A numerical model for simulating sediment routing in shallow water flow

Thesis

Reference

A numerical model for simulating sediment routing in shallow water flow

ZIA, Haseeb

Abstract

The last few decades have seen significant efforts to model erosion, transport and deposition of sediment in flowing surface water. A wide variety of models, ranging from simple empirical models to highly sophisticated physically-based models have been presented. This thesis aims to present a physically realistic but simple model to simulate sediment routing. The model uses shallow water equations for hydrodynamics, while it is coupled with sediment conservation and transport laws to enable modification of the underlying substrate. To solve the model, I present a straightforward Riemann-solver free approach, based on the explicit non-oscillatory central differencing (NOC) scheme. The used central approach becomes excessively diffusive when applied to highly nonlinear advection problems. In this thesis, I present a correction to reduce such numerical dissipation. To test the accuracy and robustness of the scheme, I present a number of test cases and the results are compared with analytical solutions and laboratory experiments.

ZIA, Haseeb. A numerical model for simulating sediment routing in shallow water flow . Thèse de doctorat : Univ. Genève, 2015, no. Sc. 4855

URN : urn:nbn:ch:unige-818659

DOI : 10.13097/archive-ouverte/unige:81865

Available at:

http://archive-ouverte.unige.ch/unige:81865

Disclaimer: layout of this document may differ from the published version.

1 / 1

A Numerical Model for Simulating

Sediment Routing in Shallow Water Flow

THÈSE

présentée à la Faculté des sciences de l'Université de Genève

pour obtenir le grade de Docteur ès sciences, mention Sciences de la Terre

par

Haseeb Zia

de Pakistan (Lahore)

Thèse N^o xxx

Genève

Atelier d'impression ReproMail, Université de Genève 2015

I would like to express my deepest gratitude to my supervisor, Dr. Guy Simpson for his advice, support and encouragement. Without his guidance, completion of this thesis would not have been possible. I also want to thank the jury members, Prof.

Sébastien Castelltort, Dr. Shiva P. Pudasaini and Dr. Annunziato Siviglia for taking out time to read my thesis and providing valuable feedback.

I especially want to thank Swiss National Science Foundation for providing the funding for research and preparation of the thesis.

I would like to extend my sincere thanks and appreciation to all my colleagues in the doctoral school for their friendship and for making my time in Geneva delightful.

Thanks to my siblings for their support and good will. Special thanks to my parents (Zia Minhas and Yasmeen Zia) for their constant patience, love and support while I was away from home. Finally, I would like to thank my lovely wife Ayeda, who missed out on a lot of my time while I was working on this thesis.

ABSTRACT xi

Résumé xiii

1 Introduction 1

1.1 Opening statement . . . 1

1.2 Background . . . 2

1.3 Physically-based sediment routing modelling . . . 5

1.3.1 Hydrodynamics . . . 5

1.3.1.1 Navier-Stokes Equations . . . 6

1.3.1.2 Shallow water equations . . . 7

1.3.1.3 Diusive Wave approximation . . . 8

1.3.1.4 Kinematic wave . . . 9

1.3.2 Sediment transport modelling . . . 10

1.3.2.1 Conservation laws for sediment transport . . . 13

1.3.2.1.1 Exner's equation : . . . 14

1.3.2.1.2 Entrainment-deposition ux formulation : . 15 1.4 The mathematical model . . . 15

1.5 Challenges . . . 19

1.6 Literature review of numerical schemes . . . 20

1.7 Thesis structure . . . 24

2 Anti-diusive non-oscillatory central dierence adNOC Scheme 25 2.1 Introduction . . . 25

2.2 Central schemes . . . 27

2.3 Anti-diusion slopes . . . 31

2.4 Stability . . . 33

2.5 Test cases . . . 36

2.5.1 Shallow water Equations . . . 36

2.5.1.1 Dam break in one-dimension . . . 37

2.5.1.2 Breach of circular dam . . . 39

2.5.2 Flow over mobile bed . . . 39

3 2D adNOC Scheme for the Shallow water-morphodynamic evolu- tion model 45 3.1 Introduction . . . 45

3.2 Governing Equations . . . 47

3.3 NOC Numerical Scheme . . . 52

3.4 Anti-diusion correction . . . 60

3.5 Results . . . 63

3.5.1 Dam Break over erodible bed in one dimension . . . 63

3.5.2 Dam Break over erodible bed in two dimensions . . . 64

3.5.3 Evolution of a conical dune . . . 65

3.5.4 Stratigraphic development of a one dimensional sediment wedge 68 3.5.5 Two dimensional ume experiment . . . 69

3.6 Notation . . . 73

4 Well ballanced adNOC scheme 75 4.1 Introduction . . . 75

4.2 The adNOC Scheme . . . 77

4.3 Balancing . . . 81

4.4 Well-balanced adNOC scheme . . . 82

4.4.1 Verication of the C-property . . . 84

4.4.2 Partially wet cells . . . 85

4.5 Positivity-preservation . . . 85

4.6 Numerical results . . . 87 4.6.1 One-dimensional quiescent ow over a bump . . . 87 4.6.2 Oscillatory ow in a parabolic bowl . . . 89 4.7 Perturbation of a lake at rest with discontinuous bottom topography 91 4.8 Conclusion . . . 93

5 Concluding remarks 99

5.1 Future perspectives . . . 101

1.1 Complete spectrum of the sediment routing models with simple mo-

dels on the left and complicated models on the right side. . . 4

2.1 Cell reconstructions and quadrature evaluations. . . 30

2.2 Finite dierence approximations to be utilized for anti-diusive scheme. 32 2.3 Maximum stable courant number for dierent combinations of γ and ε evaluated using the inequality (2.22). . . 36

2.4 Solution for 1d-dam breach simulation. . . 38

2.5 Solution for circular dam breach simulation. . . 40

2.5 Solution for circular dam breach experiment. . . 41

2.6 Initial conditions for the ow over mobile bed test case. The water depth is divided by 10 for better viewing. . . 42

2.7 The bed-form after 200, 700 and 1400 seconds of sediment transport as calculated by standard NOC, anti-diusive NOC and fourth order CWENO schemes. . . 43

3.1 Staggered grid with grid points marked by◦and•for the two dierent grids. The cell centers of the contributing sub-cells are marked by . 56 3.2 1D dam break on erodible bed. Water depth and bed morphology, 60 sec. after dam break. . . 64

3.3 Two dimensional dam break experiment. . . 66

3.4 Evolution of a conical dune. Bed topography after 100 hours. . . 68

3.5 Stratigraphic development in the 1D ume experiment. . . 70

3.6 Dynamic bed forms in a two dimensional numerical ume calculated with the adNOC scheme. . . 72

4.1 Cell reconstructions in the case of a partially wet cell. . . 86 4.2 The water height H and the Discharge q at steady state calculated

with the NOC and adNOC schemes. Test case (1) is shown at the top, test case (2) in the middle and test case (3) is shown at the bottom. 90 4.3 Flow over a parabola bed topography. The analytical and the nume-

rical solution after 1¹₂ and 2 time periods of the oscillation. . . 91 4.4 Bed topography with a sharp step in the center. . . 92 4.5 Water height and discharge for lake test case. The solution is calcu-

lated with the adNOC scheme. The arrows in the discharge gures signify the direction of the discharge while the color signies the am- plitude. . . 96

3.1 Parameter values used in the test cases. Units are shown in brackets where applicable. . . 65 3.2 List of notation used, with units in brackets where applicable. . . 74 4.1 Boundary conditions for the one-dimensional quiescent ow over a

bump test cases. . . 88

The last few decades have seen signicant eorts to model erosion, transport and deposition of sediment in owing surface water. A wide variety of models have been presented, ranging from simple empirical models to highly sophisticated physically- based models. The aim of this thesis is to present a coupled model that retains only the most essential aspects of water ow. It is therefore physically realistic but simple. The model uses the shallow water equations for ow hydrodynamics, while it is coupled with sediment conservation and transport laws to enable modication of the underlying substrate.

