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Continuous and discontinuous Galerkin

finite element methods of variationalBoussinesq water-wave models

M. Sc. thesis.

O.R. Koop

University of TwenteDepartment of Applied Mathematics

Group of Numerical Analysis and Computational Mechanics

October 4, 2006


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Supervisors:Dr. Ir. O. Bokhove (University of Twente)Ir. G. Klopman (University of Twente)

Committee:Dr. Ir. O. BokhoveIr. G. KlopmanDr. R.M.J. van DammeIr. S. RhebergenM.Sc. D. Hurdle

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Abstract

There are many reasons to investigate the propagation of surface water waves in the sea. Inthis report Boussinesq models are studied, that can be applied for modeling free-surface wavespropagating from the deep sea up into the surf zone. These models can become fairly complex

and improving linear frequency dispersion properties of these Boussinesq models often invokedifficult higher-order terms in the resulting partial differential equations [18].Among others, Broer [7, 8], Broer et al.[9]and Klopman et al.[19] propose, based on the vari-ational principle for potential flows proposed by Luke [27], variational Boussinesq modelsto describe potential flows with dispersion. The advantage of these models is that energyis conserved and guaranteed to be positive, while mixed higher order spatial and temporalderivatives are avoided. The cost of this is that we additional unknown quantities, with as-sociated elliptic partial differential equations, that are relatively simple to solve numerically[20].In this report three variational models are considered, namely those of Luke [27], Klopman etal. [19] and Whitham [37], and they are extended including surface tension effects. Addition-

ally, we integrated the motion of the fluid domain boundaries with our variational principlesdescribing the fluid motion. We compare the linear dispersion relations of the three modelsand draw the conclusions that the Klopman variational Boussinesq model approximates theexact linear dispersion obtained from Luke variational principle for potential flow, better thanthe Whitham model.We numerically solve the Whitham and Klopman variational Boussinesq models in time usinga continuous Galerkin finite element method. To simulate the propagtion of the discontinu-ous jump that occurs when waves are breaking, we propose a discontinuous Galerkin finiteelement method. In this progress of this report, a discontinuous Galerkin approach was suc-cesfully applied to the linear Klopman variational Boussinesq model. This approach can befollowed to formulate a discontinuous Galerkin finite element discretization to solve the fully

nonlinear model.In this report it is shown that breaking waves do not occur in Whithams variational Boussi-nesq model for waves propagating over a horizontal seabed. Further, we consider the weaklynonlinear Klopman VBM, i.e. neglecting higher-order waves-slope effects, and it is shown thatwaves described by this model can not break as well. Further research on the fully nonlinearKlopman variational Boussinesq model is required. In this report we draw some conclusionsand state recommendations, particularly on how to construct a discontinuous Galerkin FEMto solve the fully nonlinear Klopman variational Boussinesq model numerically.

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Preface

The research in this M.Sc. thesis has been conducted at the University of Twente. I did myresearch within the department of Applied Mathematics in the group Numerical Analysis andComputational Mechanics. I chose the Boussinesq water-wave models as the subject of my

thesis, because since I was a small child I have been interested in water motion. Waterfalls,breaking waves and waves moving up and down the beaches have always intrigued me. I enjoybeing in the water as my main hobbies are waterpolo and swimming. Moreover, I find thebeauty of solving practical problems with such an abstract tool as mathematics, combinedwith the power of numerical methods to solve practical problems, very appealing.First of all I would like to thank Onno Bokhove, for his everlasting enthusiasm, his straight-forward manner, his criticism and teaching me to become a better research worker. I wouldlike to thank Gert Klopman, for the fruitful discussions we had and his help investigatingthe model that he et al.. proposed. I would like to thank both supervisors for the pleasentco-operation and for their efforts correcting and looking over my thesis.Furthermore I would like to thank Chris Klaij, Vyjaja Ambati and Sander Rhebergen for

their helpfull suggestions on the lifting operators and getting acquinted with LATEX2e andC++. Moreover, I would like to thank Sander for overlooking the chapter on discontinuousGalerkin and being a member of the comittee. Furthermore, I would like to thank Ruud vanDamme and David Hurdle for their willingness to serve on the committee.Marieke and Renske, I really had a pleasant time with you in the afstudeerkamer, thank you.Bernhard, Chris, Diana, Fedderik, Marielle, Sander and all others with whom I spent mydaily lunch, thank you for the pleasent time.A special word of thanks to my parents, Jeanie and Henk, for their love and care, for theireverlasting support and understanding: Thank you. Arjen and Emily, I am really happy hav-ing you as brother and schone zus and I have enjoyed the time we have spent in Enschede.Which city will follow hereafter?! Shelly, thank you for everything!

