A new approach to the local time stepping problem forscalar transport

Ben R. Hodgesa,

a Center for Research in Water Resources,The University of Texas at Austin,

10100 Burnet Road, Bldg. 119, STOP R8000 Austin, TX 78758-4445, USAemail: hodges@utexas.edu

Abstract

A new scalar transport method is proposed to reduce computational timewhen a large number of scalars are transported in coupled hydrodynamic-ecosystem models. The new Local Mass Transport (LMT) method confinessubtime transport computations to regions where the local Courant-Freidrichs-Lewy (CFL) number exceeds a given numerical stability criteria for a global(large) time step, but the method does not require either contiguous regions orspecial boundary algorithms between regions as used in previous Local TimeStepping (LTS) approaches. The new method uses conservative transport ofmass rather than dissolved concentration. This approach allows different facesof a single grid cell to use different subtime steps. The new LMT method isfurther extended to include background filtering (LMTB) so that scalars belowa pre-defined background concentration are ignored in transport calculations.This new approach can further reduce computational time where large regionsare at or below an irrelevant background concentration. Both LMT and LMTBmethods can be more computationally efficient than global subtime stepping.

Keywords: mass transport, mathematical model, transport processes,advection

1. Introduction

1.1. Transporting mass instead of concentration

The hydrodynamic transport of scalars in hydrodynamic modelling is con-ventionally based on transport of dissolved concentrations. The standard 3Dadvection-diffusion equation uses volume concentration of a dissolved constituent,c, typically written as

∂c

∂t= − ∂

∂xj(ujc) + κ

∂2c

∂xj∂xj+ snet (1)

Email address: hodges@utexas.edu (Ben R. Hodges)Author’s uncorrected manuscript submitted to and published by Ocean Modelling, 77:1-19

(2014) http://dx.doi.org/10.1016/j.ocemod.2014.02.007

Preprint submitted to Ocean Modelling March 28, 2014

where uj is the velocity field, κ the diffusivity, snet a net source/sink, and theEinstein summation convention is used for repeated subscripts. In a control-volume formulation, this is commonly written as

∂

∂t

∫V

cdV = −∫S

(ujc) n̂j dS +

∫S

κ∂c

∂xjn̂jdS +

∫V

snet dV (2)

where V is a control volume and S is the area of the surrounding control surfaceand n̂j is a unit vector normal to the surface. Numerical time-marching for theconcentration from time step n to n + 1 at cell location (i) can be representedas

cn+1i = cni + ∆t f (c, κ, u, a, snet) (3)

where subscripts indicate discrete cell location, superscripts indicate time level,∆t is the time step, and f() is a discrete function representing the spatially-averaged effects of neighbor values for the independent variables. For a systemthat allows temporally-changing control volumes, an equivalent conservativefunction is

cn+1i =

V niV n+1i

cni +∆t

V n+1i

g (c, κ, u, a, snet) (4)

where g() is a discrete function representing the integrated effects of neighborvalues for the independent variables, and V n+1

i 6= V ni typically occurs for amoving free surface within a fixed 2D or 3D grid.

A slightly different approach can be developed by considering the transportof scalar mass, m as

∂m

∂t=

∂

∂t

∫V

cdV (5)

The left-hand-side of eq. (2) can be replaced with ∂m/∂t and the right-hand-side interpreted as the rate of change of mass in a control volume; it followsthat eq. (4) can be replaced with

mn+1i = mn

i + ∆t g (c, κ, u, snet) (6)

which can be recognized as eq. (4) multiplied through by V n+1i using m = cV ;

thus a change from transporting concentration to transporting mass might be arelatively trivial alteration to many numerical models. However, to close a masstransport equation set for known u, κ, and snet, two additional equations arerequired. First, the conservation of volume for an incompressible fluid, whichcan be written as

∂V

∂t= −

∫S

uj n̂j dA (7)

where subscripts are vector indexes using the Einstein summation conventionand volume sources/sinks are neglected. Second, we require a diagnostic defini-tion of concentration, mass, and volume relationships:

c =m

V(8)

2

which can be applied discretely at any time level n and grid cell i.The above observations might seem trivial; indeed, the volume change is

typically computed in hydrodynamic simulations with a free surface, and nodoubt many models actually compute m as a step in computing c. Neverthe-less, as illustrated herein, this minor change in viewpoint – from transportingconcentration to transporting mass – provides interesting new possibilities fornumerical algorithms that have not previously been exploited.

1.2. Motivation

A difficulty for coupled hydrodynamic, pollutant (e.g. oil spill) and waterquality/ecosystem models is the computational expense of transporting scalars.In particular, ecosystem models often represent multiple species of phytoplank-ton, zooplankton and speciation of nutrients, e.g. nitrate, ammonium, organicnitrogen, phosphate, and organic phosphorous (e.g. Camacho and Martin, 2013;Leon et al., 2012; Marinov et al., 2008; Robson and Hamilton, 2004). Even instrictly hydrodynamic models, a large number of transported tracers can be ef-fective in visualizing and quantifying circulation or possible pollutant behavior,thus adding to computational costs (e.g. Young et al., 2011). Indeed, herein thetransport of 41 scalars in a hydrodynamic model results in the scalar transportcomputational time ranging from 15 to 30 times greater than the hydrodynamiccomputational time when using conventional transport methods (see §3.3).

Such computational costs can be exacerbated by localized regions where theCourant-Friedrichs-Lewy (CFL) number (CFL = u∆t/∆x) at a desired largetime step exceeds a numerical stability constraint. In general, a large modeltime step is incompatible with large velocities relative to the space discretization(with the constraint quantitatively depending on the numerical method usedfor scalar transport), so the allowable transport time step must be controlledto limit the CFL everywhere in the domain. A high velocity in a single gridcell can require a reduced time step for the entire domain and thus increasemodel computational time. This constraint becomes even more problematicwhen unstructured or curvilinear grid system are used with a large range of cellsizes (Dawson et al., 2013).

Further contributing to transport costs is an apparently unrecognized prob-lem: the expense of transporting zero concentrations for phenomena that arestrictly confined to a local area (e.g. an oil spill). A similar problem arisesin transport of water quality constituents below a background concentrationthat does not affect the ecosystem behavior. For example, if a nutrient alonga coastal shelf has some background level that is invariant outside of a riverplume, why bother transporting the nutrient outside of plume waters? Whena pollutant has been diffused below detection limits and below any ecosystemconcern, why continue to track irrelevant and unconfirmable concentrations?It seems likely that extensive computational time in ecosystem and pollutantmodels is expended simply shuffling around irrelevant changes in background orzero concentrations.

In this paper, methods for Local Mass Transport (LMT) and Local MassTransport with Background filter (LMTB) are proposed as new ways to ad-dress these issues. These algorithms are radical departures from previous scalartransport methods and provide an opportunity to rethink the way we computescalar fields.

3

1.3. Background

There is a performance mismatch between most hydrodynamic and scalartransport models for localized high velocity conditions when using finite differ-ence/volume methods. Many hydrodynamic models only retain their theoreticalaccuracy order for CFL < O(1), where the exact CFL limit for accuracy dependson the discretization scheme (Hodges, 2004). However, it has long been under-stood that implicit and semi-implicit hydrodynamic models are both stable andreasonably accurate for 1 < CFL < 5, especially when high CFL numbers arelocally confined in space and/or time. Modelers concerned about computationalcosts take advantage of this robustness by setting the largest possible hydrody-namic time step such that CFL < 1 is achieved over most of the domain, butincreased error is accepted in localized regions where CFL > 1. Momentumconservation can remain stable and reasonably accurate with locally-high CFLnumbers simply because momentum equations are globally biased towards losses(i.e. dissipation). Unfortunately, common Eulerian conservative scalar trans-port methods are not as robust when locally-high CFL numbers are encountered.Conservative scalar transport at high CFL typically results in unrealistic localconcentrations that rapidly degrade simulation accuracy. The fundamental dif-ficulty for scalar transport with high CFL is the increasing number of neighborcells influencing the concentration in a single grid cell over a model time step,i.e. the “domain of dependence” illustrated in Fig. 1. These issues are wellunderstood and are often addressed by use of semi-Lagrangian schemes (e.g.Blossey and Durran, 2008; Lentine et al., 2011; Manson and Wallis, 2000) thattrack back along hydrodynamic characteristics to ensure the correct domain ofdependence is achieved. Leonard (2002) showed that these ideas could be usedto define unconditionally stable Eulerian and semi-Lagrangian schemes for highCFL numbers based on the discrete stencil “sweep point” and “balance point.”Despite these advances, many coupled hydrodynamic-water quality models stillrely on simple transport schemes where selection of a sufficiently small modeltime step is required to prevent locally-high CFL numbers that exceed the CFLstability constraint.