To solve the model I present a straightforward Riemann-solver free approach ba- sed on the explicit non-oscillatory central dierencing (NOC) scheme. The scheme has already been widely applied to hyperbolic conservation laws in other contexts.

The version of this predictor-corrector scheme presented in this thesis is second-order accurate in time and space, which enables it to resolve shocks and discontinuities of the solution reasonably well. Following the central dierencing approach, the scheme uses staggered grids for discretization. To ensure stability of the scheme, I use slope and ux, TVD (Total Variation Diminishing) limiters. The Riemann-solver free for- mulation of the scheme provides exibility to easily change the formulae used to calculate erosion, deposition and sediment uxes. To test the accuracy and robust- ness of the scheme, I present a number of test cases and the results are compared with analytical solutions and laboratory experiments.

The central approach, which forms the basis of the scheme presented in this thesis, becomes excessively diusive when applied to highly nonlinear advection problems where small time steps are necessary to maintain stability. In this thesis, I present a correction to reduce such numerical dissipation for this class of problems.

The correction is obtained by selecting the appropriate nite dierence approxima-

tions for calculating slopes utilized to reconstruct the solution from the cell averages.

The proposed correction is applied to the widely used Nessyahu-Tadmor scheme to demonstrate the utility of the correction. The stability of the corrected scheme is also discussed and the condition for the scheme to become TVD (Total Variation Diminishing) is presented. The corrected scheme is tested for its ability to eectively resolve sharp discontinuities while using small time steps with a number of test cases and the results are compared with analytical solutions and published results.

The solution of the shallow water equations with variable bed slope is numeri- cally challenging, mainly posing challenges on two fronts. First, the solution often requires a delicate balance of the ux gradient and the source terms. An incorrect calculation of this balance can result in articial waves that have comparable am- plitude to the solution. Second, the moving wet-dry boundary poses challenge for the stability of the scheme as the calculation may break down if the water depth becomes negative. In the latter part of the thesis, a modied anti-diusive central scheme is presented, which adequately meets both the above mentioned challenges of well-balancing and positivity preservation. The performance of the scheme is tes- ted for correct evaluation of the source term and robustness of the scheme in the presence of wet-dry fronts. The results are compared with published and analytical solutions.

This thesis demonstrates the success of a modied central dierencing approach when applied to coupled problems involving shallow water hydrodynamics and mor- phodynamic evolution.

D'importants eorts ont été fournis récemment pour modéliser l'érosion, le trans- port et le dépôt des sédiments dans les écoulements d'eau de surface. Une grande variété de modèles ont été présentés, allant de modèles empiriques simples à des mo- dèles très sophistiqués, basés sur des principes physiques. Le but de cette thèse est de présenter un modèle qui ne retient que les principaux aspects de l'écoulement d'eau.

Il est donc physiquement réaliste, mais simple. Ce modèle utilise les équations de ux hydrodynamique pour des eaux peu profondes, tout en étant associé à aux lois du transport et de la conservation des sédiments pour permettre une modication du substrat sous-jacent.

Pour résoudre ce modèle, je présente une approche Riemann-solver free , basée sur le schéma explicite non-oscillatory central dierencing (NOC). Ce schéma a déjà été largement appliqué à des lois de conservation hyperboliques dans d'autres contextes. La version de ce schéma de prédiction-correction présenté dans cette thèse est précis au deuxième degré dans le temps et l'espace, ce qui lui permet de résoudre correctement des chocs et des discontinuités de la réponse. En suivant l'approche de la diérenciation centrale, le schéma utilise des grilles décalées pour la discrétisation.

Pour assurer la stabilité du schéma, j'ustilise un limitateur MinMod TVD (Total Variation Diminishing). La formulation Rieman-solver free du shéma permet assez de exibilité pour changer les formules utilisées pour le calcul de l'érosion, de la déposition et du transport des sédiments. Pour tester la précision et la solidité du schéma, je présente un certain nombre de cas et les résultats sont comparés avec des solutions analytiques et des expériences en laboratoire.

L'approche centrée, qui forme la base du schéma présenté dans cette thèse, de- vient très diuse lorsqu'elle est appliquée à des problèmes d'advection non linéraires où des échelles de temps très petits sont nécessaires pour maintenir la stabilité. Dans

cette thèse, je présente une correction pour réduite de telles dissipations numériques dans des problèmes de ce type. La correction est obtenue en séléctionnant les ap- proximations de diérence nale appropriées pour calculer les pentes utilisées dans la reconstruction des solution à partir des cellules moyennes. La correction proposée est appliquée au schéma Nessyahu-Tadmor, largement utilisé, pour démontrer son utilité. La stabilité du schéma corrigé est aussi discutée et la condition pour que le schéma deviennet TVD (Total Variation Diminishing) est présenté. Le schéma cor- rigé est testé pour sa capacité à résoudre eectivement des discontinuités abruptes tout en utilisant de petites échelles de temps dans un certain nombres de cas tests et les résultats sont comparés avec des solutions analytiques et des résultats publiés.

Apporter une solution aux équations pour les eaux peu profondes avec des pentes variables est un dé numérique, principalement en rapport avec deux aspects. Tout d'abord, la solution correspond souvent à un équilibre délicate entre le gradient de courant et les termes sources. Un calcul incorrect de cet équilibre peut produire des vagues articielles avec une amplitude comparable avec la solution originale. Ensuite, la frontière uctuante sec-humide est un dé pour la stabilité du schéma puisque le calcul peut mal fonctionner si la profondeur d'eau devient négative. Dans la dernière partie de la thèse, un schéma central modié anti-diusif est présenté, qui répond de manière adéquate aux deux dés décrits ci-dessus ( well-balancing / positivity preservation ). Les performances du schéma sont testées quant à l'évaluation correct des termes sources et à la solidité du système en présence d'une limite humide/sec.

Les résultats sont comparés avec des solutions analytiques et publiées.

Cette thèse démontre le succès d'une approche centrale modiée lorsqu'elle est appliquée à des problèmes reliés, qui impliquent une hydrodynamique d'eaux peu profondes et une évolution morphodynamique.