Olger Koop Enschede, Octobre 2006

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Contents

Abstract iii

Preface v

List of Figures ix

Nomenclature xi

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Numerical computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Variational principles for water waves 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Lukes variational principle for free-surface potential flow . . . . . . . . . . . 3

2.3 Klopmans variational Boussinesq model (KVBM) . . . . . . . . . . . . . . . 7

2.4 Whithams variational Boussinesq model (WVBM) . . . . . . . . . . . . . . . 10

2.5 Comparison of dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Moving boundaries in the KVBM . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Numerical modeling: continuous Galerkin 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Whithams Boussinesq variational principle . . . . . . . . . . . . . . . . . . . 19

3.3 Assembly of global nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Klopmans variational Boussinesq model . . . . . . . . . . . . . . . . . . . . . 243.5 Assembly of global nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Numerical Modeling: Discontinuous Galerkin 314.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Linear shallow water system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Numerical Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.6 Linearized Klopman model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Verification 435.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Shallow water equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.3 Whithams variational Boussinesq model . . . . . . . . . . . . . . . . . . . . . 47

5.4 Klopmans variational Boussinesq model . . . . . . . . . . . . . . . . . . . . . 49

5.5 Small amplitude waves in the KVBM . . . . . . . . . . . . . . . . . . . . . . . 505.6 Large amplitude waves in the KVBM . . . . . . . . . . . . . . . . . . . . . . . 52

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Investigation into wave breaking and linear dispersion 556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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6.2 Wave breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.3 Linear dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Conclusions 61

8 Recommendations 63

References 65

A Nonlinear shallow water equations with discontinuous Galerkin 68A.1 Non-linear shallow water equations . . . . . . . . . . . . . . . . . . . . . . . . 68A.2 DG discretization with polynomials of degree 1 . . . . . . . . . . . . . . . . . 68A.3 Algebraic system of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B Klopmans variational Boussinesq discontinuous Galerkin discretization 71B.1 DG discretization with polynomials of degree 1 . . . . . . . . . . . . . . . . . 71B.2 Algebraic system of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

C Brezzis approach 73C.1 Fixed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C.2 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

D Periodic wave solutions for WVBM model 75D.1 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75D.2 Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75D.3 Integrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75D.4 Model closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

D.5 Further investigation into the periodic wave solutions . . . . . . . . . . . . . . 77D.6 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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List of Figures

2.2.1 Parameters describing potential flow. . . . . . . . . . . . . . . . . . . . . . . . 32.3.1 Parameters describing the fluid (KVBM/WVBM). . . . . . . . . . . . . . . . 8

2.5.1 Dispersion relations for two types of Boussinesq-models . . . . . . . . . . . . 122.5.2 Dispersion relations with surface tension for various (small) water depths . . 143.2.1 Subdivision into N elements Kj of the horizontal domain Dh. . . . . . . . . . 193.2.2 Linear basis functions for continuous Galerkin finite elements . . . . . . . . . 204.2.1 Discretization for the DG FEM . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.1 Initial sinusoidal wave profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2.2 Three time integration methods for linear SWE. . . . . . . . . . . . . . . . . 455.2.3 Nonlinear SWE (deep water, CG FEM). . . . . . . . . . . . . . . . . . . . . . 465.2.4 Nonlinear SWE (deep water, DG FEMs). . . . . . . . . . . . . . . . . . . . . 465.3.1 Nonlinear WVBM, small amplitude (CG FEM). . . . . . . . . . . . . . . . . . 485.3.2 Nonlinear WVBM, cnoidal wave . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3.3 Nonlinear WVBM, matlab BVP . . . . . . . . . . . . . . . . . . . . . . . . . . 495.5.1 Small amplitude waves KVBM; Surface height and energy . . . . . . . . . . . 515.6.1 Large amplitude waves in linear KVBM; Surface height and velocity potential 536.2.1 Nonlinear WVBM; sinusoidal wave . . . . . . . . . . . . . . . . . . . . . . . . 566.2.2 Weakly nonlinear KVBM; sinusoidal wave . . . . . . . . . . . . . . . . . . . . 576.3.1 Linear KVBM and linear SW; (DG) . . . . . . . . . . . . . . . . . . . . . . . 59D.6.1Numerical solution of the WVBM . . . . . . . . . . . . . . . . . . . . . . . . . 80