Parallel to development of the finite-difference methods described above, ap-proaches for handling locally-high CFL numbers in finite element models havebeen constructed following the ideas of Osher and Sanders (1983), which are nowknown as Local Time Stepping (LTS) algorithms. These methods have a richfinite element literature, where the effects of locally-refined unstructured gridshave made their use almost a necessity (Constantinescu and Sandu, 2007; Coquelet al., 2010; Crossley and Wright, 2005; Dawson and Kirby, 2001; Krivodonova,2010; Sanders, 2008; Seny et al., 2013; Tang and Warnecke, 2006; Zhang et al.,1994). These methods typically involve separating a domain into regions sharingsimilar CFL constraints such that small time steps are used only where locallynecessary. LTS requires identifying and treating the interface between regionswith specialized algorithms to handle the change in time step, so the numberof regions affects the efficiency of the system. Much as domain decompositionmethods are used for separating a large computational space into separate sub-regions with different grid resolution, LTS can be considered an approach totime decomposition (Shishkin and Vabishchevich, 2000). The LTS idea has alsobeen extended to multiscale modeling (Muller and Stiriba, 2007; Schlegel et al.,2012). A similar regionalization approach is also found in the finite-difference

4

ci, j+1

ci, j

ci-1, j ci-2, j ci-3, j

ci, j-1

ci+1, j

x

y

Figure 1: Illustration of the domain of dependence problem for a 2D diverging flow intorightmost cell i, j with an inflow at CFL = 2.4 and outflows at CFL = 0.8 through each face.Darker shading represents higher concentrations. Any single-step transport scheme that doesnot include cni−3,j in the computational stencil cannot achieve its theoretical accuracy order

because cn+1i,j depends on cni−3,j .

literature under the name “dynamic subtime-stepping” (Misra et al., 2012).Much of the literature in LTS methods involves solution of Burgers equation ora general hyperbolic conservation law, although LTS has recently been adaptedfor the shallow water equations in coastal ocean modeling (Dawson et al., 2013;Trahan and Dawson, 2012).

1.4. Mass transport as an approach for subtime stepping

This paper provides an entirely new approach to address the problems oflocally-high CFL numbers in a transport domain. The approach involves alocal time step – or more appropriately a “subtime” step relative to the globalhydrodynamic time step for u. Despite the similarity in using localization tosolve high CFL problems, the new method is entirely distinct from the LTSapproaches based on Osher and Sanders (1983). The new Local Mass Transport(LMT) approach dispenses with the need to identify contiguous regions andspecial numerical schemes at interfaces between regions, which are the principalcomputational complexities of previous LTS methods.

LMT requires rethinking the conventional method of transport scalar con-centrations – herein, scalar mass and fluid volume are separately transportedvariables with the concentration recovered diagnostically as their ratio. Thekey feature is that each face of a single computational cell can have a differentnumber of subtime steps in a single global hydrodynamic time step. This isan entirely new approach: under previous LTS methods only the interface cellsbetween different time-step regions are allowed to have different time steps ontheir faces, and these interface cells required special numerical treatment. Inthe present method, the identical algorithm is used across all cell faces, butdifferent faces can operate at different time steps.

5

For the new LMT method, the computational volume of a grid cell is al-lowed to fluctuate across subtime intervals as both mass and volume are fluxedseparately through different faces at different intervals. That is, over the courseof the global time step ∆t, different faces of a grid cell see different fluxes indifferent subtime intervals, which implies the volume of the grid cell is changingduring the subtime steps. Despite the subtime fluctuating volume, the sum-mation of all subtime steps in/out of a grid cell retains both mass and volumeconservation over the global time step. The new scheme is built with threeideas: a Subtime Mass Transport (SMT) concept that is invoked over all spaceat fixed subtime intervals, a Local Mass Transport (LMT) concept that is in-voked locally with different subtime intervals, and a Local Mass Transport withBackground filtering (LMTB) that confines transport computation to regionswith concentrations above a background level. These methods are compared toa Concentration Subtime Transport (CST) algorithm using conventional con-centration transport with global subtime stepping.

2. Methods

2.1. Concentration subtime transport (CST)

A simple approach for transporting scalar concentrations under high CFLnumbers with an Eulerian discretization is to use global subtime stepping; i.e.,where the hydrodynamic field has one or more grid cells whose CFL exceedsthe scalar transport algorithm’s constraint, scalars through the entire domainare transported at a subtime step smaller than the hydrodynamic time step.The method has been used in other models (e.g. Hodges, 2000), but a simpleillustrative example is useful for comparison with the new mass transport meth-ods proposed herein. Neglecting any source/sink term (snet), a conservative,two-level θ-discretization in 1D of eq. (2) on a uniform staggered Cartesian gridcan be written as

V n+1i cn+1

i = V ni cni −∆t

(q∗i+1/2c

n+θi+1/2 − q

∗i−1/2c

n+θi−1/2

)+

∆t

∆x

([κa]

n+θi+1/2

[cn+θi+1 − c

n+θi

]− [κa]

n+θi−1/2

[cn+θi − cn+θ

i−1

])(9)

where q∗i+1/2 represents the effective volume flux rate across cell face i+1/2 overthe time step, ai+1/2, ai−1/2 are face areas of the grid cell, and concentrations(c) are stored on the cell centers. The θ in the time superscript allows gen-eral consideration of different temporal discretization methods: θ = 0 providesexplicit Euler, θ = 1 is implicit Euler, and 0 < θ < 1 can be used in either two-level weighted implicit or predictor-corrector discretizations (Hodges and Rueda,2010). Note that V n+1

i , q∗, and an+θ are considered known from a hydrody-namic solution before computation of cn+1. That is, we are specifically limitingour method to passive scalars or those active scalars (i.e. influencing density)whose baroclinic contributions to the hydrodynamic solution are obtained ei-ther with fully explicit or predictor-corrector algorithms (Hodges and Rueda,2010; Rueda et al., 2007). Concentrations on the cell faces (e.g. ci+1/2) can beobtained with any standard interpolation method, such as QUICK (Leonard,1979). Eq. (9) can be written in a more familiar form by considering the case

6

where V n+1 = V n and κ is constant, which simplifies to

cn+1i = cni −

∆t

∆x

(u∗i+1/2c

n+θi+1/2 − u

∗i−1/2c

n+θi−1/2

)+κ∆t

∆x2

(cn+θi+1 − 2cn+θ

i + cn+θi−1

)(10)

where we have used q∗ = au∗ and V = a∆x. This result is a simple discreteform of the original transport PDE, eq. (1), neglecting any source/sink terms.

The flux rate through the cell face, q∗i+1/2, in eq. (9) requires a time levelthat is consistent with the continuity discretization of the hydrodynamic so-lution from time step n to n + 1, as discussed by Gross et al. (2002). Forexample, continuity in a 1D hydrodynamic model for the shallow-water equa-tions is ∂η/∂t = ∂q/∂x, where η is the free surface elevation and q is a flux rateper unit width (e.g. m2/s). This continuity equation can be enforced by a θtime-stepping method as

ηn+1i − ηni

∆t= − θ∗

∆x

(qn+1i+1/2 − q

n+1i−1/2

)− 1− θ∗

∆x

(qni+1/2 − q

ni−1/2

): 0 ≤ θ∗ ≤ 1

(11)

In general, the discretization methods for continuity and scalar transport canbe separately chosen so θ∗ need not be equal to θ of eq. (9). However, “consis-tency with continuity” (Gross et al., 2002) for eq. (11) requires the q∗ in scalartransport eq. (9) to be defined from

q∗ ≡ θ∗qn+1 + (1− θ∗) qn (12)

Using the standard approach to demonstrating numerical consistency and ac-curacy through a Taylor series expansion (e.g. Hirsch, 1988) it can be easilyshown that consistency and first-order temporal accuracy are obtained evenwith θ∗ 6= θ. However, for the above two-time-level system, second-order accu-racy is only obtained with θ = θ∗ = 1/2; i.e. when both continuity and scalartransport are time-centered discretizations.

If the flux field (q∗) from the hydrodynamic model provides one or more cellfaces with CFL > 1 (or greater than some other constraint) an explicit subtime-stepping algorithm can be applied to ensure that CFL < 1 is achieved in everysubtime step over all space. That is, we can subdivide the hydrodynamic timestep, ∆t, into R smaller scalar substeps of ∆t/R with the approximation ofconstant q∗ over all subtime steps (this approximation is not strictly necessary,as shown in Appendix A). It is useful to define the substep set as r = {1, 2, ...R}for R defined as

R =

⌈CFL∆t

CFLM

⌉(13)

where d e is the ceiling operator rounding to the next larger integer, CFL∆t

is the largest CFL in the simulation at the hydrodynamic time step ∆t, andCFLM the maximum allowable CFL associated with both the time-marchingscheme and the spatial interpolation stencil for ci+1/2 in eq. (9). Although itappears possible to invoke any time-marching form within a subtime transportalgorithm, our derivation will focus on the simplest explicit algorithm: forward-time (θ = 0) and centered space. Furthermore, the basics of the subtime idea

7

can be readily grasped from discretization of the advective term alone, so wewill focus the following on a non-diffusive transport equation. For completeness,discretization of diffusion terms are provided in Appendix B. Applications of themethod have been developed in both 2D and 3D and include diffusive terms (§3).