INTRODUCTION

1.1 Opening statement

Earth's landscape is a dynamic surface shaped by the interplay between tec- tonics and climate-mediated surface processes. One of the most important factors shaping Earth's morphology is the erosion and transport of sediment in streams of water owing on the surface. The sediment is sourced mainly in the upper part of a river catchment and is transported along the ow before being deposited mainly in downstream regions. Collectively, this has been termed the sediment routing system [72, 88, 341]. The erosion and transport of sediment has far reaching economic and social implications for humans, in addition to be a fundamental topic containing several poorly understood research questions. Due to the scientic and economic ramications of the sediment routing system, the study of erosion, transport and deposition of sediment continues to gain increasing importance. The last few decades have especially seen a signicant rise in the number of empirical and numerical mo- dels to better understand the sediment routing phenomena. A large number of mo- dels have been developed and applied in diverse areas (see e.g. [234, 87, 255, 327]).

The purpose of this thesis is to develop a sediment routing model, and a numerical scheme to solve it, which is detailed enough to accurately model the sediment rou- ting phenomena but is not too complicated. Finding the right balance between the amount of detail of the phenomena accommodated in the model and its simplicity is not trivial. A sediment routing model consists of many components (see e.g. [327]).

In this chapter, I will introduce various options available to model the phenomena of sediment routing. The aim is to select the sub-models for each component that will maintain the right balance between an accurate depiction of the phenomena and the complexity of the model.

1.2 Background

The second half of the 20th century saw a large eort by engineers and geolo- gists alike to better understand sediment transport. Early works focused on nding empirical relations to predict sediment uxes depending on the ow characteristics such as velocity, discharge, width of the channel, slope of bed and sediment charac- teristics (e.g., Schoklitsch (1934), Kalinske (1947), Meyer-Peter and Muller (1948), Einstein (1950), Laursen (1958), Rottner (1959) and Colby (1964)). These empirical formulas provide an estimate for the local sediment ux at any point in the channel, given known ow parameters and sediment properties.

Sedimentation over larger areas such as catchments and alluvial plains is ob- viously more complicated and estimation of local sediment uxes does not provide a complete picture of the sediment transport problem. To estimate the movement of sediment over large areas and to study its eects on landform evolution, catchment simulating models were subsequently developed (see e.g. [372, 339, 86, 234] for a re- view). Such models are used to estimate the amount and extent of the movement of sediment on spatial and temporal scales which typically extend beyond that of indi- vidual rainfall events. Catchment simulating models can be classied into three main categories [234]. The rst type are the empirical models that are based on the ana- lysis of observations mostly on a stochastic basis (e.g. USLE [353], SEDNET [261]).

These models are simple and require few data inputs. The second category are the conceptual models, which incorporate a general description of catchment processes without the ner details. These models represent the catchment area as a series of storages where the sediment routing is predicted by calculating the inow, out-

ow and storage of sediment from individual sub-catchments (e.g. [339, 350]). These two model types ( i.e. empirical and conceptual) require relatively little computing power to solve. The information about the catchment area such as precipitation, soil characteristics, land morphology and channel descriptions are given as input and the model calculates characteristics such as the sediment yield, the sediment particle size distribution, erosion and deposition of sediment at dierent positions within the landscape. Some examples of such models are HSPF [32], AGNPS [372], IHACRES-WQ [167, 169, 168, 166], IQQM [291], EMSS [337, 346], DRAINAL [22], GILBERT [80], CEASAR [86]. Although such models give a good estimate of sedi- mentation, they lack accuracy as detailed behaviour of the ow is not calculated.

The third type of models are the physically-based models, describing the ow of water and sediment. Typical equations solved in these models are the mass and mo- mentum balance equations. Empirical relations are mostly used for calculation of erosion and sediment uxes. Examples of such models are CHILD [326, 328], SIBE- RIA [349], DELIM [147], GOLEM [329], CASCADE [40], ZSCAPE [97], EROS [89], ANSWERS [21], CREAMS [188], GUEST [373], PERFECT [224], WEPP [204] and MIKE-11 [139]. In practice, a distinction between the dierent types is often ambi- guous and many of the models are hybrids of these categories. This thesis focuses entirely on physically-based models.

Although all of the above mentioned physically-based models solve conservation equations for the sediment, they do not all calculate detailed hydrodynamics. Re- cently, physically-based models with sophisticated hydrodynamics have been deve- loped to accurately capture the behaviour of the owing water (including turbulent ow in 3D). Such models are mostly used for engineering applications such as design of structures standing in owing water, designing of reservoirs and dams, prevention of silting of water canals and dams, etc. The temporal and spatial resolutions of these computational models are small and is governed by the hydrodynamics. In contrast to the landscape evolution models, these models operate with smaller tem-

o Large temporal and spatial scales o No hydrodynamics

o Empirical

o Short temporal and spatial scales o Sophisticated hydrodynamics o Physics-based

e.g. USLE [156], SEDNET [123], CHILD [145] e.g. Delft3D, FLOW-3D, REEF3D target range

Figure 1.1 Complete spectrum of the sediment routing models with simple models on the left and complicated models on the right side.

poral resolution of seconds or minutes. Accurate modelling of water ow allows these models to calculate detailed ow characteristics such as velocity eld, water depth, discharge, energy, turbulence in the ow and wave dynamics, etc. Sedimentation is then calculated, based on this more detailed, high resolution data. An example of such models is the widely used open source software package called Delft3D [96].

Delft3D is a comprehensive tool with a collection of modules for hydrodynamic ow, morphological, wave, water quality and nutrient production modelling. Other models capable of sophisticated hydrodynamics are the Flow-3D sediment scour mo- del [115], which is an extension of a comprehensive computational uid dynamics package called Flow-3D, the REEF3D model [271], which is an open-source com- putational uid dynamics program with a focus on hydraulic, coastal, oshore and environmental engineering, and the CCHE3D model [78], which is capable of simu- lating free surface turbulent ows with sediment transport, pollutant transport, and water quality analysis. Although these models are highly accurate and are capable of realistic replication of hydrodynamics and coupled sedimentation behaviour, they suer from the drawback of very high computational cost. These models are of- ten run on supercomputers and require very large resources to run simulations on

practical scales.

The motivation for this thesis is to develop a exible and computationally less expensive physically-based model which lies somewhere between these two extremi- ties of the complete spectrum of possible sediment routing models (see Figure 1.1).

On the one hand, it should not be as simplistic as the empirical and conceptual models and, on the other hand, it should not be as complicated and computatio- nally extensive as the 3D physically-based models. The ultimate aim is that such a model might be useful to address some morphologically relevant problems that are not accessible with more simplied models.

1.3 Physically-based sediment routing modelling

A physically-based model for sediment routing consists of the conservation equa- tions for water and sediment. The equations represent conservation of quantities such as mass, momentum and energy for water and/or sediment. These equations are written as a coupled system of equations since both processes are intimately connected. Hydrodynamics is heavily inuenced by the movement of sediment due to the exchange of mass and momentum between the owing stream and the sedi- ment. The erosion and deposition of sediment change the bed morphology, which change the ow characteristics. This change in ow characteristics, in t urn, changes the erosion and deposition itself forming a complicated interdependent system. The interplay between the erosion, sediment movement, deposition and the hydrodyna- mics results in the complicated bed-forms and ow patterns observed in nature. In what follows, I discuss the possible models for hydrodynamics, sediment transport and how the two models can be coupled to get a complete model.

1.3.1 Hydrodynamics

Hydrodynamics is the study of uid motion. In the context of the sediment rou- ting models, hydrodynamics is the description of the behaviour of water or water-

sediment mixture in the liquid form. The behaviour is obtained by writing conser- vation equations for mass and momentum of the water sediment mixture. The level of detail can be selected depending on the application and what is aimed to be achieved in the study. Below, I discuss some of the common hydrodynamic models characterizing dierent levels of detail for the uid behaviour.