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Nomenclature

(x,y ,z ,t) The velocity potential of a particle located at (x,y,z) at time t. Wavelengthc Wave speed

T Time period Wave number Wave elevationd and h0 Waterdepthhb(x) Vertical p osition of the seabedh(x) Height of the water with respect to the seabedEk Kinetic energyEp Potential energy Wave frequency

C(0)e , C

()e Phase speed without or with surface tension effects

LLagrangian functional

C0 () Set of testfunctions on with compact support in u Horizontal flow velocity

nx =n

xn n th order partial derivative (same for y and zdirection).ddt [f(x(t), t)] Total derivative off with respect to t.t[f(x, t)] Partial derivative off with respect to its second argument t. = (x, y, z) in the 3D case or (x, z) in the 2D case = (x, y) First order horizontal derivatives (in the 3D case).S Surrounding surface of a volumeSf Free surfaceDh Horizontal domainDh Boundary of the domain Dh

Lp() Space of Lebesque integrable functions; {v |v|p d1/p < }H1g () Hilbert space {v L2()

xv L2(), v|Dh = g}N Number of elements (= number of nodes 1) Global basisfunction Trial function for h Trial function for hK Reference element; K = { (1, 1(} Local basis functionM1 Invese of matrix MIN Unity matrix of size [N N], i, b Set of all, internal and boundary faces respectively[[.]] The jump at a face{{.}} The average at a faceFj() Map from the reference element to the element-interval Kjvx =

dvdx First order spatial derivative with respect to its only argument

R() Global lifting operatorRs Local lifting operatore Local lifting operator weight (dependent on the element size)Ns Number of faces adjacent to an elementS, SL, SR Face, Face left or right of an element

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1 Introduction 1

1 Introduction

1.1 Motivation

There are many reasons to investigate free-surface water waves propagation in the sea, suchas the necessity to predict the motion of the water-surface at a certain place and time andto deduce the effects when waves reach shores which is of much interest for coastal defenses.Waves reaching the entrance of a harbor are of much interest for both harbor designers andvessel constructors. Propagating waves may threaten offshore oil rigs.The recent devastation of the shores and cities of South-Asia and East-Africa by a tsunami(December 2004) is a striking example where the origination and propagation of waves isof particular interest. The tsunami was excited by a fault movement1 in the Indian Oceannear the Indonesian island of Sumatra. Reconstruction of the seabed movement at the timeof fault motion is crucial for scientific understanding of tsunami excitation and developingcoastal warning systems. Furthermore, predicting the propagation of tsunamis and tsunamiwave trains may indicate which regions are relatively safe and which coastal regions arethreatened the most when tectonic activity is measured.Wave prediction and reconstruction can be achieved in various ways. Field data can beobserved to gain real scale statistical information. A laboratory setting can be used to simulatea full-scale problem and to validate theoretical models. Lastly, mathematical models can bedeveloped to understand and predict water waves.In this report we consider Boussinesq models that aim to accurately model waves propagationfrom the deep sea to the surf-zone [25]. An important question is, whether waves in theBoussinesq models are capable to break, i.e. develop jumps in the surface elevation. Furtherchallenges will then be to cope with the breaking of the waves numerically and to modelflooding and drying of waves reaching the shore.

