided the numerical scheme into smaller
steps suitable to implement in pixel shaders. It was also required to fit the problem in the texture
data structures. For example, the conservative variables vector in a cell, Uij=[wij, Pij, Qij]T, are
stored in red, green, blue, color components of a texel (i.e., a texture cell, similar to a pixel in a
digital image). Therefore, a texture of size (nx+4)×(ny+4) on the GPU stores the conservative
variables vectors for the solution domain of size nx×ny, where four cells are added in each
direction to account for the two layers of ghost cells on each boundary side. Other values in cells
are also stored similarly in texture data structure, while globally constant variables are stored in
constant buffers.
The explicit steps of the computation are relatively easy to implement on the GPU. That is
because pixel shaders are designed to apply a kernel function on every texel of input textures and
store the results in output textures. For example, the reconstruction step in KP07 is an explicit
step, where the result depends only on some known values from the previous time step on the
cell itself and its neighbors. But solving implicit equations such as Eq. (19) and Eq. (20) on the
GPU can be challenging because the output on each cell is tied to the output of the other cells at
the same time step and therefore the system needs to be solved simultaneously. We employed
Cyclic Reduction (CR) algorithm to solve these equations on the GPU[33]. CR consists of two
phases: forward reduction and backward substitution. In the forward reduction phase, the system
is successively reduced to a smaller system with half the number of unknowns, until a system of
2 unknowns is achieved which can be solved trivially. In the backward substitution phase, the
other half of the unknowns are found by substituting the previously found values into the
equations. This process is well illustrated in Figure 4 of [33].
5. Model tests
5.1 Solitary wave run-up on a conical island
As the first validation test, we run a case which we previously used to validate Celeris Advent
(v.1.0.0) with fixed time step in [33]. The experimental data of Briggs et al. [4] for solitary wave
interaction around a conical island is frequently used to validate numerical models [11, 21, 38,
40] and has become a standard benchmark for Boussinesq-type models. The experimental setup
consists of a circular island with 7.2 m base diameter and ¼ side slope, s, located in a 30m×25m
wave tank with 0.32 m depth. Out of several cases in this set of experiments, we only test the
case with target relative waveheights of H/d=0.20 which is expected to be more difficult for
numerical models to simulate because of the higher non-linearity and the wave breaking
condition. Briggs et al. [4] recorded the wave maximum run-up on the island and surface
elevation time series on several gauges, which are used in this study for validation.
Our numerical setup for the conical island experiments consists of a 30m×30m domain with
the conical island in the center and a solitary wave placed as an initial condition near the west
boundary. The basin is extended by 5m to accommodate the solitary wave as an initial condition.
The west and east boundaries are set to the sponge layer condition, while the north and south
boundaries are fully reflective solid walls. The domain is discretized by 601×601 cells. We used
adaptive time stepping with an initial time step of 0.0033s, corresponding to a CFL number of
0.145. No bottom friction is applied. The test case is performed with a slightly smaller relative
waveheight at H/d=0.18. This reduction in waveheight ratio is used in several other studies such
as [21], [40], and [11], as it is closer to the waveheight ratio observed downstream of the
wavemaker.
Figure 5.1 shows the experimental setup and the gauges locations. Gauge #6 and #9 are in
front of the island, while gauge #16 is on the side, and gauge #22 is behind the island. The
numerical surface elevation compared to the experimental results are shown in Figure 5.2. The
initial waveheight and subsequent draw-down are predicted well, which is consistent with
numerical results from Celeris Advent with fixed time step [33] as well as results from other
Boussinesq-type solvers [11, 21, 38, 40].
Figure 5.1: Experimental setup of the conical island. The gauge locations are shown by
dots and the wave approaches the island from the left.
Figure 5.2: Experimental (– –) and numerical (–) time series for Briggs et al. [4]
benchmark at gauges #6, #9, #16, and #22 (a-d).
Figure 5.3 compares the numerical and measured horizontal maximum run-ups on the island,
scaled by the initial shoreline radius (2.32 m). Similar to [33] we used a threshold of δ = sΔx/3
for water depth to determine the inundated area. The agreement between numerical data and
measurements is very good even for the run-up on the back face of the island. As mentioned
earlier, the wave breaks on the island and the strong agreement of data validates that numerical
dissipation in TL17 successfully imitates wave breaking.