1.3.1.1 Navier-Stokes Equations

The Navier-Stokes equations give the most detailed hydrodynamic behaviour.

These equations and their simplied versions are the most widely used system of equations in computational uid dynamics. The equations describe the general 3- dimensional behaviour for velocity, pressure, temperature and density of any moving gaseous or liquid, viscous compressible uid. They are written in the form of par- tial dierential equations. The full set of equations consists of a mass conservation equation and three equations for momentum conservation in each spatial dimension.

The equations governing the motion of an incompressible, Newtonian viscous uid in Cartesian coordinates are given by :

Mass balance equation

∂ρ

∂t +

v_x∂ρ

∂x +v_y∂ρ

∂y +v_z∂ρ

∂z

+ρ ∂vx

∂x +∂vy

∂y +∂vz

∂z

= 0,

Equation of motion

ρ ∂v_x

∂t +v_x∂v_x

∂x +v_y∂v_x

∂y +v_z∂v_x

∂z

=−∂P

∂x +µ ∂²v_x

∂x² +∂²v_x

∂y² +∂²v_x

∂z²

+ρg_x,

ρ ∂v_y

∂t +v_x∂v_y

∂x +v_y∂v_y

∂y +v_z∂v_y

∂z

=−∂P

∂y +µ ∂²v_y

∂x² + ∂²v_y

∂y² +∂²v_y

∂z²

+ρg_y,

ρ ∂v_z

∂t +v_x∂v_z

∂x +v_y∂v_z

∂y +v_z∂v_z

∂z

=−∂P

∂z +µ ∂²v_z

∂x² +∂²v_z

∂y² +∂²v_z

∂z²

+ρg_x, where ρ is the density, v_x, v_y and v_z are the velocity components in x, y and z dimensions, respectively, P is the pressure, µ is the dynamic viscosity of the uid

and g is the gravitational acceleration.

A desirable characteristic of the Navier-Stokes model is that it can capture turbu- lence in the ow which is important for highly detailed scour modelling. To capture turbulence with relatively low computational cost, turbulence models are used in conjunction with time-averaged equations such as Reynold-averaged Navier-Stokes equations. These models have less computational cost compared to the so called direct numerical simulation where a separate turbulence model is not used and the whole range of spatial and temporal scales of turbulence are resolved. Nu- merous sediment routing modeling studies have been carried out using dierent forms of the Navier stokes equations, along with dierent turbulence models (e.g., [347, 376, 47, 357, 34, 81, 171]).

As mentioned before, the solution of 3D Navier-Stokes equations is computa- tionally very expensive and analytical solutions of the equations are not possible.

Numerical solutions must be obtained by simulation codes running on supercompu- ters, which is the major drawback when investigating these equations. A common alternative is to simplify the Navier-Stokes equation, resulting in a simpler set of equations that are computationally less expensive to solve.

1.3.1.2 Shallow water equations

Most hydraulic ows in nature are shallow ows, i.e., the vertical dimension of the ow is very small compared to its horizontal dimension. Under such conditions, the depth of the ow is small as compared to the wavelength of the disturbances. Al- though, the absolute depth might not be small in some cases (e.g., in ocean tsunamis where the water depth can be many kilometers), the ow can often still be conside- red shallow ow because the wavelength of the waves can be hundreds of kilometers.

For such ows, the Navier-Stokes equations can be simplied by the assumption that the vertical velocity component is negligible compared to the horizontal components.

Mathematically, this is done by depth-integrating the Navier-Stokes equations, re-

sulting in simplied mass and momentum conservation equations called the shallow water equations. It can be shown from the momentum conservation that for shal- low ows, the pressure is hydrostatic, which means that for incompressible uids like water, pressure can be represented by the weight of the column of uid above the selected area and is proportional to the depth at that point. The shallow water equations in cartesian horizontal-vertical coordinates are written as :

∂h

∂t +∂(hv_x)

∂x + ∂(hv_y)

∂y = 0,

∂(hv_x)

∂t + ∂

∂x(hv_x²+ 1

2gh²) + ∂

∂y(hv_xv_y) = −gh(S_x+S_{f x}),

∂(hv_y)

∂t + ∂

∂x(hv_xv_y) + ∂

∂y(hv_y²+ 1

2gh²) = −gh(S_y+S_{f y}),

where h is the water depth, v_x and v_y are the velocity components in x and y dimensions respectively, g is the acceleration due to gravity, S_x and S_y are the slopes inxand ydimensions, respectively andSf x andSf y are the friction slopes in x and y dimensions respectively. Due to their realistic modeling and relatively less computational cost, the shallow water equations have been used by a large number of studies for modelling hydrodynamics in sediment routing models. Some examples are [361, 69, 91, 55, 53, 98, 26, 153, 302, 342, 58, 358, 76, 227, 63, 67, 24, 6, 117, 342, 26, 69, 292, 170, 301, 66, 65, 64, 237]

1.3.1.3 Diusive Wave approximation

The shallow water equations can themselves be simplied by neglecting some of the terms in the momentum equation. The resulting diusive wave equation is suitable for low velocity ow scenarios where the acceleration is negligible compared to basal friction. The diusive wave equation is derived by neglecting the convective and local acceleration terms in the shallow water momentum equations, assuming that the friction and the water surface slopes are equal [263]. An empirical open channel ow formula such as the Manning's or Chézy's formula is then taken and

the bottom slope is replaced by the water surface slope (see e.g. [110, 148] for the derivation). The diusion wave equation is written as :

∂H

∂t = ∂

∂x

(H−z)^α G| ^∂H_∂s |^1−γ

∂H

∂x

! + ∂

∂y

(H−z)^α G| ^∂H_∂s |^1−γ

∂H

∂y

! ,

where, H is the water level, z is the bottom elevation and s is the direction of ow. The parametersα, γ and G depends on the wetland conditions. The diusive wave model is generally used in simulating wetland ows under uniform turbulent ow conditions mainly driven by gravity. These conditions are often encountered in marshes and in overland ow in vegetated areas. The equation cannot accurately model scenarios where hydrodynamics is highly dynamic due to the lack of inertia in the equation. Some numerical studies of ow simulated with the diusive wave equation are [110, 360, 148, 377, 99, 7, 286, 240, 260, 285, 181, 131, 241].

1.3.1.4 Kinematic wave

The kinematic wave equation is a further simplication of the diusive wave equation, obtained by neglecting the pressure term in addition to the accelerative terms from the momentum equations of the shallow water equations. Thus, only friction and gravity forces are considered, which essentially balance each other. The equation is written as (see e.g. [232, 146] for derivation).

∂h

∂t +α ∂

∂x(h^m) = i−f

where h is the water depth, i is the rainfall intensity, f is the inltration rate, α and m are parameters given by open channel empirical formulas depending on the problem. For example, for fully turbulent ow, Manning's formula

α= 1 n

√

S, m = 5 3

or Chézy's

α=C√

S, m= 3 2

formula can be used, where n is Manning friction coecient, C is the Chezy co- ecient and S is the bed slope. The kinematic wave equation is fairly simple and easy to solve but the hydrodynamics is completely ignored. Special atten- tion has to be taken in using it for a particular problem (see [297] for a detailed explanation). There have been many studies to determine the suitability of the kinematic wave equation for dierent types of hydraulic and geological settings [238, 295, 150, 240, 257, 149, 296, 298, 299, 165, 263]. Some examples of studies using kinematic wave theory for overland ow are : [112, 309, 181, 131, 226, 343, 316].