1.2 Mathematical modelWaves of Boussinesq type account for dispersion2 and nonlinearity [17]. Ideally, Boussinesqmodels can be applied to water waves occurring from deep water to the surf zone, without anyextensions of the model. The model is valid for long waves with a depth to wavelength ratio ofh/


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2 1 Introduction

Lukes variational principle. Within this model energy is conserved and is p ositive-definite.This model is applicable for fully non-linear water waves of Boussinesq-type and has im-proved dispersion characteristics for propagating wave-groups such as tsunami wave-trains ascompared to classical Boussinesq-type equations.The third variational Boussinesq model is derived by Whitham [37], resulting (under addi-

tional approximations) in the original Boussinesq (partial differential) equations formulatedby Peregrine [30]. It is a further simplification relative to Lukes and Klopmans variationalprinciple.In this report surface tension effects are included in the three variational models. Theseeffects are not negligible in small water-depths [23] and are easily included in a variationalprinciple. This has the advantage that the mathematical models can be validated eventuallywith a laboratory test in which surface tension is imporatant.In this report we introduce moving boundaries for the Klopmans variational Boussinesqmodel in order to capture flooding and drying effects in the model.

1.3 Numerical computation

The wave propagation described by the two variational Boussinesq models is solved numer-ically. The variational models lead to weak variational forms, that are used to conduct theweak formulations of the continuous Galerkin finite element method. We compare the numer-ical results with approximate the original Boussinesq equations [11], called cnoidal waves3.Instead of investigating shoaling effects we will simulate breaking waves by initially consideringvery steep linear wave solutions in our non-linear model [36]. When a wave is breaking,the water surface shows a discontinuous jump. To be able to solve the propagation of adiscontinuous jump numerically we propose a discontinuous Galerkin method.

3A cnoidal wave is a periodic analytical solution to the Korteweg-de Vries (KdV) euations, which is apropagating, shape invariant solution in both space and time.


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2 Variational principles for water waves 3

2 Variational principles for water waves

2.1 Introduction

Considering fluids with the assumption of conservation of energy enables us here to formulate

a variational principle. We assume the fluids to be incompressible. Neglecting air pressure,Lukes principle [27] varies the total energy for potential flow, which is the addition of thekinetic energy Ek and potential energy Ep as the basis of his principle. This principle com-pletely describes the fluid motion at the free surface and incorporates appropriate boundaryconditions for free surface flows. Taking variations results in partial differential equations,which are called the Euler-Lagrange equations and which are the governing equations of mo-tion for our system. In the next section 2.2, we consider Lukes principle including surfacetension for nonlinear potential flow. Thereafter, we will derive the exact linear dispersionrelation for potential flow.In section 2.3, we consider a variational principle for Klopman et al.s [19] nonlinear con-servative Boussinesq equations extended with surface tension. This variational principle is

deduced by approximating the vertical structure of the velocity potential. As a result, thekinetic energy term is changed relative to Lukes variational principle. Surface tension effectsare included in the model and the linear dispersion relation is derived.In section 2.4 we consider the simpler Whitham Boussinesq model. We again include surfacetension in the model and we derive the linear dispersion relation.Finally, in section 2.5 the three dispersion relations are compared for deep water and forrelatively small water depth, as compared with the wave length. For long waves the surfacetension effects can be neglected, for short waves the effects of surface tension must be in-cluded.Finally, in section 2.7 some conclusions are drawn.

2.2 Lukes variational principle for free-surface potential flowWe consider a fluid layer with surface tension in three dimensions extending over a horizontaldomain (x, y) Dh and in the vertical direction z, see figure 2.2.1. The solenoidal4 velocity is

x0

fluid

air

z

0 y

d

0(x,y,t)

y

h0

x(x,y,z,t)

seabedseabed

Figure 2.2.1: Parameters describing the free surface (x,y ,t) of the fluid moving in a hori-zontal velocity v = (x,y ,z) above a varying seabed z = d(x, y) in an horizontal domain(x, y) Dh.

given by v =(x,y ,z ,t), where (x,y ,z ,t) is the velocity potential and = (x, y, z)T

the three dimensional gradient. We write ||2 = (x)2+(y)2+(z)2. Surface tension isa force that tends to minimize the area of the free surface. The surface tension is measuredin Newton per meter [N m1] and is defined as the intensity of the molecular attraction per

4The divergence of a vector in a solenoidal field v is equal to zero; the fluid is irrotational.