To compare our proposed adaptive scheme to the fixed time step scheme, we ran this
experiment with Δt = 0.0008s which was the largest fixed time step that resulted in a stable
simulation. Figure 5.4 shows the adaptive time step variation over time and compares it to the
fixed time step. As seen in this figure, adaptive time stepping let us run this experiment with a
much larger time step during most of the simulation, saving a lot of computational effort.
Figure 5.3: Numerical (solid line) and measured (x) maximum horizontal run-up in
Briggs et al. [4] benchmark.
Figure 5.4: Largest adaptive (solid line) and fixed (dashed line) time step for a stable
simulation of Briggs et al. [4] benchmark.
5.2 Breaking solitary wave runup on a slope with a conical island (Lynett et
al., 2019)
We further test our proposed scheme by simulating the experiments of Lynett et al. [20] which
have unfavorable hydrodynamic conditions for numerical models such as wave breaking and a
moving shoreline. Such configuration is a reliable test for the robustness of our scheme. Lynett et
al. [20] investigate the three-dimensional hydrodynamics of breaking solitary waves, traveling
over shallow waters and their interaction with a conical island on a shelf. In a 26.5m×48.8m
wave basin with 2.1m depth, a solitary wave is generated to propagate over a triangle-shaped
shelf with a conical island as shown in Figure 5.5. This experiment is repeatedly simulated as a
benchmark case for various Boussinesq-type models [9, 10, 29].
In our numerical experiment we constructed the bathymetry by combining the measured data
of the shelf and the analytical equation of the conical island. The wave basin is discretized using
a grid size of 0.1m×0.1m. The computational domain is further extended to the leeside of the
wave maker, from x=0m to x=-10m, to accommodate the entire solitary wave as an initial
condition. The waveheight is set to 0.39m at the wave generator location (x=0) where still water
depth is set to 0.78m as in the laboratory experiment.
Figure 5.6 shows three snapshots of the simulated water surfaces at different times along
with the corresponding local CFL numbers calculated from Eq. (22). The initial time step is set
to Δt=0.0025s which later varies according to the evolving hydrodynamic conditions. The local
CFL number at the corresponding time instants are shown in Figure 5.6. This figure shows that
the local CFL number is entirely maintained below the critical value of 0.25 because of the
adaptive time stepping. The free surface elevation simulated by Celeris Advent shows that the
physical processes resulting from the wave evolution over the shallow shelf, such as wave
steepening at the shelf front, wave scattering due to the island, wave runup, bore generation,
wave breaking, and wave merging at the lee side of island, are well-captured in the simulation.
The time step variation during the simulation is shown in Figure 5.7 where the time instants
corresponding to the snapshots in Figure 5.6 are shown by vertical dashed lines. The time step
decreases when the solitary wave collides with the apex of the shelf and the island (i.e., Figure
5.6a). Then it reaches its minimum roughly at t=9s (i.e., Figure 5.6c and d) as the diffracted wave
merges behind the island and thus the highest local Froude number is produced. Afterward, the
wave condition becomes milder and time step is rapidly recovered up to 0.004s (i.e., Figure 5.6e
and f). We also ran this experiment with a fixed time step and found Δt=0.0001 as the largest
time step which still results in a stable simulation. This value is compared in Figure 5.7 with the
time step of the adaptive scheme. As can be seen, the fixed time step requires a significantly
smaller time step to keep the simulation stable, while the adaptive scheme only decreases the
time step when the wave hits the island and is able to keep a relatively large time step for rest of
the simulation.
Figure 5.8 compares calculated surface elevations with the laboratory data. For offshore-most
four gauges (G1, G2, G4, G5), good agreement is seen between model and experimental data
indicating that the model successfully predicts the wave transformation over three-dimensional
shelf in front of the island. However, the more complicated hydrodynamic processes at G3, G6,
G7, G8 and G9 due to the combined effects of wave breaking, bore formation, hydraulic jumps
around the island and shoreline makes the predictions less reliable. Overall, the comparison
between modelled and measured data shows reasonable agreement which is comparable to
results from other studies [9, 10, 29].