1.3.2 Sediment transport modelling

Modelling sediment behaviour consists of three complicated processes : namely erosion, transport and deposition of sediment. All three may be interdependent on each other. In the last half century, the three processes have been modelled in a wide variety of ways, both individually or all combined in a single model.

Erosion is usually categorised as one of three types [234]. The rst is the general loss of soil, called overland ow erosion or sheet erosion. This type of erosion is assumed to be solely caused by the shear stress applied on the soil due to the impact of rain drops (e.g. [155]). Agricultural and catchment simulating models calculate overland erosion by empirical soil loss formulations (e.g. Universal Soil Loss Equation [353] and its variations), which are typically a function of the surface area, rainfall measurements and parameters depicting soil characteristics (see e.g. [272]). This type of erosion is usually calculated on low temporal resolutions with time step of several days. The second type of erosion is gully erosion, which happens when overland water ow concentrates in a narrow zone resulting in localised erosion. The amount of sediment eroded in gullies is often added to that caused by overland ow erosion to get a more accurate estimate of the total sediment loss. Some examples

of gully erosion formulations are [290, 354, 35].

In this thesis, I focus on in-stream erosion, which is the third type of erosion. This is the erosion that occurs on the bed and sides of a owing stream. In-stream erosion is important for physically-based models as all three types of erosion can be modelled as this type. Most of the physically-based models solve the conservation equations for uid ow to get the velocity eld and discharge for the whole domain being simulated. With this information, the local in-stream sediment erosion and sediment discharge are estimated, mostly using empirical functions. This local erosion is then added with the conservation principle to obtain the global eect of the erosion. A large amount of empirical formulations have been developed for in-stream erosion in the last half century. These can be broadly subdivided into cohesive and non- cohesive laws, since the mechanics for both are very dierent. Cohesive sediments usually have clay sized particles that have strong inter-particle forces due to ionic charges. Erosion of cohesive sediment is usually calculated with power law formula depending on the velocity or discharge of the ow. A comprehensive review of the erosion modelling and empirical relations available for cohesive sediment erosion can be found in [369]. Non-cohesive sediment can be modelled by physically-based formulas but the process is very complicated and these formulas often fail to produce accurate results. All of the relations used to calculate non-cohesive erosion are also partially or fully empirical.

An important concept regarding non-cohesive erosion is the so-called incipient motion. Particle motion is induced when either the lift force acting on it due to drag is more than it's weight or the drag force is more than the friction force of the bottom. Particles can also roll on the bed due to dierences in the moments produced by the forces acting on their surface. Calculating the forces on particles is very computationally extensive and also very dicult as the particles have dierent shapes and sizes. As a result, empirical formulae are normally used to calculate the incipient motion, based on the criteria such as the shear stress (e.g. Shields diagram

[288, 206, 348]) and ow velocity (e.g. [116, 145, 333, 368]). One therefore commonly speaks of critical values of shear stress and ow velocity below which erosion does not occur.

Deposition of sediment happens when lift forces on particles applied by the ow become less than gravitational forces. This can happen for example due to decreases in the ow velocity or due to aggregation of the particles (in the case of cohesive sediment [123]). Deposition is also usually treated by empirical relations for the physically-based models. These relations are usually functions of sediment concen- tration in the sediment-water mixture and the fall velocity of the sediment particles (see e.g. [160, 63, 254]). The fall velocity or settling velocity is the constant velocity with which a sediment particle falls in still water when the weight of the particle becomes equal to or more than the drag force in an upward direction. Although, the fall velocity can be exactly calculated for spherical particles, it is usually ap- proximated with empirical relations since particles are usually not of symmetrical shape. A large number of empirical relations estimating fall velocity can be found in literature (see [369, 123]).

Apart from modelling erosion and deposition separately, a popular approach is to model all three processes of erosion, transport and deposition combined in the form of the sediment discharge generated by the owing stream. The erosion and deposition is determined implicitly by the conservation principle i.e., if more sediment is locally going out of a region than the amount of sediment coming in, erosion will occur and the bed height will decrease. On the other hand, if more sediment is coming in than going out, deposition will occur. The sediment discharge is calculated from empirical relations depending on a wide variety of characteristics such as water discharge, ow velocity, energy of the stream (water surface slope), shear stress, stream power, sediment grain diameter, sediment concentration, fall velocity and density of the sediment, etc. Some commonly used empirical relations to calculate sediment discharge are :

Grass equation [133] :

q_s=Au|u|^m−1,

where q_s is the sediment discharge, u is the ow velocity, 0 ≤ A ≤ 1 and 1 ≤ m ≤ 4 are constants, usually obtained from experimental data, depending on the characteristics of the sediment. The Grass formulation is the simplest of the sediment discharge formulations and does not involve any critical shear stress, which means that the sediment starts moving as soon as the water-sediment mixture on top starts moving.

Meyer-Peter & Müller's equation[235] : q_s

p(G−1)gd³_i =sgn(u)8(τ∗−τ∗c)^3/2,

where G is the relative density of the sediment ρ_s/ρ_w, d is the sediment grain dia- meter, τ∗ is the non-dimensional shear stress given by _(G−1)dR^gη2u21/3

h

(see [100]) andτ∗c

is the critical shear stress often taken as 0.047. The relation presented here is a simplied version of the original relation (see [268]).

Van Rijn's equation[331, 276, 332] : qb

p(G−1)gd³_i = 0.005 C_D^1.7

d h

0.2

τ_∗^0.5(τ_∗^0.5−τ_∗c^0.5)^2.4,

whereC_D is the dragging coecient (see [100]).

Examples of other popular formulas for sediment discharge are the Stream power formula [17, 3, 107], Kalinske's Bedload Formula [179], Rottner's bedload formula [282], Einstein's Bedload Formula [105] and Laursen's Bed-Material Load Formula [207].

1.3.2.1 Conservation laws for sediment transport

Modelling sediment transport in physically-based models is done by writing conservation laws for the sediment. Equations for the conservation of mass and mo-

mentum are usually written while the equation for energy balance is ignored. Many models also ignore the momentum of sediment because most natural streams contain very little momentum in the sediment compared to that associated with the water.

Notable exceptions are dam breaks, storms and tsunamis. A single momentum equa- tion conservation is written for both the sediment and uid. Most physically-based models with detailed hydrodynamics use one of two types of conservation laws for the sediment :

1.3.2.1.1 Exner's equation : Exner's equation formulates the sediment trans- port with a continuity equation for the bed material that is transported in the form of sediment uxes calculated by empirical relations (outlined above). The equation is given by :

∂z

∂t + 1 1−φ

∂q_sx

∂x + ∂q_sy

∂y

= 0,

whereq_sx and q_sy are the sediment uxes in x andy directions respectively,φ is the porosity and z is the bed elevation. The uxes are calculated on the basis of ow characteristics such as the velocity, discharge, energy and slope etc., which determine the sediment transport capacity of the stream. The possible modes of transport for sediment in a owing stream are creeping (rolling on the bed), saltation (frequent bouncing on the bed) and suspension. The ux formulations used in the Exner's equation does not explicitly specify the transport mode. Rather a single ux value, based on the equilibrium capacity of the stream is calculated, which represents all three possible forms of transport. The bed topography is calculated based on the divergence of the uxes, i.e. if more sediment is entering locally as compared to leaving, the bed height will increase and vice versa. As discussed in the previous section, there are a large number of empirical formulas available for the calculation of sediment ux. Exner's equation has been widely used in the literature (see e.g.