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4 2 Variational principles for water waves

unit length along any line in the surface [29]. In the following we will write = /. Thekinetic Ek and potential energy Epot are of the form

Ek = Dh(x,y,t)

d1

2||2 dz dx dy (2.1)

Epot =

Dh

(x,y,t)d

gz dz + (

1 + (x)2 + (y)2 1) dx dy (2.2)

with the constant density of the fluid, (x,y,t) the surface elevation, d(x, y) the still waterdepth at (x, y), and g the gravitational acceleration. The first order horizontal derivativesare denoted as = (x, y)T, with the transpose ()T. We define Hamiltons principle forpotential flow with the action functional Lf:

0 = Lf(, s, ) =

t1

t0

Lf(, s, ) dt (2.3)

with Lf =

Dh

st dx dy H (2.4)

and H = Ek + Epot. (2.5)Here, s = []z= is the evaluation of on the free surface z = [27] and Lf is the Lagrangian.This action functional is known as Lukes principle (Luke, 1967) for = 0, in the form asgiven by Miles [27]. We vary (2.3):

0 =

t1

t0

Lf(, s, ) dt (2.6)

= lim0

1

t1

t0

Lf( + ,s + s, + ) Lf(, s, ) dt (2.7)

= lim0

1

t1

t0

Dh

[(s + s)t( + ) A

+d

1

2|( + )|2

B

dz

1

2

g( + )2 C

1 + |( + )|2 D ] dx dy (2.8)

Dh

[st A

d

1

2||2 B

dz 12

g2 C

+

1 + ||2 D

] dx dy dt

.The linearity property of variational problems allows us to write out expressions per termmore elaborately [35]. Integrating with respect to time, terms A in (2.8) can be rewritten as:

t1

t0 Dhst + st dx dy dt =

t1

t0 Dhst ts dx dy dt, (2.9)

where we used that (t0) = (t1) = 0, since belong to the set of test functions C0 (Dh),

which is the set of admissible elements. We rewrite terms B by splitting the integration


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domain [d, + ] in the two intervals [d, ] and [, + ]. We get for the first interval[d, ] using Gausss law:

Dh

d ( ) dV = Dh

d 2 dV + S n dS, (2.10)

with S the (closed) surrounding boundaries of the volume V = [Dh] [d, ] and withdV = dx dy dz. The boundary S includes the seabed, the vertical boundaries on the edgesof the horizontal domain Dh (for example a wall at x = x0), and the free surface boundarySf at z = . We adjust our infinitesimal surface element dS with respect to the surface weare integrating over. We again evaluate at the free surface with s. The free surfaceparametrized as the surface s = 0 with s = z and we obtain for the normal vector on thefree surface the following expression:

n =

s

|s| =(x, y,

1)T(x)2 + (y)2 + 1 . (2.11)

Combining this with evaluating on the free surface, the boundary integral of equation(2.10) becomes:

S

n dS =S\Sf

n dS+Sf

s [ z] dx dy (2.12)

For the second contribution to B, i.e. the interval [, + ], we obtain:

lim0

1

Dh

+

1

2|( + )|2 dV (2.13a)

= lim0

1

Dh

+

1

2||2 dV + lim

0

Dh

+

dV (2.13b)

=

Dh

2||2z= dx dy, (2.13c)

where we applied Taylor expansions around z = in going from (2.13b) to (2.13c). Terms Din equation (2.8) are approximated with Taylor expansions around = 0:

lim0

1

Dh

1 + |( + )|2

1 + ||2 dx dy (2.14a)

= lim0

Dh

( + ) ()

1 + |( + )|2

dx dy (2.14b)

=

Dh

()

1 + ||2 dx dy (2.14c)

= Dh

x

1 + ||2x

+ y

1 + ||2y

dx dy, (2.14d)


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where x = /x. We used Gauss law from (2.14c) to (2.14d) and used that is arbitrary.With the (derived) expressions for A to D, we obtain for equation (2.8):

0 =

t1

t0 Dh ts + st g 1

2|

|2z=

+s [ z]z= +

d

2 dz [ n]S\s (2.15)