The subtime version of the advective terms of eq. (9) can be written as

Vn+r/Ri c

n+r/Ri = V ni c

ni −

∆t

R

r−1∑k=0

(q∗i+1/2 c

n+k/Ri+1/2 − q

∗i−1/2 c

n+k/Ri−1/2

)(14)

where k in the summation on the right-hand side has been lagged from r on theleft-hand side to maintain an explicit algorithm. For R = 1, i.e. no substeps,eq. (14) reduces to an explicit Euler (θ = 0) version of eq. (9) with κ = 0. Eq.(14) can also be written as a recursive algorithm for the concentration at anysubtime level (r) using an explicit function of the prior subtime level (r − 1),which can be written as

Vn+r/Ri c

n+r/Ri = V

n+(r−1)/Ri c

n+(r−1)/Ri − ∆t

R

(q∗i+1/2 c

n+(r−1)/Ri+1/2 − q∗i−1/2 c

n+(r−1)/Ri−1/2

)(15)

Thus, each substep is effectively an explicit Euler time-advance from the previ-ous substep, so this CST scheme is numerically consistent but only first-orderaccurate. Higher-order subtime schemes could be defined using either time n−1data, multi-step approaches (e.g. Runge-Kutta), or with implicit solution be-tween substeps r − 1 to r.

Using either eq. (14) or (15), transport between all grid cells requires subtimestepping when CFLi > CFLM for any i cell in the domain. The advantage ofthis CST method is that it is easily implemented in an explicit scheme using anadditional substep loop of a conventional time-marching algorithm. However,the requirement to substep all space for high CFL numbers occurring over onlya small region leads to high computational costs.

2.2. Subtime mass transport (SMT) concept

The conservative discrete versions of mass transport using eqs. (6), (7) and(8) for a non-diffusive (κ = 0) scalar without sources or sinks (snet = 0) can bewritten for a forward-time, centered-space approach on a conventional staggeredgrid as

mn+1i = mn

i −∆t(q∗i+1/2 c

ni+1/2 − q

∗i−1/2 c

ni−1/2

)(16)

V n+1i = V ni −∆t

(q∗i+1/2 − q

∗i−1/2

)(17)

cn+1i =

mn+1i

V n+1i

(18)

8

Similar to eq. (14), a conservative explicit subtime-stepping algorithm can beapplied to eqs. (16) – (18) over substeps r = {1, 2, ...R} as

mn+r/Ri = mn

i −∆t

R

(r−1∑k=0

q∗i+1/2 cn+k/Ri+1/2 −

r−1∑k=0

q∗i−1/2 cn+k/Ri−1/2

)(19)

Vn+r/Ri = V ni −

∆t

R

(r−1∑k=0

q∗i+1/2 −r−1∑k=0

q∗i−1/2

)(20)

cn+r/Ri =

mn+r/Ri

Vn+r/Ri

(21)

For this SMT algorithm, the number of substeps (R) is a global settingthat ensures CFL < 1 everywhere for all substeps; thus SMT is similar to theconventional CST method in that the entire domain is sub-stepped for highCFL numbers in a small region. SMT is theoretically more expensive than CSTbecause SMT requires separate computations for mass and volume transportalong with a diagnostic concentration computation. However, a single volumetransport solution is applicable to all scalars and the concentration calculationis trivial, so CST and SMT have similar scales of computational cost when alarge number of scalars is transported. SMT is useful principally as a simplerexpository step in understanding the LMT algorithm described below.

2.3. Local mass transport (LMT)

The LMT algorithm is developed using a slight modification of the SMTequations above, such that the number of substeps (R) can be different onthe different faces of a single grid cell. This property is useful so that onlyfaces with high CFL numbers need to have subtime transport computations.The interesting consequence is that the volume of the ith cell in the r substep,

Vn+r/Ri , will not vary linearly from V ni to V n+1

i as occurs in the SMT algorithmwith eq. (20); that is, during any subtime interval a mismatch of inflow andoutflow volumes is allowed, but the net volume change over a complete timestep remains conservative and consistent with continuity – i.e. unchanged for afixed grid cell or exactly equal to the net volume change for a free surface cell.We can imagine the grid cell is a balloon with multiple orifices that increasesand decreases its volume over the substeps as volumes are fluxed in and outthrough different orifices at different substeps; the balloon will naturally returnto its conservative volume by the end of the last substep since the sum of thefluxes through all the orifices over all the substeps is exactly conservative. Thisidea will be made clearer in the example associated with Fig. 2, below.

To develop the LMT method from the SMT method described above, con-sider the case of cell faces i+ 1/2 and i−1/2 of the ith cell with Ri+1/2 subtimesteps on the former and Ri−1/2 on the latter. For convenience, let us define the

set of computed subtime levels for the ith cell as γi, where

γi ≡{

1

Ri+1/2,

2

Ri+1/2...Ri+1/2

Ri+1/2

}∪{

1

Ri−1/2,

2

Ri−1/2, ...

Ri−1/2

Ri−1/2

}(22)

9

which must be sorted in ascending order. Then the j substep occurs at subtimelevel γi(j), such that

mn+γi(j)i = mn

i −∆t

Ri+1/2q∗i+1/2

fp(j)−1∑k=0

cn+k/Ri+1/2

i+1/2 +∆t

Ri−1/2q∗i−1/2

fm(j)−1∑k=0

cn+k/Ri−1/2

i−1/2

(23)

where fp(j) and fm(j) represent the number of substeps less than the fractionγi(j) that have been executed on the plus and minus faces, respectively; i.e.

fp(j) = bγ(j)Ri+1/2c (24)

fm(j) = bγ(j)Ri−1/2c (25)

where b c is the floor function that truncates down to the nearest smaller integer.The volume of the grid cell consistent with continuity at subtime level γi(j) is

Vn+γi(j)i = V ni −∆t

(fp(j)

Ri+1/2q∗i+1/2 −

fm(j)

Ri−1/2q∗i−1/2

)(26)

The concentration at any subtime level γi(j) is recovered diagnostically, witheq. (21) written as

cn+γi(j)i =

mn+γi(j)i

Vn+γi(j)i

(27)

The above describes the underlying LMT algorithm; however, implementa-tion is simplified using recursion in place of eq. (23), such that mass and volumetransport are

mn+γi(j)i = m

n+γi(j−1)i −∆t

(λp(j)

Ri+1/2q∗i+1/2c

n+γi(j−1)i+1/2 − λm(j)

Ri−1/2q∗i−1/2c

n+γi(j−1)i−1/2

)(28)

Vn+γi(j)i = V

n+γi(j−1)i −∆t

(λp(j)

Ri+1/2q∗i+1/2 −

λm(j)

Ri−1/2q∗i−1/2

)(29)

where

λm(j) =

0 : mod(γ(j), 1

Ri−1/2

)> 0

1 : mod(γ(j), 1

Ri−1/2

)= 0

(30)

with a similar definition for λp using Ri+1/2. Thus, γi is the set of substeps thatwill change the concentration of cell i, while λp and λm are non-zero for substepsof γi that have fluxes across the plus or minus faces of cell i, respectively. Itfollows that cell face i− 1/2 is associated with λm for cell i and λp for cell i− 1.Similarly, cell face i+ 1/2 is associated with λm for cell i+ 1 and λp for cell i.

As an example, consider three fixed-volume grid cells isolated from a largersystem shown in Fig. 2a. For simplicity in illustration, the flux rate, q∗ istaken as uniform in the positive x direction (i.e. from i − 1 towards i + 1).

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We arbitrarily require two substeps on the i − 1/2 face, three substeps on thei + 1/2 face, and only one (global) time step on the i − 3/2 and i + 3/2 faces,as indicated in Table 1. Note that for this example the cell faces beyond thediagram (e.g. i + 5/2) might have other levels of subtime stepping, which willonly play a role at the global time step for fluxes across the i+ 3/2 and i− 3/2faces, and thus need not be a priori specified for the local time stepping. Thisbehavior is an important strength of the method: any region of cells that isbounded by faces with R = 1 will naturally be disconnected from all adjacentregions during subtime stepping. The discrete substep equations for the threecells corresponding to this example and Fig. 2 are provided in Table 2.