[160, 53, 227, 98, 26, 55, 342, 302]).

1.3.2.1.2 Entrainment-deposition ux formulation : In the entrainment- deposition ux formulation, rather than calculating the sediment ux (as with Ex- ner's equation), one computes local entrainment and deposition uxes on the bed of the owing water-sediment mixture. In this case, two continuity equations are written. One for the suspended sediment, which is represented by the sediment concentration in the water-sediment mixture, while the second equation is written for the bed height, which is used to update the bed topography. The bed topogra- phy evolves depending on which of the two uxes is higher. If the entrainment ux is more than the depositional ux, the bed height increases and vice versa. The equations are written as :

∂(hc)

∂t +∂(hcu_x)

∂x +∂(hcu_y)

∂y =E−D,

∂z

∂t = D−E 1−φ ,

where c is the volumetric sediment concentration, E and D are the erosional and depositional uxes respectively. This type of formulation advects the sediment in suspension with the velocity of the ow. Although the equations are written assu- ming that all of the sediment is transported in suspension, which seems to be un- realistic, in practice the model behaves dierently. If the erosion and deposition ux formulae are properly calibrated, the eroded sediment is very often deposited very close to the location from where it was eroded, producing the same behavior as the creeping of the sediment (bed-load transport). Some examples of the models using entrainment-deposition uxes are [63, 160, 67, 24, 69, 292, 66, 65, 299, 156, 67, 160]

1.4 The mathematical model

The dierent hydrodynamics and sediment transport models discussed in the previous sections can be coupled to get a complete sediment routing model. Among

the available choices for hydrodynamic models, I have selected the shallow water equations due to their computational simplicity while being able to capture realistic physics. The Navier-stokes equations, although modelling more detailed hydrodyna- mics, are not selected due to their excessive computational cost. The diusive wave model, on the other hand, is not selected due to the oversimplication, especially because of the lack of momentum conservation, which is very important for the modeling of dynamic bed-forms commonly found in nature.

For the sediment transport model, this study has used both the suspended (entrainment-deposition) and bed load (Exner) transport models since both forms of transport have dierent mechanics and are modeled dierently. Accommodating both forms of transport is necessary as the amount of sediment being transported in either form varies greatly. Typically, the ratio between bedload and suspended load is close to 10%, but it can vary depending on ow conditions [283, 259]. Ignoring one form would therefore reduce the ability of the model to depict realistic sediment transport behaviour.

Below, I derive the mass conservation equations for the water-sediment mixture owing on top of a mobile bed and the mass conservation equations for the sediment forming the bed (similar equations have been derived in [62]). For the purpose of clarity, I derive the system in one dimension only. A two dimensional version can be derived in a straight forward manner following the same reasoning. I begin with the basic conservation law for the water-sediment mixture :

∂hρ

∂t + ∂

∂x(huρ) =S, (1.1)

whereh is the water depth,ρ is the density of the water sediment mixture given by ρ=ρ_w(1−c) +ρ_sc=ρ_w+ (ρ_s−ρ_w)c,cis the volumetric sediment concentration,ρ_s and ρ_w are the densities of sediment and water respectively, uis the velocity and S is the source strength function. For a water-sediment mixture, the source strength

function is quantied by the net volumetric ux from the bed to the mixture :

S = ρ₀F (1−φ),

whereρ₀ =ρ_wφ+ρ_s(1−φ) = ρ_s−(ρ_s−ρ_w)φ is the density of the bed material, F is the sediment ux from the bed to the water-sediment mixture and φ is the bed porosity. Similarly, the continuity equation for the sediment in the water-sediment mixture can be written as :

∂

∂t(hcρ_s) + ∂

∂x(hcρ_su) =F ρ_s.

Assuming the sediment and water density is constant and quantifying the net volu- metric ux from bed to the water-sediment mixture as the dierence between the local erosion and deposition(E−D), the previous equation is rewritten as :

∂hc

∂t + ∂

∂x(hcu) = E−D, (1.2)

The continuity equation for the bed material is written as :

∂z

∂t + 1 1−φ

∂

∂x(q_s) =−(E−D)

1−φ , (1.3)

whereq_s is the bedload sediment ux. Expanding Eq. 1.1 gives :

∂

∂th(ρ_w + (ρ_s−ρ_w)c) + ∂

∂xhu(ρ_w + (ρ_s−ρ_w)c) = ρ₀(E−D) 1−φ

ρ_w ∂h

∂t + ∂

∂x(hu)

+ (ρ_s−ρ_w) ∂hc

∂t + ∂

∂x(huc)

= ρ0(E−D) 1−φ Substituting Eq. 1.2 gives :

ρ_w ∂h

∂t + ∂

∂x(hu)

+ (ρ_s−ρ_w)(E−D) = (ρ_wφ+ρ_s(1−φ))(E−D) 1−φ

ρ_w ∂h

∂t + ∂

∂x(hu)

− ρw

1−φ(E−D) = 0 Finally, substituting Eq. 1.3 gives :

∂h

∂t + ∂

∂x(hu) = −∂z

∂t − 1 1−φ

∂q_s

∂x.

The bedload sediment ux divergence is often ignored from the continuity equation of the water-sediment mixture( see e.g. [361, 292]). Momentum conservation for the water-sediment mixture is derived similarly [66]. The full system of equations in two dimensions is given by :

∂h

∂t + ∂(hu)

∂x + ∂(hv)

∂y =−∂z

∂t, (1.4)

∂(hu)

∂t + ∂

∂x

hu²+ 1 2gh²

+ ∂

∂y(huv) =B_x, (1.5)

∂(hv)

∂t + ∂

∂x(huv) + ∂

∂y

hv²+ 1 2gh²

=B_y, (1.6)

∂(hc)

∂t + ∂(hcu)

∂x +∂(hcv)

∂y =E−D, (1.7)

∂z

∂t + 1 1−φ

∂qx

∂x + 1 1−φ

∂qy

∂y = D−E

1−φ , (1.8)

whereBx and By are source/sink terms dened as

B_x =−gh∂z

∂x −ghS_{f x}− (ρ_s−ρ_w)gh² 2ρ

∂c

∂x + (ρ₀−ρ)u ρ

∂z

∂t, (1.9)

B_y =−gh∂z

∂y −ghS_{f y}− (ρ_s−ρ_w)gh² 2ρ

∂c

∂y +(ρ₀−ρ)v ρ

∂z

∂t. (1.10) In these equations,tis the time, xandyare horizontal coordinates,h is the ow

depth,uand v are depth-averaged velocities in thexandydirections respectively, z is the bed elevation,cis the ux-averaged volumetric sediment concentration,gis the gravitational acceleration, S_{f x} and S_{f y} are the friction slopes inx and y directions respectively, φ is the bed sediment porosity, E and D are substrate entrainment and deposition uxes across the bottom boundary of ow (representing sediment exchange between the water column and bed),ρ =ρ_w(1−c) +ρ_scis the density of the water-sediment mixture, ρ₀ = ρ_wφ+ρ_s(1−φ) is the density of the saturated bed, ρ_s, ρ_w are the densities of water and sediment respectively and q_x, q_y are bed load uxes in the x and y directions respectively. To close the system of equations, formulae have to be introduced to calculate the friction slopes, erosion, deposition and bedload sediment uxes. See Section 3.2 for the formulae used in this thesis.