+

x1 + ||2

x

+

y

1 + ||2

y

dxdy dt.Note that the subscript Sf indicates the free surface, while S denotes all the surroundingsurfaces. The variational problem states that L(, s, ) is stationary with respect to theindependent variables , s and [35], and thus we obtain the following expressions:

t1t0

Dh

d

2 dz dx dy + [ n]S\Sf dt = 0

: 2 = 0 for z [d, ] (x, y) Dh (2.16)and : n = 0 at S\Sf (2.17)

t1t0

Dh

s(t + z) dx dy dt = 0

s : t + z = 0 at z = ; (2.18)t1t0

Dh

{ts g 12||2z=

+

x1 + ||2

x

+

y

1 + ||2

y

} dxdy dt = 0 : t + g + 1

2||2

x

1 + ||2x y

1 + ||2y = 0 at z = . (2.19)Expression (2.16) implies that the fluid is incompressible (Laplaces equations) and denotesconservation of mass. Expression (2.17) states that no particle of the fluid can move throughthe seabed and the wall. The natural boundary condition (2.17) is also known as Neumannsboundary condition. In expression (2.18) the free surface kinematic free surface boundarycondition [16] is given, which expresses that matter does not leave the fluid through the freesurface (for example, there is no evaporation). Expression (2.19) represents the free surfacedynamic free boundary condition, which is the so-called Bernouilli equation, evaluated at thefree surface [16].

2.2.1 Linear dispersion for potential flow

We sketch the derivation of the exact linear dispersion relation for potential flow over ahorizontal seabed, as in [23], but now with surface tension. We translate the surface elevation


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to h + hb, see figure 2.3.1, and the seabed d to hb in order to compare the models later.The two boundary conditions in one dimension regarding the free surface in potential flowread, as seen in section 2.2:

th + xxh z = 0 at z = h0, (2.20)

t + gh +1

2(x)

2 hx1 + (hx)2

x

= 0 at z = h + hb. (2.21)

We rewrite the term

hx

1 + (hx)2

x

= hxx

(1 + (hx)2)3

2

(2.22)

and we linearize around the free surface z = h0:

th z = 0 at z = h + hb (2.23)t + g(h h0) hxx = 0 at z = h0. (2.24)

Combining the two linear boundary conditions and with a constant seabed, such that thb = 0,we obtain

2t + gz z2x = 0 at z = h0. (2.25)With Stokes first-order theory [23] we can write the velocity potential as

(x,z ,t) = (z)ei(xt), (2.26)

which satisfies Laplaces equation 2 = 0 and we may write

(x,z ,t) = 0 cosh[z], (2.27)

since as is a solution to Laplcaces equation, so is its complex conjugate. Substituting thisin the free-surface boundary condition (2.25) yields the exact linear dispersion relation

+ g tanh[z] + 3 tanh[z] = 0 at z = h0 (2.28)2 = (g + 2) tanh(h0). (2.29)

We get the following Taylor expansion in , around the long-wave limit = 0:

2 = gh02 ( 1

3gh30 h0)4 + (

2

15gh50

1

3h30)

6

( 17315

gh70 215 h50)

8 + O(10). (2.30)

2.3 Klopmans variational Boussinesq model (KVBM)

We consider the fluid layer in two dimensions with horizontal xaxis and vertical zaxis, seefigure 2.3.1. The fluid velocity is now given by (u, v) (x, z) and we change the definitionfor the free surface elevation (x, t) to the total water height h(x, t)+hb(x) by translation overh0, such that d equals the seabed height hb(x). This choice is convenient when modelingfree surface waves that may run up, for example, beaches, or where dry patches of land mayappear. Now we make the following Ansatz for the potential (x,z ,t), motivated by the

parabolic vertical structure of the velocity potential found in conventional Boussinesq modelsfor water waves over a horizontal bed [19]:

(x,z ,t) = (x, t) + f(z)(x, t), (2.31)


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z

x

0

0

h0fluid x(x, t)

seabed

h(x, t)

hb(x)

air

Figure 2.3.1: Parameters describing the free surface h(x, t) of the fluid moving with horizontalvelocity x above a varying seabed hb(x) in an horizontal domain x Dh for the Klopmanand Whitham variational Boussinesq model.

where = at the free surface z = h(x, t) + hb(x) and

f(z) = (z h hb)(z + h hb). (2.32)