Substep 1 is from n to n + 1/3 (Fig. 2b), where the mass, volume, andconcentrations are only changed in cells i and i + 1. The volumes are changedeven though these cells are nominally fixed volume since the flux through thei+ 1/2 face during the substep is not balanced by fluxes through other grid cellfaces. Note that the discrete mass equations of Table 2 are based on eq. (28),which requires a center-to-face interpolation method to obtain ci+1/2 from cellcenter concentrations (e.g. ci, ci+1). For simplicity in illustration, Table 2 usescentral interpolation, although the present implementation also provides optionsfor 1st-order upwind or quadratic 3rd-order upwind interpolation. For Substep1 the time levels for the concentration interpolation in the mass equations areall cn.

feature R γ λp λm

face (i− 3/2) 1 depends on cell i− 2 {0, 1}cell (i− 1)

{12 , 1}

face (i− 1/2) 2 {1, 1} {0, 1, 0, 1}cell (i)

{13 ,

12 ,

23 , 1}

face (i+ 1/2) 3 {1, 0, 1, 1} {1, 1, 1}cell (i+ 1)

{13 ,

23 , 1}

face (i+ 3/2) 1 {0, 0, 1} depends on cell i+ 2

Table 1: Cell and face values for LMT example in Fig. 2 that provides discrete equations ofTable 2

Substep 2 is from time n to n + 1/2 for the i − 1 cell and from n + 1/3to n + 1/2 for the i cell, while no substep is applied to the i + 1 cell (Fig.2c). The cell-centered concentration interpolation for mass transport across cell

face i − 1/2 uses the latest available data, i.e. cn+1/3i and cni−1. The mixing of

different substep levels in the interpolation is a consequence of imposing differentsubsteps on different faces. Where a CFL constraint is used to set the substeplevels, the different time-level interpolations occur when faces have different flux

rates, thus a central interpolation of (cn+1/3i + cni−1)/2 implies that the cell i

concentration is changing faster than the cell i−1 concentration due to a higherflux rate (CFL) into cell i through a cell face not shared with cell i− 1.

Substep 3 is from time n+ 1/2 to n+ 2/3 for the i cell and from n+ 1/3 ton+ 2/3 for the i+ 1 cell, while no substep is applied to the i− 1 cell (Fig. 2d).Again, mixed substep levels are used in the concentration interpolation, basedon latest available data at each cell center.

In Substep 4 (Fig. 2e), the full time step is completed and cells i − 2 and

11

step

cell

(i−

1)ce

ll(i

)ce

ll(i

+1)

mn

+1/3

i=mn i−

∆t

3q∗ i

+1/2

( cn i+

1+cn i

2

)mn

+1/3

i+1

=mn i+

1+

∆t

3q∗ i

+1/2

( cn i+

1+cn i

2

)1

no

chan

geVn

+1/3

i=Vn i−

∆t

3q∗ i

+1/2

Vn

+1/3

i+1

=Vn i+

1+

∆t

3q∗ i

+1/2

cn+

1/3

i=

mn+

1/3

i

Vn+

1/3

i

cn+

1/3

i+1

=m

n+

1/3

i+

1

Vn+

1/3

i+

1

mn

+1/2

i−1

=mn i−

1−

∆t

2q∗ i

−1/2

( cn+1/3

i+cn i−

1

2

)mn

+1/2

i=mn

+1/3

i+

∆t

2q∗ i

−1/2

( cn+1/3

i+cn i−

1

2

)2

Vn

+1/2

i−1

=Vn i−

1−

∆t

2q∗ i

−1/2

Vn

+1/2

i=Vn

+1/3

i+

∆t

2q∗ i

−1/2

no

chan

ge

cn+

1/2

i−1

=m

n+

1/2

i−

1

Vn+

1/2

i−

1

cn+

1/2

i=

mn+

1/2

i

Vn+

1/2

i

mn

+2/3

i=mn

+1/2

i−

∆t

3q∗ i

+1/2

( cn+1/3

i+

1+cn+

1/2

i

2

)mn

+2/3

i+1

=mn

+1/3

i+1

+∆t

3q∗ i

+1/2

( cn+1/3

i+

1+cn+

1/2

i

2

)3

no

chan

geVn

+2/3

i=Vn

+1/2

i−

∆t

3q∗ i

+1/2

Vn

+2/3

i+1

=Vn

+1/3

i+1

+∆t

3q∗ i

+1/2

cn+

2/3

i=

mn+

2/3

i

Vn+

2/3

i

cn+

2/3

i+1

=m

n+

2/3

i+

1

Vn+

2/3

i+

1

mn

+1

i−1

=mn

+1/2

i−1−

∆t

2q∗ i

−1/2

( cn+2/3

i+cn+

1/2

i−

1

2

)mn

+1

i=mn

+2/3

i−

∆t

3q∗ i

+1/2

( cn+2/3

i+

1+cn+

2/3

i

2

)mn

+1

i+1

=mn

+2/3

i+1

+∆t

3q∗ i

+1/2

( cn+2/3

i+

1+cn+

2/3

i

2

)+

∆tq

∗ i−3/2

( cn+1/2

i−

1+c` i−

2

2

)+

∆t

2q∗ i

−1/2

( cn+2/3

i+cn+

1/2

i−

1

2

)−

∆tq

∗ i+3/2

( c` i+

2+cn+

2/3

i+

1

2

)4

Vn

+1

i−1

=Vn

+1/2

i−1−

∆t

2q∗ i

−1/2

+∆tq

∗ i−3/2

Vn

+1

i=Vn

+2/3

i−

∆t

3q∗ i

+1/2

+∆t

2q∗ i

−1/2

Vn

+1

i+1

=Vn

+2/3

i+1

+∆t

3q∗ i

+1/2−

∆tq

∗ i+3/2

cn+

1i−

1=

mn+

1i−

1

Vn+

1i−

1

cn+

1i

=m

n+

1i

Vn+

1i

cn+

1i+

1=

mn+

1i+

1

Vn+

1i+

1

Tab

le2:

Exam

ple

of

3ce

lls

usi

ng

LM

Talg

ori

thm

for

Tab

le1

sub

step

con

dit

ion

s,as

illu

stra

ted

inF

ig.

2

12

i-1Vn

iVn

i+1Vn

n

i+1i-1 i VV nVn

n+1/3

i+1ViV n+1/3i-1V n

n+1/3i+1ViV n+1/2

i-1Vn+1/2

n+2/3

i+1ViV n+2/3

i-1Vn+1/2

i-1Vn+1iV n+1

i+1Vn+1

(a) Time n volumes

(b) Substep 1: flux from n value across i+1/2 face for time n+1/3 results

(c) Substep 2: flux from n value across i-1/2 face for time n+1/2 results

(d) Substep 3: flux from n+1/2 value across i+1/2 face for time n+2/3 result

(e) Substep 4: flux across all faces for time n+1 result

(f) End of substeps; volumes conserved

Figure 2: Volume fluxes during LMT substeps in Table 2. Gray areas are volumes movedbetween grid cells, in (b) and (d) flux areas are ∆t q∗/3; in (c) flux area is ∆t q∗/2; in (e) fluxareas are ∆t q∗, ∆t q∗/2 , ∆t q∗/3, and ∆t q∗ respectively, from left to right.

i + 2 contribute through fluxes across faces i − 3/2 and i + 3/2, respectively(these faces did not have subtime steps, see Table 1). Note that the discretemass equations in Substep 4 use c`i−2 and c`i+2, where c` is taken as the latestavailable concentration data in the cell. Our example has not specified thenumber of substeps on the i−5/2 and i+5/2 faces for cells i−2 and i+2. If thesefaces do not have substeps, then cn values would be used for the placeholder c`.

Although the above is illustrated for a structured grid cell with flux throughtwo sides, the basic functionality can be extended to grid cells with any numberof flux faces using either structured or unstructured grids. This LMT example issomewhat unwieldy because the subtime levels for the i cell of γi =

{13 ,

12 ,

23 , 1}

are non-uniform, which is awkward to implement in an algorithm. Implemen-

tation is simplified if we have uniform intervals of γi ={

1Fi, 2Fi, 3Fi, ...1

}where

Fi = max(Ri−1/2, Ri+1/2

)along with the requirement that Ri−1/2 and Ri+1/2

are integer powers of 2, which is similar to the time step constraint used in theLTS method of Sanders (2008).

The LMT method of eq. (26) - (28) is a conservative formulation that can

13

be applied with any interpolation scheme that provides cell face (ci−1/2, ci+1/2)concentrations from cell center concentrations (ci), e.g. QUICK (Leonard, 1979).Formally, the required number of subtime steps (R) depends on the stabilityregime associated with the interpolation method, typically given using a maxi-mum allowable constraint (CFLM ) compared to the local CFL computed withthe time step ∆t. For a simple 1D system, the number of substeps for the i+1/2cell face is

Ri+1/2 =

⌈ui+1/2

∆t∆x

CFLM

⌉(31)

The LMT method described above is entirely equivalent to a conventionalforward-time, centered space numerical scheme when R = 1 everywhere (i.e.identical to the CST and SMT methods for R = 1). However, R > 1 on one ormore faces affects the truncation error and numerical consistency of the method,as discussed in Appendix C.