Figure 5.9 compares the model results to the experiment for the flow velocity recorded at the
measurement locations. For ADV1 which is deployed offshore of the island, the proposed
scheme predicts x direction velocity component, u, very well both in magnitude and phase, while
keeping y direction velocity component, v, close to zero consistent with the experiment. For
ADV2, modelled u and v are generally consistent with the measurement, bearing in mind that
some measurement data are not available due to the extremely dynamic wave motions. The
comparison of the model and experiment at ADV3 shows good agreement, both for u and v as
well.
Figure 5.5: (a) Perspective view of bathymetry in the experiment of Lynett et al. (2019)
and (b) bathymetric contours with apparatus setup. In a contour plot, red circles
represent bottom pressure gauges while blue crosses indicate ADVs.
Figure 5.6: Sequential snapshots of simulated free surface elevation (a, c, e) and
corresponding map of local CFL number (b, d, f).
Figure 5.8: Time series of free surface elevation at gauge locations.
Figure 5.9: Time series of velocity u and v at ADV location.
5.3 Rip current with regular wave (Haller et al., 2002)
Another good benchmark for validating our proposed scheme is simulating a rip current
experiment in which wave-current interactions are significant. It is widely known that
Boussinesq-type models are capable of modelling wave-current interactions without
incorporating short crested wave models [5, 6, 30, 31]. We simulate the laboratory experiments
of Haller et al. [13] in this section.
Haller et al. [13] carried out laboratory experiments in a wave basin of size 18.2m×17m to
investigate the rip current induced by wave breaking on the biplanar slope with a rip channel.
The bathymetry as used in the experiment is set up from the detailed dimensions of the
experimental geometry while an analytical expression is borrowed for the submerged bar
creation [1]. The reconstructed topography is shown in Figure 5.10. In the numerical experiment,
the water depth is maintained at 0.678m at the wave-generation boundary and decreases until the
water level sets shoreline at x=14.3m. A monochromatic wave with H=8.26cm and T=1.0s is
generated on the left boundary and propagated onto the slope where two 7.32m-long alongshore
submerged bars are located at x=12m with a spacing of 1.82m. Offshore current is anticipated to
be generated through the rip channel due to the wave breaking induced momentum imbalance. A
uniform grid size of 0.02m×0.02m is used for discretizing the domain while the adaptive time
stepping with initial time step of 0.001s is applied. A bottom friction factor of 0.0025 is also
considered for quadratic friction formula and the simulation time is set to 600s. For calculating
time-averaged quantities of water level and velocity, simulated results are averaged over the last
500s which is equivalent to 500 wave periods. Figure 5.11 shows a snapshot of Celeris Advent
simulating this experiment while visualizing vortical flows and wave breaking.
Figure 5.12 compares the alongshore variation of the calculated mean water level (MWL) at
different locations with the measurements, showing good agreement. Figure 5.12a and Figure
5.12b indicate that wave setdown caused by increasing waveheights in the front of the bar system
is well captured in the simulation and is consistent with the measurements. Figure 5.12c and
Figure 5.12d show that the breaking induced wave setup taking place over the bar and setdown
persisting through the rip channel are both well predicted by the model.
Figure 5.13 depicts the cross-shore variation of waveheight and MWL at the bar and at the
rip channel. As pointed out by [12] the waveheight increases at the gap location due to the
interaction between the incoming wave and offshore-directed rip current. Even though some
discrepancy is noticed in the waveheight variation through the rip channel, the overall pattern of
wave setup and setdown is well predicted by the model.
Figure 5.14 compares the time averaged cross-shore and longshore velocities at four different
longshore transects with the laboratory data. The model results are generally consistent with the
measurement as they well predict both cross-shore and longshore velocity variations over the rip
channel system. Rip currents through the channel are clearly generated by the model as
represented by offshore directed velocity (or negative value of uavg) in Figure 5.14c and Figure
5.14d. The generated rip current also leads to longshore velocity changes which in turn form two
opposite vortical flow pattern behind the rip channel [12, 13, 44]. These modelling results on rip
current system including wave setup, wave setdown, and vorticity generation process confirm the
capability of Celeris Advent and our proposed adaptive scheme.