1.5 Challenges

Many natural phenomena are modelled by, and extensive research has been car- ried out on hyperbolic conservation laws (see e.g. [211, 127, 93, 190]). The shallow water equations are one set of hyperbolic conservation laws for which a large number of numerical techniques have been developed (See e.g. [319, 266] for a comprehen- sive review). The coupled shallow water-morphodynamic evolution equations model poses a number of specic challenges for numerical schemes. Among the challenges are :

• The system of equations allows solutions with propagating shocks, which can produce oscillations if solved with naive schemes. Any scheme solving the shallow water equations has to robustly capture shocks, while avoiding spu- rious oscillations in their vicinity. Numerical dissipation is another drawback a scheme can suer, which would result in excessive smearing of the solution.

•Calculation of steady state ows requires a delicate balance between the ux gradient and bed slope terms in the momentum equations. Accurate calculation of these terms demands special attention in the design of a successful numerical

scheme.

•A successful scheme has to robustly handle wet-dry fronts. Many natural ow scenarios involve drying and wetting of the bed, which may cause singularities in the solution at the boundaries as the water depth diminishes. A successful scheme should avoid negative water depths that may arise due to the numerical oscillations as the water depth approaches zero.

•A good numerical scheme has to account for dierences in the characteristic time scales between hydrodynamics and changes in the bottom topography.

Hydrodynamics is a highly dynamic and rapid process compared to evolution of the bed. Calculating a solution for the two processes with the same scheme may cause stability problems.

•Models involving coupling between the shallow water equations and Exner's equation may suer from exibility. Some schemes require detailed knowledge about the wave-structure of the problem (e.g. the upwind schemes described later). Such schemes can be used only in those cases where the wave (eigenva- lue) structure is available. Due to the use of empirical formulae in the shallow water-Exner's equation model, a exible scheme, which allows easy switch bet- ween dierent formulations, is desirable so that a proper sediment ux function can be selected on a case by case basis.

1.6 Literature review of numerical schemes

A large number of numerical schemes for the shallow water equations with source terms have been published in literature. Godunov type nite volume me- thods [128], also called upwind nite volume methods, are the most widely used numerical schemes for this purpose. These schemes utilize Riemann-solvers to ob- tain an accurate estimate for inter-cell uxes before updating the solution in an explicit fashion. Godunov schemes are able to successfully resolve shocks in the solution but may have problems accurately balancing uxes and source terms in

some situations. The original Godunov scheme is not capable of maintaining the ba- lance for steady state conditions [136]. Research by Bermudez and Vazquez [28], and Greenberg and LeRoux [136] were among the earliest studies that presen- ted Godunov type schemes that were able to successfully balance ux and source terms. These early studies labelled those schemes "well-balanced schemes or "C- property" maintaining schemes. The schemes of Bermudez[28] and LeRoux[136] were based on Roe's approximation of the Riemann-solver. There are many other stu- dies that have used the Roe's approximation to solve the shallow water equations (see [209, 334, 124, 152, 176, 278, 325, 253, 191, 227, 247, 95]). Zhou et al. in [380, 379] presented the surface gradient method in which the water surface level was reconstructed instead of the conserved variables. These methods utilize HLL or HLLC approximate Riemann-solvers for the solution of Riemann-problems (see also [185, 12, 262]). Other Godunov type methods utilizing the HLL approximation include [159, 119, 371, 137, 151]. Rogers et al. [277, 278] presented a Roe's approxi- mation based Godunov's method that is based on the hierarchical quadtree (Q-tree) grids, adapting to inherent ow parameters such as the magnitude of the free surface gradient and depth-averaged vorticity. Other studies with Quadtree schemes include [221, 208]. Gallouët et. al in [122] presented the VFRoe approximate Riemann-solver, which also has been adapted by some other studies [233, 30]. George, in [125] presen- ted a Godunov scheme with adaptive mesh renement for complicated geometries.

ADER schemes introduced by Toro and Millington [317] solve higher-order Riemann- problems (also called the generalized Riemann-problems) to solve the advection problems with arbitrary accuracy. ADER schemes for the shallow water equations with geometrical source terms are presented in [338, 101, 73]. Studies published in [251, 359, 29] provide exact Riemann-solvers for the shallow water equations. Some other Godunov type nite volume schemes to solve shallow water equations with variable bed topography are given in [381, 126, 242, 300, 144, 4, 205, 378, 15].

Another widely used class of nite volume methods for the solution of shallow

water equations are the ENO (Essentially Non-Oscillatory) and WENO (Weigh- ted Essentially Non-Oscillatory) schemes [289]. These schemes were introduced by Harten, Engquist, Osher and Chakravarthy in [142]. ENO/WENO schemes provide an approach to reduce the spurious oscillations in the vicinity of discontinuities that is dierent from the other widely used approaches. Publications by Nuji¢ [250]

and later Vukovic et. al [340] were among the earliest studies using ENO/WENO reconstructions for the shallow water equations with the source terms. Higher or- der versions were later presented by [90, 363, 50]. There are a large number of other studies using ENO/WENO schemes for the shallow water equations (see e.g.

[91, 74, 53, 248, 48, 366, 216, 230, 215]).

More recently, discontinuous Galerkin methods, which are often described as a hybrid of nite volume and nite element methods have become quite popular for the solution of shallow water equations. The technique was introduced by Reed and Hill [270] for the neutron transport equation. Xing and Shu in [364, 365] presented the rst discontinuous Galerkin scheme to solve the shallow water equations with source terms. A large number of studies solving shallow water equation with discontinuous Galerkin methods have been recently presented (see e.g. [274, 273, 45, 183, 49, 102, 362, 313, 352, 38, 351]). Other notable schemes for the solution of shallow water equations are presented in [130, 256, 324, 287, 325, 330, 315, 231, 16, 223, 25, 275, 225, 114, 104, 370].

The Riemann-solver based methods mentioned above require the so called wave- structure or eigenstructure of the underlying problem. This wave-structure, although available for the clear water shallow water equations, is dicult to obtain for the coupled shallow water-morphodynamic evolution equations model due to the widely varying bed load ux and erosion-deposition empirical functions. Central schemes are a sub-class of nite volume schemes that do not require the eigenstructure, hence, they do not suer from this limitation. The scheme of Lax and Friedrich presented in [120] was the rst central scheme, which was later on developed further by Nessyahy

and Tadmor [244]. Among these schemes, the so called central-upwind schemes have been quite popular for the shallow water equations. These schemes only require the largest eigenvalue and are Riemann-solver free. The central-upwind schemes were rst introduced by Kurganov and Levy [194] for the shallow water equations and a second-order version of the scheme was subsequently presented in [200]. Some other studies presenting central-upwind schemes for the shallow water equations with geometric source terms include [43, 18, 36, 42, 294, 228, 83]. The PRICE central scheme, also following the centred approach, was introduce in [318]. Higher- order schemes with exible-grids for the shallow water equations are presented in [58, 56]. FORCE scheme is another related central scheme introduced in [321, 103], with application to the shallow water equations in [304, 60, 57, 267].