2.4. LMT with Background filter (LMTB)A common but heretofore neglected issue in scalar transport is the compu-

tational cycles devoted to large regions of background values, which is arguablywasted effort. For example, if 90% of a water body has salinity (S) of 33 psu andthe dynamics associated with small perturbations (±ε) from this backgroundvalue are unimportant, one might like to limit salinity transport to regions thathave S > 33 + ε or S < 33 − ε psu. Otherwise, substantial computational ef-fort is wasted in transporting 33 psu from one cell into an adjacent cell withno change to the system dynamics. This idea does not appear to have beeninvestigated, likely because previous numerical schemes do not readily admitany simple approach to masking unimportant cells. We can think of this ideaas applying a background filter to the scalar field. Arguably such a filter canhave any complexity desired, but the most obvious uses are for masking out lowconcentration regions of transported scalars. Implementing a background filterto eliminate unnecessary computations is straightforward using a minor modi-fication of the LMT method. A set of transport cells is identified at the start ofa time step to include only cells whose concentration is outside the backgrounddefinition and those neighbor cells in the interpolation stencil that are withinthe distance dCFLM∆xe of a cell. Clearly, the background filtering can becomecomputationally expensive for higher-order interpolation stencils or ubiquitoushigh CFL numbers, so it appears best suited for systems where large regions ofbackground values are expected. The idea of the background filter is presentedherein as an obvious and natural extension of the LMT method, with its useillustrated in some simple test cases below. However, it should be noted thatthe consistency analysis of the LMT method (Appendix C) is only strictly ap-plicable to cells in the LMTB mask that include active transport. It is not clearwhether existing approaches are adequate for consistency/accuracy analysis ofa background filter. Further development and analyses of this LMTB methodand its practical advantages and limitations remain subjects for future research.

2.5. Hydrodynamic model couplingThe CST, SMT, LMT, and LMTB methods were implemented in the Fine

Resolution Environmental Hydrodynamics model (Frehd), which is a 3D Cartesian-grid, conservative finite-difference model for free-surface flows based on the

14

semi-implicit hydrodynamic TRIM model of Casulli (1990) and the ELCOMmodel of Hodges (2000). These models use two-level time-advance algorithms(i.e. only time n and n + 1 levels are involved) with barotropic (free-surface)terms computed through solution of a linear matrix inversion while baroclinic(density stratification) and momentum terms are discretized explicitly. Frehdincludes improvements for stability and accuracy (Hodges, 2004; Rueda et al.,2007) as well as optional inclusion of non-hydrostatic pressure (Wadzuk andHodges, 2009). An optional predictor-corrector approach in Frehd (Hodges andRueda, 2010) allows the baroclinic, momentum, and scalar transport terms tobe time-centered (i.e. θ = 1/2) rather than the strictly explicit approach inHodges (2000) or the Euler-Lagrangian approach in Casulli (1990). However,to prevent these more advanced features from complicating the results, thispaper uses Frehd in its simplest hydrostatic form: barotropic terms and conti-nuity are discretized with backwards Euler, while baroclinic, momentum, anddiffusive terms are discretized with forward Euler. Consistent with the back-wards Euler continuity discretization, the velocity flux (q∗) for scalar transportuses q∗ = qn+1. Spatial momentum gradients in nonlinear terms use quadraticdiscretization.

3. Results

3.1. Overview

Two test cases are used to illustrate the different scalar transport methods. Asimple 2D domain with time-invariant velocities is used to examine the baselineperformance for two scalar plumes over a wide range of CFL numbers (§3.2).As a more comprehensive test case, 41 passive scalars and one active scalarare transported in 3D model of Lake Kinneret, which includes a complicatedhydrodynamic field due to the presence of internal waves (§3.3).

3.2. Scalar plume in 2D

The transport method is illustrated with a simple 2D computational domainof 6× 104 grid cells with two parallel channels connected by a common outflow(Fig. 3). The flow is converged to steady-state using the Frehd model. Thechannels have different flow rates and depths, resulting in the velocity fieldshown in Fig. 4. Strong velocities and gradients occur locally at the channelconnection with peak velocities of 0.6 m/s. The model grid cells are 100 x 100m, with the global time step varied for different model runs to achieve a rangeof CFL numbers. Initial conditions for two different scalars plumes are createdwith 2D Gaussian concentration distributions with peak values of unity thatmerge into uniform background concentrations of 0.01. Each scalar plume isinitially 3 km in diameter. Although the hydrodynamic flow is steady, the apriori knowledge of the velocity field is not used in the transport solution; i.e.the transport solution computes the CFL numbers and subtime step masksanew for each time step. The progression of the scalar plumes is shown in Fig.5.

The four different methods, CST, SMT, LMT and LMTB were each runwith five different time steps, ∆t = {120, 360, 720, 1200, 1680} s, resulting inmaximum CFL numbers near the channel connection of {0.7, 2.2, 4.3, 7.2, 10.1},respectively. For the LMTB method, the background filter was invoked to mask

15

40 km20 km

10 km

18 km

Figure 3: Test domain with 100 x 100 m grid cells.

out concentrations below 0.02, which is twice the background concentration ofthe initial plume. The statistical distribution for the wall-clock transport timeof the two scalars in each global time step is shown in Fig. 6. These data are forthe three simulated days during which both plumes were entirely in the domain.During the runs used for timing, all output data other than computational timewere suppressed.

Root-mean-square (RMS) errors for simulations are presented in Fig. 7.These errors were computed by comparison to a reference CST simulation usingan order of magnitude smaller time step (∆t = 12 s), which provides a maximumCFL number of 0.07. To avoid contaminating the RMS statistics with relativelymeaningless background concentrations, the errors were computed using onlycells with concentrations greater than the background filter concentration of0.02. It is clear that the introduction spatial partitioning and local substeppingin LMTB (for all CFL) and for LMT (CFL > 1) leads to increased diffusion,which is expected from the order of accuracy analysis in Appendix C. The errorbehavior of the SMT and LMT methods at CFL = 0.7 for Scalar B is somewhatanomalous, and is discussed further in §4.2.

16

km

km

0 10 20 30 400

5

10

15

20

25

30

flow

spe

ed (m

/s)

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 4: Steady-state 2D velocity field used for scalar transport.

0 2 4 6

simulation days

0 0.75 1.52.25

3.5

km

km

0 10 20 30 400

5

10

15

20

25

30

conc

entra

tion

0

0.2

0.4

0.6

0.8

1

Figure 5: Transport of scalars over 6 days of simulation.

17

0

2

4

6

8

10

CST

SMT

LMT

LMTB

norm

aliz

ed s

cala

r tra

nspo

rt tim

e

CFL = 0.7

CST

SMT

LMT

LMTB

CFL = 2.2

CST

SMT

LMT

LMTB

CFL = 4.3

CST

SMT

LMT

LMTB

CFL = 7.2

CST

SMT

LMT

LMTB

CFL = 10.1

transport method

Figure 6: Distributions of wall-clock computational time for scalar transport at each time stepwith different maximum CFL numbers. Normalization scale is the mean computational timefor a single time step of the CST method at CFL = 0.7. Box provides median, 25th and 75th

percentile values, whiskers provide maximum and minimum values.

0 2 4 6 8 1010−3

10−2

10−1

Scalar A (fast)

RM

S er

ror

CFL0 2 4 6 8 10

10−3

10−2

10−1

Scalar B (slow)

CFL

RM

S er

ror

CTSSMTLMTLMTB

Figure 7: RMS errors for scalar concentrations after three days for simulations with the samerange of maximum CFL numbers as Fig. 6. Scalars A and B correspond to the lower andupper scalars in Fig. 5, respectively.

18

3.3. Application to Lake Kinneret

The CST, LMT and LMTB have been applied in a model simulation of LakeKinneret as a more complex test case. Lake Kinneret, Israel, also known as theSea of Galilee or Lake Tiberias, has been a subject of intensive study due to itsimportance as a source of fresh water in the Middle East. The active internalwave field in Lake Kinneret poses a hydrodynamic challenge due to the range ofscales and influences of the earth’s rotation (Dallimore et al., 2003; Hodges et al.,2000; Laval et al., 2003a,b; Gomez-Giraldo et al., 2006; Marti and Imberger,2006). For ecosystem models, computational challenges are associated with theneed to simulate seasonal-to-annual time scales at fine grid resolution for a largenumber of scalars; e.g. Hillmer et al. (2008) used the CAEDYM model (Bruceet al., 2006) with transport of 5 phytoplankton groups, 3 zooplankton groups,1 bacteria group along with carbon, nitrogen, phosphorous and silica in bothfilterable and particulate forms.

A model of Lake Kinneret was developed using the Frehd code applied ona 200× 200× 0.5 m grid for a total of 2.5× 105 grid cells (Fig. 8). The modelconfiguration and driving data were the same as used in Hodges et al. (2000)with the exception that the 3-day spin-up time used in the prior work was ne-glected; the present model was simply started from rest to simplify the initialcondition specification for scalars. In addition to the active scalar (tempera-ture) that affects density and internal waves, 41 passive scalars (tracers) weretransported by the model. These scalars were initialized as uniform horizontallayers at different depths with a concentration of 1.0 in the layer and 0.0 outside.Six of the scalars initially covered large physical regions of the domain: Surface,Thermocline, and Bottom scalars (each with 2 separate copies) were distributedover the entire surface mixed layer, the entire thermocline, and the entire hy-polimnion (i.e. volume below the thermocline), respectively. The additional 35tracers were initially confined to 1 m layers (Table 3). The model hydrody-namic time step was set at 120 s, with substeps occurring whenever the CFLexceeds 0.2, which results in two to seven substeps through most of the threedays of model simulation. The low advective CFL constraint for substeppingwas selected to test a range of substep conditions while preventing baroclinicinstabilities (as opposed to advective instabilities) that occur for CFL > 1.5for this particular test case. These instabilities are due to the use of simpleexplicit-Euler baroclinic terms (rather than predictor-corrector) in the modelconfiguration. The models were run on a desktop workstation using 2 × 2.66GHz 6-Core Intel Xeon processors with 16 GB of RAM running OSX 10.9 andMatlab R2012b. The total wall-clock run times were between 36 and 62 hoursfor all models, with the LMT and LMTB methods performing better than theCST and SMT methods.