Unlike previous experiments, the time step in this experiment did not significantly change
during the simulation and only fluctuated within ±5% of its initial value. We attribute this small
variation to the regular wave condition which was used in this experiment and the submerged bar
which limited the run-up on the beach. Using adaptive scheme might not save us any
computational time in experiments like this case, where extreme situations do not happen.
However, the overhead of using the adaptive scheme is negligible and therefore we recommend
always using this scheme over the fixed one.
s
Figure 5.10: Bathymetry of rip channel experiment in Haller et al. (2002). Dashed lines are
contours at 0.1m depth intervals.
Figure 5.11: Snapshot of Celeris Advent simulating the experiment of Haller et al. (2002)
while visualizing the vorticities and wave breaking.
Figure 5.12: Variation of mean water level (MWL) at four different longshore transects.
Solid line represents calculated result and asterisk symbol denotes experimental data.
Figure 5.13: Variation of the waveheight and mean water level (MWL) at two different
cross-shore transects. Solid line represents calculated result and dashed line with circle
symbol denotes experimental data. Note that in the plots cross-shore distance x and
alongshore distance y are nondimensionalized into x` and y` , respectively, according to
[13].
-1.5 -1 -0.5 0 0.5
x`
0
5
10
15
H(cm)
y`=0.01 (at bar crest)
Numerical
Experimental
-1.5 -1 -0.5 0 0.5
x`
0
5
10
15
H(cm)
y`=0.5 (at rip channel)
-1.5 -1 -0.5 0 0.5
x`
-0.5
0
0.5
1
1.5
2
MWL(cm)
-1.5 -1 -0.5 0 0.5
x`
-0.5
0
0.5
1
1.5
2
MWL(cm)
Figure 5.14: Variation of cross-shore (uavg) and longshore (vavg) time-averaged velocity
component at four different longshore transects. Solid line represents calculated result,
while asterisk symbol denotes experimental data. Panels (a-d) are for uavg and (e-h) for vavg.
5.4 Rip current with irregular wave (Hamm, 1992)
The final case that we simulate is similar to the one in the previous section, but it includes
irregular waves. We choose this test because the interaction of waves with a wide range of
periods and the complex bathymetry is expected to result in challenging situations for a
numerical model. For such situations, our adaptive scheme is more suitable for phase-resolving
models since it enables them to control possible instabilities through spontaneous adjustment of
the time step based on local Courant number [29]. This benchmark demonstrates how robust our
proposed scheme is even when the wave condition is relatively harsh and random.
Hamm [14] experimentally investigated breaking-induced nearshore circulation in the wave
basin where a plane beach with a rip channel was installed. Using both monochromatic and
random waves, overall process of rip current generation by breaking waves was analyzed. The
bathymetry in the experiment was created using the following equation.
0 5 10 15
-0.2
0
0.2
uavg(m/s)
(a) uavg at x=10m
0 5 10 15
-0.2
0
0.2
uavg(m/s)
(b) uavg at x=11.25m
0 5 10 15
-0.2
0
0.2
uavg(m/s)
(c) uavg at x=12.3m
0 5 10 15
y(m)
-0.2
0
0.2
uavg(m/s)
(d) uavg at x=13m
0 5 10 15
-0.2
0
0.2
vavg(m/s)
(e) vavg at x=10m
Numerical
Experimental
0 5 10 15
-0.2
0
0.2
vavg(m/s)
(f) vavg at x=11.25m
0 5 10 15
-0.2
0
0.2
vavg(m/s)
(g) vavg at x=12.3m
0 5 10 15
y (m)
-0.2
0
0.2
vavg(m/s)
(h) vavg at x=13m
(48)
where z is the bottom elevation, x is the onshore ward distance from the wave generator, y is the
longshore distance from the centerline of the channel. The bathymetry is shown in Figure 5.15.
The domain is discretized by cells of size 0.02m×0.02m. The initial time step is set to 0.003s.