The original Nessyahu-Tadmor scheme, commonly referred to as the Non-oscillatory central dierencing NOC scheme (see e.g. in [266, 264, 265]), is of interest in this thesis as it only requires the largest eigenvalue to calculate the maximum stable time step and no knowledge of eigenstructure is required for the calculation of uxes. Since the time step in the coupled hydrodynamic-morphodynamic model presented in this thesis is controlled only by the hydrodynamics, the morphodynamic evolution model can be solved without any knowledge of the eigenstructure. The only drawback of the scheme is that it becomes highly diusive due to numerical dissipation if a small time step is used. This drawback becomes very severe for the morphodynamic evo- lution model as it is solved with a small time step calculated for the hydrodynamic model. This might be the reason that this scheme (to the best of our knowledge) has not been used to solve the coupled hydrodynamic-morphodynamic models, such as the one presented here. An approach which can mitigate this drawback of the NOC scheme is developed in this thesis.

1.7 Thesis structure

The rest of the thesis comprises four chapters, three of which have been submit- ted to international journals for publication. Chapter 2 introduces an anti-diusive non-oscillatory central dierencing (adNOC ) scheme, which is suitable to solve the coupled shallow water-morphodynamic evolution model. A brief description of the original Nessyahu-Tadmor central scheme, which forms the basis of the new adNOC scheme is presented and a description of the approach to reduce numerical dissipa- tion by an anti-diusion correction is then discussed. The stability of the corrected scheme is also discussed and nally, test cases and results are presented.

Chapter 3 presents a two dimensional version of the adNOC scheme that is im- plemented to solve the coupled shallow water-morphodynamic evolution equations.

The chapter provides an account of the governing equations and the formulae used for erosion, deposition and the sediment uxes to close it. A detailed derivation of the two-dimensional NOC scheme and the anti-diusion correction to tackle nume- rical dissipation is then presented. Five numerical test cases and their comparison with the analytical and experimental results are nally presented.

Chapter 4 deals with the challenge of well-balancing and positivity preservation, faced by numerical schemes solving the shallow water equations with source terms.

This chapter describes how the ux gradient and the source terms can be balanced and presents a well-balanced adNOC scheme. The issue of positivity preservation by the scheme is also discussed and a condition that is necessary for the scheme to be positivity preserving is presented. Numerical results and their comparison with benchmarks and analytical solutions are also presented for the well-balanced, positivity preserving scheme.

Chapter 5 provides concluding remarks and some comments regarding future prospects.

ANTI-DIFFUSIVE,

NON-OSCILLATORY CENTRAL DIFFERENCE (adNOC) SCHEME

Submitted : Journal of Computational Physics

2.1 Introduction

Advection dominated problems described by hyperbolic conservation laws are of major importance in many areas of science and engineering. Because these problems are often highly nonlinear, for example due to the presence of propagating shock waves, analytical solutions are often complicated or not available. Therefore, the development of accurate and robust numerical methods to solve hyperbolic conser- vation laws is a domain of intense ongoing research.

Many of the schemes developed to solve hyperbolic conservation laws have their origin in Godunov's method [128]. These schemes are essentially upwind conserva- tive nite volume methods where numerical uxes are computed at cell interfaces based on local Riemann problems (e.g., see [320, 211] for more details). While these methods can be very accurate, they may suer from several potential drawbacks.

First, for some very nonlinear problems, the wave structure of the governing equa-

tions may not be known, which makes it dicult to accurately compute interface uxes (e.g. see [53, 20]). Second, conservation laws with source terms are often trea- ted with operator splitting, which can in some cases lead to signicant numerical errors (e.g. see [108, 174, 20]).

An alternative to the Godunov approach are central dierencing nite volume schemes, due largely to the work of Nessyahu and Tadmor [244]. Central dierencing schemes are based on the Lax-Friedrichs (LxF) scheme that is normally modied to include higher order accuracy [229, 157, 31] and several dimensions [10, 174, 180, 19].

These schemes have become known as non-oscillatory central dierencing (NOC) methods. Central schemes involve no Riemann problems (and therefore require no knowledge of the eigenstructure of the governing equations) and necessitate no opera- tor splitting. Thus, they are relatively simple and especially suitable for the solution of highly nonlinear hyperbolic conservation laws involving sti source terms (e.g., see application of central schemes in [312, 44, 9, 201, 85, 264, 265, 267]). A major factor limiting the utility of the central dierencing schemes has been that they introduce excessive numerical diusion [202, 158, 2, 195, 302, 59]. In the original central dierencing schemes, this diusivity was shown to be of orderO((∆x)^2r/∆t) [202], where r is the order of the scheme, showing that the numerical diusion be- comes more important as the time step is reduced. Thus, for very nonlinear systems where small time steps are of paramount importance to ensure stability, the solu- tion may be partially or completely destroyed by articial diusion [202, 195, 305].

Less diusive modied schemes utilizing partial knowledge of eigenstructure have been proposed [202, 194], but excessive diusivity remains a limitation to central dierencing schemes when small time steps are necessary.

In this article we present a simple anti-diusion correction to the classic NOC scheme [244] in an eort to reduce numerical dissipation when small time steps must be used. As the original scheme, our method utilizes a staggered grid where the solu- tion is approximated by reconstructing piece-wise polynomials within the cells from

the evolving cell averages. The staggered approach enables the central scheme to have smooth cell interfaces, which makes evaluation of numerical uxes particularly straight forward. Unlike previously proposed modications [202, 194] which require partial knowledge of the eigenstructure, this scheme does not involve the solution of Riemann problems and does not require any knowledge of the eigenstructure for the calculation of uxes. Here we demonstrate the ability of the anti-diusive, non- oscillatory central dierence scheme (adNOC) to solve the shallow water equations coupled to substrate erosion and sedimentation.

The article is structured as follows. In section 2.2, a brief description of Nessyahu- Tadmor central scheme is presented. Section 2.3 describes how the diusion can be eliminated by using appropriate nite dierence approximations. Section 2.4 dis- cusses the stability of the corrected scheme and nally, test cases and results are presented in section 2.5.

2.2 Central schemes

Consider the following scalar hyperbolic conservation law

∂u

∂t +∂f(u)

∂x =s(u), (2.1)

where u is the conserved quantity, f is the ux and s is the source term, both functions of u. To explain the centred approach, we will use the Nessyahu-Tadmor scheme [244], the most widely used second order method as the standard central scheme. The development of this section follows closely that presented by [266].

The method is a high-order extension of the Lax-Friedrichs solver which operates in predictor corrector fashion. The predictor step involves evaluation of rst order approximations at half time steps. The second order solution is then realized in the corrector step which utilizes the calculations from the predictor step to evaluate the solution on the staggered cells. Below, we present a description of the procedure