The daily sea breeze over Lake Kinneret develops a strong internal wavefield as previously demonstrated with the ELCOM model (Hodges et al., 2000).A set of time/space slices of the modeled temperature field using Frehd areshown in Fig. 9. Above the thermocline the lake tends to be well mixed, asevidenced by the the results for the Violet and Indigo scalars that start abovethe thermocline and are mixed over most of surface mixed layer by the end ofthe simulation (Table 3). The CST and LMT scalar distributions for Surface,Thermocline, Bottom, and Green scalars can be compared using Figs. 10 – 17.The scalar distributions are qualitatively similar in their major features, results

19

easting (km)

north

ing

(km

)

0 2 4 6 8 10 12

0

2

4

6

8

10

12

14

16

18

20

depth (m)0 10 20 30 40

Figure 8: Model bathymetry of Lake Kinneret (data courtesy of Yigal Allon Kinneret Limno-logical Laboratories).

which are consistent across all other scalars and the SMT method (not shown).The LMTB method performs similarly to the LMT method for scalars throughthe bulk of the system, as can be seen by comparing Figs. 13 and 17 to Figs. 18and 19, respectively. Unfortunately, the LMTB method appears have an incom-patibility with the turbulence model of Hodges et al. (2000) near boundaries,which substantially distorts modeling of the Surface, Red, and Orange scalars(not shown). The cause of this incompatibility is under investigation, but thesuccessful performance of the LMTB method with the Thermocline, Bottom,Violet, Inidigo, Blue, Green and Yellow scalars indicates the problem is notinherent in the LMTB method itself.

For Lake Kinneret simulations, it was not practical to compute RMS errorin the same manner as Fig. 7, i.e. with reference to a small time-step CSTsimulation. Instead, the RMS differences between the scalar concentrations forthe three mass-transport methods (SMT, LMT, LMTB) and the CST methodat the same time step are shown in Fig. 20. The mass transport methods aregenerally with 10% of the CST method, excepting the Surface, Orange and Redscalars for the LMTB method as discussed above. The Violet scalar in Fig.20 shows an extremely small RMS difference compared to the others, which isconsistent with a scalar that is nearly uniformly dispersed by turbulent mixingin the surface mixed layer. As shown in Table 3, the Violet scalar at the endof the simulation covers 62% of the domain, quite close to the initial 66.5%initially covered by the Surface scalar that was initially distributed over theentire surface mixed layer.

20

scalar initial depth final depth initial # final # initial % of final % ofname range (m) range (m) cells cells domain domain

Surface 0 – 14.3 0 – 26.8 165929 199505 66.5 79.9Thermocline 14.3 – 16.8 0 – 26.8 28393 155996 11.4 62.5

Bottom 19.3 – 41.8 9.8 – 41.8 65435 107079 26.2 42.9

Violet 9.8 –10.8 0 – 20.3 21800 154804 8.7 62.0Indigo 14.8 –15.8 0 – 23.8 19069 101546 7.6 40.7Blue 19.8 – 20.8 12.8 - -34.8 15468 75887 6.2 30.4

Green 24.8 – 25.8 14.3 – 38.3 11354 72818 4.6 29.2Yellow 29.8 – 30.8 14.3 – 39.8 7813 54190 3.1 21.7Orange 34.8 – 35.8 27.3 – 39.8 4549 13079 1.8 5.2

Red 39.8 – 41.8 39.8 – 41.8 1311 1369 0.5 0.6

Table 3: Initial and final conditions of a subset of the transported scalars (CST method).Final depth range and number of cells is based on concentrations greater than 0.02 (i.e. abovethe background filter cutoff). The upper set of scalars (Surface, Thermocline, Bottom) aredistributed over the identifiable physical regions of the initial temperature structure. The lowerset of scalars are distributed over 1 m thick layers. The 31 scalars not list above consisted ofduplicates of Surface, Thermocline and Bottom, along with 4 additional scalars of each colorthat are initialized at 1 m intervals between the layers identified above.

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Tem

pera

ture

(C)

15

20

25

30

Step = 1636Time = 55 h 32 m

Figure 9: Typical isotherm displacements on three cutting planes for Lake Kinneret modelresults.

21

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.2

0.4

0.6

0.8

1

Step = 1636Time = 55 h 32 mCST Surface Scalar

Figure 10: Typical concentration of Surface scalar using the CST method.

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.2

0.4

0.6

0.8

1

Step = 1636Time = 55 h 32 mLMT Surface Scalar

Figure 11: Typical concentration of Surface scalar using the LMT method.

22

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.2

0.4

0.6

0.8

1

Step = 1636Time = 55 h 32 mCST ThermoclineScalar

Figure 12: Typical concentration of Thermocline scalar using the CST method.

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.2

0.4

0.6

0.8

1

Step = 1636Time = 55 h 32 mLMT ThermoclineScalar

Figure 13: Typical concentration of Thermocline scalar using the LMT method.

23

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.2

0.4

0.6

0.8

1

Step = 1636Time = 55 h 32 mCST Bottom Scalar

Figure 14: Typical concentration of Bottom scalar using the CST method.

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.2

0.4

0.6

0.8

1

Step = 1636Time = 55 h 32 mLMT Bottom Scalar

Figure 15: Typical concentration of Bottom scalar using the LMT method.

24

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.1

0.2

0.3

0.4

0.5

Step = 1636Time = 55 h 32 mCST Green Scalar

Figure 16: Typical concentration of Green scalar using the CST method.

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.1

0.2

0.3

0.4

0.5

Step = 1636Time = 55 h 32 mLMT Green Scalar

Figure 17: Typical concentration of Green scalar using the LMT method.

25

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.2

0.4

0.6

0.8

1

Step = 1636Time = 55 h 32 mLMTB ThermoclineScalar

Figure 18: Typical concentration of Thermocline scalar using the LMTB method.

easting (km)

north

ing

(km

)

C

C’

B B’

A−A’

2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

easting (km)

dept

h (m

)

B−B’

A’A

0 2 4 6 8 10 12

0

5

10

15

20

25

30

35

40

northing (km)

dept

h (m

)

C−C’

A A’

02468101214161820

0

5

10

15

20

25

30

35

40

Con

cent

ratio

n

0

0.1

0.2

0.3

0.4

0.5

Step = 1636Time = 55 h 32 mLMT Green Scalar

Figure 19: Typical concentration of Green scalar using the LMTB method.

26

0 0.1 0.2 0.3

Red

Orange

Yellow

Green

Blue

Indigo

Violet

Bottom

Thermocline

Surface

RMS difference

SMTLMTLMTB

Figure 20: RMS difference between mass transport and CST methods for scalars transportedin Lake Kinneret at the final model time step, computed for cells with concentrations greaterthan the background filter cutoff.

27

One way to think about the computational effort required by the differentmethods is to compare the number of grid cells used in the transport computa-tions for each time step. Because the hydrodynamic field and high CFL regionevolves through time, the number of cells requiring subtime stepping will alsoevolve. When subtime stepping is required, the CST method requires multipletransport computations for each grid cell, which can be thought of as increas-ing the total number of grid cells requiring transport computations, e.g. threesubsteps over the entire domain of Ncells is the same as computing transportover 3Ncells, as shown in Fig. 21. The LMT and LMTB methods reduce thenumber of grid cells in the transport computations because subtime stepping isnot required for all cells. For the LMTB method, each scalar is subtime-steppedover a different subset of cells, as shown with separate lines for each scalar inFig. 21. For the LMT method each scalar is subtime-stepped over the same cellset, so that all scalars collapse to a single line in Fig. 21. method provides somefurther improvement (Fig. 22).

0 20 40 60

10−1

100

101

simulated hours

Nco

mpu

te /

Nce

lls

YellowGreenBlue

Violet

Indigo

All LMT

All CST

Figure 21: Number of computational cells (Ncompute) for a single scalar in each model timestep (normalized by total number of cells in the domain, Ncells). Each of the Yellow, Green,Blue, Indigo and Violet scalars in LMTB has a different number of computational cells. AllLMT scalars have exactly the same number of computational cells. The number of CST scalarcomputational cells includes the entire domain that is iterated multiple times when subtimestepping is required.