Irregular waves are generated based on the JONSWAP spectrum with a significant waveheight
of Hs=0.13m, and peak period of Tp=1.6s. The spectrum shown in Figure 5.16 is used to generate
68 wave components with discretized frequencies of f=0.01Hz. Quadratic formula is applied
with a friction factor of 0.0025. Simulation time is set to 600s of which the beginning 100s is for
spin up period and is not used in analyses.
Figure 5.17 shows the cross-shore variation of the significant waveheight, Hs, both at the
plane beach and at the rip channel. It also shows the return current velocity, uavg, at the rip
channel. Calculated waveheights at two transects agree well with the measurement with some
discrepancies, most possibly due to the imperfect reproducibility of the physical bathymetry [15].
Comparing Figure 5.17a and Figure 5.17b, the waveheight along the cross-shore transect at plane
beach decreases more rapidly after x=11m than that along the rip channel, mostly due to the
wave breaking taking place farther offshore along the plane beach transect. The returning current
(i.e., rip current) induced by breakers at the rip channel is also well predicted by the model as
shown in Figure 5.17c. The rip current interacts with the incoming waves in the opposite
direction and consequently makes the waveheight larger around the channel as shown in Figure
5.17b.
Figure 5.18 shows the variation of the time step, Δt, during the simulation. It initiates from
the starting value of 0.00325s and then steadily decreases as the wave propagates on to the shore.
Unlike the regular wave case in which Δt varies within a very limited bound and gets stabilized,
Δt of random wave simulation fluctuates continuously within a larger bound with occasional but
drastic changes. This unpredictable variation is attributed to the random nature of irregular
waves. The largest fixed time step that resulted in a stable simulation was t = 0.0015s which is
almost half the average adaptive time step. This experiment did not have extreme events such as
collision of a solitary wave on an island, yet the random nature of the waves required occasional
drops in the time step to keep the simulation stable, which was possible thanks to the adaptive
scheme.
Figure 5.15: Bathymetry of rip current experiment in Hamm (1993). Dashed lines are
contours at 0.1m interval.
Figure 5.16: JONSWAP wave spectrum discretized by 68 frequency components.
0.5 0.6 0.7 0.8 0.9 11.1
Frequency (Hz)
0
1
2
3
4
5
6
Wave Spectral Density (m2/Hz)
10-3
Figure 5.17: Cross-shore validation of the waveheight, Hs at (a) plane beach and (b) rip
channel, and return flow, uavg at (c) the rip channel.
0 2 4 6 8 10 12 14 16 18
x (m)
0
0.1
0.2
Hs(m)
(a) Hs at plane beach ( y = -13m)
Numerical
Experimental
0 2 4 6 8 10 12 14 16 18
x (m)
0
0.1
0.2
Hs(m)
(b) Hs at rip channel ( y = 0m)
0 2 4 6 8 10 12 14 16
x (m)
-0.4
-0.2
0
uavg(m/s)
(c)uavg at rip channel ( y = 0m)
Figure 5.18: Temporal variation of the time step. Solid and dashed-dot lines refer to
adaptive and fixed time steps, respectively.
6. Conclusion
We explained our development of the third order Adams Bashforth time stepping equations
using the Lagrange formula for polynomial interpolation and assuming a variable time step
value. We then employed these equations to solve the extended Boussinesq equations in time by
developing the second order finite difference discretization equation for variable time steps and
incorporating them in the rearrangement of the Boussinesq-type equations suggested by Wei and
Kirby [42]. We implemented the resultant numerical scheme in the latest version of Celeris
Advent (v1.3.4) and briefly explained this implementation. We validated the proposed adaptive
scheme against several benchmarks proving the software’s accuracy in modeling wave breaking,
wave runup, irregular waves, and rip currents. The adaptive time stepping makes the model more
robust by allowing it to keep the CFL number constant throughout the simulation. This is
especially beneficial where the superposition of a wide range of wave conditions and a complex
bathymetry (e.g., in field sites) creates occasional extreme conditions with large local Froude
numbers. These brief extreme moments in the simulation are gracefully handled by the adaptive
scheme using an accordingly small time step. As high-speed events diminish, the time step size
is then recovered, letting the simulation continue efficiently.
7. Acknowledgements
This research was partially funded by the Office of Naval Research (ONR) award numbers
N00014-13-1-0624 and N00014-17-1-2878.
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