Summing the computational cells for each scalar (i.e. from Fig. 21) overall the scalars provides a comparison of the total number of cells that requirecomputation for each method, which evolves over time as shown in Fig. 22.Comparing CST to LMT in Fig. 22 indicates the LMT method reduces thenumber of computational cells by a factor of ∼3 for the Lake Kinneret testcase, where at least 3 substeps are required for a portion of the domain throughmost of the simulation period. This reduction in computational cells resultsin a reduction of the scalar computational time per time step (Tscalar) by afactor of ∼2, as shown in Fig. 23. This figure also illustrates the overwhelming

28

0 20 40 600

1

2

3

4

5

6

7

LMTB

LMT

CST

simulated hours

Y N

com

pute

/ (N

scal

ar N

cells

)

Figure 22: Total number of cells requiring transport computations in each time step, normal-ized by the total number of cells times the number of scalars; i.e. 4 indicates four computationcycles over all 2.5 × 105 cells for all 41 scalars during a single time step; thus, the y-axis alsoindicates the total number of substeps for the CST algorithm.

computational costs, relative to hydrodynamics, associated with transportinga large number of scalars. The scalar transport time for the LMTB methodis initially low for each time step, but slowly increases during the simulationand eventually surpasses that of the LMT method. This result occurs due tothe increasing complexity of the scalar distributions over time (e.g. Fig. 17),which increases the computational burden of creating the scalar backgroundmask. Initially this burden is offset by the decreased effort associated withthe reduced number of computational cells for LMTB (Fig. 22), but as thescalars diffuse the number of cells increases for each scalar (Fig. 21) and theadvantage of the background filter is lost. For scalars covering large swaths ofa domain (e.g. Thermocline, Bottom, Violet and Indigo scalars at the end ofthe simulation, Table 3), the LMTB algorithm is disadvantaged as it obtainslittle computational savings for widely distributed scalars but must pay thecomputational expense for individual background masks; in contrast LMT onlyrequires a single substep mask (based on CFL) that is applicable to all scalarsand is not affected by scalar distribution. Note that the CST computationaltime in Fig. 23 substantially increases around 30 h, where correspondingly Fig.22 shows the number of CST substeps increasing from 3 to 6 due to a locally-high CFL number in the simulation. This increase does not significantly affectthe LMT computational time or number of computational cells, which implies

29

that the number of grid cells affected by the increased substep requirement isrelatively small. Despite the small number of cells involved, the computationaltime for a single global time step of the CST method nearly doubles, and remainsquite variable over the entire simulation.

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40

simulated hours

T scal

ar /

T hydr

o

LMTB

LMT

CST

Figure 23: Total computational time per time step for transport of 41 scalars, normalized byhydrodynamic computational time Thydro in the same step.

30

4. Discussion

4.1. Computational efficiency

The simulation results show that the LMTB and LMT methods are gener-ally more computationally efficient than the conventional CST method for scalartransport. Unfortunately, there does not seem to be any simple a priori methodfor quantitatively predicting the computational speed-up of either method. TheLMT method gains its computational advantage when a small region requiressubstepping and there are multiple scalars requiring transport. This method in-creases in effectiveness as the number of scalars increases, the number of substepsincreases, and the volume of the substepped region decreases. In contrast, theLMTB method gains its advantage when a large region of scalars can be consid-ered “background,” an effect that neither depends on the number of scalars northe number of substeps. Thus, the LMTB method increases in effectiveness onlywith decreasing volumes for transported scalar concentrations. Furthermore, asdiffusion (either physical or numerical) tends to increase the non-backgroundvolume of a scalar, the effectiveness of the LMTB method will generally tendto decrease over the course of a simulation. It follows that LMT directly de-pends on variability in the hydrodynamic field as well as the number of scalars,whereas LMTB depends principally on the scalar distribution (as influenced byhydrodynamics).

The computational time results for the simple example (Fig. 6) show thatthe LMTB method is vastly superior to LMT when using only two scalars thatoccupy a small portion of the domain. However, the results of the more com-plex Lake Kinneret example (Fig. 23) indicate there are practical limits, whichdepend on how dispersed the scalars are over the domain (Table 3). If the trans-portable scalar volume increases with time then LMTB becomes less effective:the costs of masking the background will exceed the computational savings asso-ciated with spatially limiting the transport algorithm. It follows that the LMTBmethod might be best suited to systems with significant boundary exchanges(e.g. not lakes), where influxes of below-background scalar concentrations willcontinually renew the volume that is masked out of the transport computation.

4.2. Computational accuracy

The practical accuracy of the new methods requires further investigation.Error results for scalar plumes (Fig. 7) are consistent with expectations forScalar A, which is within the fast-moving (lower) portion of the domain in Fig.5. For this region the SMT and LMT errors collapse to the CST error forCFL < 1 (i.e. when there is no local region for subtime stepping). For theLMTB simulations and LMT with CFL > 1, the Scalar A results are also asexpected, showing increased diffusion when the scalar is partitioned betweentransported and non-transported regions (see Appendix C). However, the errorbehavior of Scalar B is anomalous at low CFL conditions for both SMT andLMT methods in that the errors do not collapse to the CST error for CFL < 1,although they are (as expected) smaller than errors for Scalar A. The principaldifference between Scalar A and Scalar B is that the former is in a stronglyadvective region, while the latter is dominated by diffusion. The failure of themass-transport SMT and LMT schemes to collapse to the CST scheme at lowCFL most likely indicates an issue with the diffusive term in the mass transport

31

scheme (see Appendix B). It is not clear whether there is a latent bug in thecode, or increased numerical diffusion is inherent in the discretization of thediffusion term in the mass transport scheme.

4.3. Open questions

The theoretical and practical efficiencies of the new LMT method comparedto conventional LTS methods remain subjects for future investigation. Thecomputational expense of conventional LTS methods is in regionalization: i.e.,identifying contiguous regions to be solved with a similar time step and theinterface between different regions. In contrast, the LMT method identifies thesubtime step for each cell face and uses a simple mask to identify all cell facesthat have the same subtime interval, regardless of their contiguous relationship.The principal computational expense in the present LMT method is identifyingneighbors that are required for the convective discretization on a cell face; i.e.simple 1st order upwind and 2nd order central interpolation schemes requireonly the cells on either side of the face, but 3rd order requires three cells (ormore if a multi-dimensional stencil is applied). Because a single cell may beinvolved in the convection stencil for multiple cells, this neighbor identificationinvolves a logical union operation to remove duplicate cells from the computa-tional set. Profiling with the present code shows that the union operation isa substantial computational expense. Beyond this general discussion, a moredirect comparison of efficiencies from LMT and LTS methods would be prema-ture: LTS methods have a long history and significant efforts have been made todevelop efficient regionalization schemes, in contrast the present LMT methodhas been simply coded with off-the-shelf (Matlab) functions; no doubt there aremore efficient approaches than used herein.

The proposed LMTB method is a radical departure from previous transportmethods that theoretically has the potential to dramatically reduce computa-tional time for scalar transport (e.g. Fig. 6). However, there is an open questionas to whether the computational costs associated with separately masking indi-vidual scalars can be sufficiently reduced in more complicated simulations (e.g.Fig. 23). Furthermore, in the present implementation the LMTB method ap-pears to interact poorly with the turbulence model near boundaries, for reasonsthat are not yet clear. It remains to be seen whether the LMTB method can bemade sufficiently robust for practical use.

5. Conclusions

The numerical schemes for Local Mass Transport (LMT) and Local MassTransport with Background filtering (LMTB) are entirely new approaches toscalar transport. These methods are conceptually simple: instead of transport-ing concentration, both volume and mass are transported with concentrationrecovered diagnostically. The subtime intervals for transport through differentfaces of a single cell are allowed to be different, which makes it possible to splitthe domain into volumes with different subtime step sizes without requiring adhoc boundary conditions. Although only tested for a simple explicit transportscheme herein, the methods appear to be suitable for implementation with anynumber of established methods. Likewise, although tested for a structured gridthere does not appear to be any impediment to using an unstructured grid.

32

These mass transport methods have two major advantages over conventionalconcentration transport: 1) the number of substeps used on each face is inde-pendent, so that the increased computational effort associated with locally-highCFL numbers at the global (large) time step depends upon the number of cellsin the high CFL region rather than the overall number of grid cells; and 2)background filters can be defined so that scalar transport computations are notconducted for irrelevant background concentrations. The effectiveness of thesemethods in reducing computational costs appears to be principally a functionof the efficiency of the algorithms used to identify localized regions (either byCFL or concentration), which remains a subject for further research.

6. Acknowledgements

This material is based upon work partially supported by the U.S. NationalScience Foundation under Grants No. 0710901, CFF-1331610, and in part by theBP/The Gulf of Mexico Research Initiative and the Research and Developmentprogram of the Texas General Land Office Oil Spill Prevention and ResponseDivision under Grant No. 13-439-000-7898.

7. Appendix A: Substep with linearly varying fluxes

In eq. (14), the subtime flux rates are represented by q∗, the effective uniformflux over the time step ∆t. However, it is also possible to use time-varying fluxrates as

Vn+r/Ri c

n+r/Ri = V ni c

ni −

∆t

R

r−1∑k=0

(qn+k/Ri+1/2 c

n+k/Ri+1/2 − q

n+k/Ri−1/2 c

n+k/Ri−1/2

)(32)

Note that for consistency with continuity (Gross et al., 2002), the subtime stepfluxes, qn+k/R for k = {0...R− 1}, must identically satisfy

∆t

R

R−1∑k=0

qn+k/R −∆tq∗ = 0 (33)

where q∗ is the flux between time steps n and n + 1 that is consistent withdiscrete continuity in the hydrodynamic solution. Using eq. (12), it follows thatthe subtime-step flux must satisfy

1

R

R−1∑k=0

qn+k/R −[θqn+1 + (1− θ) qn

]= 0 (34)

For θ = 1/2, i.e. Crank-Nicolson hydrodynamic discretization, a definition forsubtime-step values of q that provides for linearly increasing or decreasing fluxrates over the subtime intervals is

qn+k/R =2R− (2k − 1)

2Rqn +

2k − 1

2Rqn+1 (35)

However, if rapid velocity changes are unimportant, consistency with continuitycan be obtained by using a temporally-uniform flux, i.e.

qn+k/R = q∗ : 0 ≤ k ≤ R− 1 (36)

will identically satisfy eq. (33).

33

8. Appendix B: Discretization with diffusion terms

The CST method with diffusion terms added to eq. (14) can be written as

Vn+r/Ri c

n+r/Ri = V ni c

ni −

∆t

R

r−1∑k=0

(q∗i+1/2 c

n+k/Ri+1/2 − q

∗i−1/2 c

n+k/Ri−1/2

)+

∆t

R∆x

([κa]

n+1/2i+1/2

r−1∑k=0

[cn+k/Ri+1 − cn+k/R

i

]− [κa]

n+1/2i−1/2

r−1∑k=0

[cn+k/Ri − cn+k/R

i−1

])(37)

The SMT method with diffusion terms added to eq. (19) is identical to eq. (37)above with the identities mn = V ncn and mn+r/R = V n+r/Rcn+r/R. The LMTmethod with diffusion terms added to eq. (23) is

mn+γ(j)i = mn

i −∆t

Ri+1/2q∗i+1/2

fp(j)−1∑k=0

cn+k/Ri+1/2

i+1/2 +∆t

Ri−1/2q∗i−1/2

fm(j)−1∑k=0

cn+k/Ri−1/2

i−1/2

+∆t

Ri+1/2 ∆x(κa)

n+1/2i+1/2

fp(j)−1∑k=0

(cn+k/Ri+1/2

i+1 − cn+k/Ri+1/2

i

)

− ∆t

Ri−1/2 ∆x(κa)

n+1/2i−1/2

fm(j)−1∑k=0

(cn+k/Ri−1/2

i − cn+k/Ri−1/2

i−1

)(38)

Note that we have included an approximation of [κa]n+r/R ≈ [κa]

n+1/2in the

diffusive term, which is appropriate where surface displacements are small, theflow is dominated by advection, and κ is a turbulence model approximation thatis inherently uncertain. This approximation is not required for the algorithm;

i.e. the subtime values of [κa]n+r/R

could be computed by linear interpolationfrom the time n to n + 1 hydrodynamic solution, but its inclusion here wouldclutter this presentation without providing further insight.

9. Appendix C: Consistency of subtime-stepping discretization

The 1D advection equation for concentration c can be written as

ct + ucx = 0 (39)

where subscripts represent derivatives with time t and space x. As a point ofreference, the forward-time, first-order upwind discretization for u > 0, withoutsubtime stepping can be written as

[c]n+1i − [c]ni

∆t+ u

[c]ni − [c]ni−1

∆x= 0 (40)

where brackets are used to provide subscripts that are discrete locations andsuperscripts that are discrete time. Using Taylor series expansions, it can beshown that

[ct]ni + u[cx]ni = −∆t

2[ctt]

ni + u

∆x

2[cxx]ni +O

(∆t2

)+O

(∆x2

)(41)

34

In contrast, the forward-time, first-order upwind discretization for the LMTmethod for a uniform, constant-volume grid can be presented as

[c]n+1i − [c]ni

∆t+

u

∆x

{1

F

F−1∑k=0

([c]n+k/Fi

)− 1

G

G−1∑k=0

([c]n+k/Gi−1

)}= 0 (42)

where F and G are a simpler notation for substep size Ri+1/2 and Ri−1/2 for asingle grid cell. Taylor series expansion provides

[ct]ni + u[cx]ni =u

∆t

∆x[ct]

ni

{1

F 2

F−1∑k=0

k − 1

G2

G−1∑k=0

k

}− ∆t

2[ctt]

ni + u

∆x

2[cxx]ni

+O

(∆t2

∆x

)+O

(∆t2

)+O

(∆x2

)(43)

Note that

1

F 2

F−1∑k=0

k =1

2

(1− 1

F

)(44)

which evaluates to zero for F = 1, and to 1/2 as F →∞. It follows that

[ct]ni + u[cx]ni =u

∆t

∆x[ct]

ni

{1

2G− 1

2F

}+O

(∆t2

∆x

)+O (∆t) +O (∆x) (45)

which appears to be conditionally consistent for finite F 6= G and collapses tounconditionally consistent eq. (41) for F = G, including as F,G → ∞. It canbe shown that the O

(∆t2/∆x

)term will similarly vanish for F = G. Of key

importance in comparing eqs. (41) and (45) is the introduction of the truncationerror term that scales on the CFL and the mismatch between substeps on thefaces (F,G).

Although eq. (45) appears to be conditionally consistent for finite F 6= G,for a fixed CFL constraint in the LMT method the number of face substeps Fand G are functions of the CFL, which changes with refinement of ∆t or ∆x.Thus as ∆t → 0 for a fixed ∆x and a fixed CFL constraint, the CFL musteverywhere decrease such that F,G → 1 for all cells and eq. (45) collapses tounconditionally consistent eq. (41). Conversely, as ∆x → 0 for fixed ∆t anda fixed CFL constraint, the CFL must everywhere increase linearly towardsinfinity as will the required number of substeps, i.e. F,G→∞ and their effectgoes linearly to zero; the equation again collapses to unconditionally consistenteq. (41). Thus, despite the new truncation error term, the LMT method isformally unconditionally consistent under refinement of either ∆t or ∆x.

It can similarly be shown that Taylor series expansion of a forward-timequadratic upwind scheme with LMT leads to

[ct]ni + u[cx]ni =u

∆t

∆x[ct]

ni

{1

F 2

F−1∑k=0

k − 1

G2

G−1∑k=0

k

}+O (∆t) +O (∆t∆x) +O

(∆x2

)(46)

which reduces to

[ct]ni + u[cx]ni =u

∆t

∆x[ct]

ni

{1

G− 1

F

}+O (∆t) +O (∆t∆x) +O

(∆x2

)(47)

35

Thus, the quadratic upwind scheme has a similar leading error as the 1st-orderupwind for F 6= G, with the term evaluating to exactly zero when F = G; i.e.for the CST or SMT algorithms. Consistency for the discrete diffusion termsof Appendix B can be shown to be similar; the conditional terms arise in the

Taylor series expansion in time for cn+k/Fi and c

n+k/Gi−1 , which can only formally

cancel with when F = G, but also disappear in the limit of refinement.

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Constantinescu, E. M., Sandu, A., 2007. Multirate timestepping methods forhyperbolic conservation laws. Journal of Scientific Computing 33 (3), 239–278.

Coquel, F., Nguyen, Q. L., Postel, M., Tran, Q. H., 2010. Local time step-ping applied to implicit-explicit methods for hyperbolic systems. MultiscaleModeling & Simulation 8 (2), 540–570.

Crossley, A. J., Wright, N. G., 2005. Time accurate local time stepping forthe unsteady shallow water equations. International Journal for NumericalMethods in Fluids 48 (7), 775–799.

Dallimore, C. J., Hodges, B. R., Imberger, J., 2003. Coupling an underflowmodel to a three-dimensional hydrodynamic model. Journal of HydraulicEngineering-ASCE 129 (10), 748–757.

Dawson, C., Kirby, R., 2001. High resolution schemes for conservation laws withlocally varying time steps. SIAM Journal on Scientific Computing 22 (6),2256–2281.

Dawson, C., Trahan, C. J., Kubatko, E. J., Westerink, J. J., 2013. A parallellocal timestepping Runge-Kutta discontinuous Galerkin method with appli-cations to coastal ocean modeling. Computer Methods in Applied Mechanicsand Engineering 259, 154–165.

Gomez-Giraldo, A., Imberger, J., Antenucci, J. P., 2006. Spatial structure ofthe dominant basin-scale internal waves in Lake Kinneret. Limnology andOceanography 51 (1), 229–246.

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