We take this choice because we only want time derivatives of h(x, t) and (x, t) to appearin the Euler-Lagrange equations. Further, the form of the vertical structure function f(z) ismotivated by the fact, that for long waves over a horizontal bed, the coshfunction of equation(2.27) becomes, to a good approximation, a parabolic function with its derivative equal tozero at the sea bed. The assumption that z = 0 at z = hb for a (locally) horizontal seabed isa good approximation for small bottom slopes [19]. Under the assumption of a mildly slopingbottom, i.e. assuming spatial derivatives of hb(x) are small and thus neglected, and underthe assumption that wave slopes are small x(h + hb h0))


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Taking variations with respect to , h and gives:

0 =

t1t0

Dh

th x

hx 23

h3x

(2.37a)

+th h12

(x)2 2h2xx + 43

h4(x)2 + 2h22 + g(h h0) (2.37b)+x

2

3h3x 8

15h5x

4

3h3 dx dt. (2.37c)

Partial integration with respect to time and space results in the variational derivative:

0 =

t1t0

Dh

th + x

hx 2

3h3x

(2.38a)

h

t +1

2(x)

2 2h2xx + 43

h4(x)2 + 2h22 + g(h h0)

(2.38b)

x 23

h3x 815

h5x

+4

3h3

dx dt, (2.38c)

where we used that and h are elements of C0 (Dh). The variational principle states thatLb is stationary with respect to the independent variables , h and . We replace the flowvelocity variable u x and obtain the Euler-Lagrange equations, from (2.38a)

: th + x

hu 2

3h3x

= 0, (2.39)

from (2.38b), and differentiating with respect to x,

h : tu + x 1

2

u2

2h2ux +

4

3

h4(x)2 + 2h22 + g(h

h0) = 0, (2.40)

and from (2.38c)

: x

2

3h3u 8

15h5x

+

4

3h3 = 0. (2.41)

2.3.1 Klopman Variational Boussinesq model including surface tension

We now consider the expressions derived previously including surface tension. Note that = h = xh in the one dimensional case. The energy equation (2.35a) now reads:

H=Dh

h+hb

hb1

2[(x+f x)

2 + (f)2]dz +1

2g(h

h0)

2 + (1 + (xh)2 1)dx. (2.42)Analogous to Lukes variational principle, due to the linearity property [35], this contributesin a term which occurs in the first variation of h. The governing equations now read (withthe respective variated variables):

: th + x

hu 2

3h3x

= 0, (2.43a)

h : tu + x

12

u2 2h2ux + 43

h4(x)2 + 2h22

+g(h

h0)

hx

1 + (xh)2 = 0 (2.43b) : x

2

3h3u 8

15h5x

+

4

3h3 = 0. (2.43c)


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2.3.2 Linear dispersion for Klopman Variational Boussinesq model

We obtain the following linearized equations of the Boussinesq model of subsection 2.3.1, fora constant mean-water depth h0:

th + h0xu 2

3 h20

3x = 0, (2.44a)

tu + gxh 3xh = 0, (2.44b)2

3h30xu

8

15h50

2x +

4

3h30 = 0. (2.44c)

We introduce solutions h(x, t) = hei(xt), u(x, t) = uei(xt) and = ei(xt) to thelinear system (2.44), and to find the dispersion relation:

h02

g=

1 +2

g

(h0)

2 1 +115(h0)

2

1 + 25(h0)2

. (2.45)

We get the following Taylor expansion in

2 = gh02 ( 1

3gh30 h0)4 + (

2

15gh50

1

3h30)

6

( 475

gh70 2

15h50)

8 + +O(10). (2.46)

2.4 Whithams variational Boussinesq model (WVBM)

Finally, we consider the following one dimensional simplified variational principle Ls intro-duced by Whitham [37], equation (1.7):

Ls(h, th, x) =t1

t0

Dh

th 12

h(x)2 +

1

6h0(th)

2 12

g(h h0)2 dx dt (2.47)

with h0(x) the height of the water when the fluid is at rest. Taking variations with respectto and h, we obtain the Euler-Lagrange equation:

0 =

t1t0

Dh

th 12

h(x)2 +

1

6(th)

2 12

g(h h0)2 dx dt

=

t1

t0

Dh

th hxx + th 1

2 h(x)2 +1

3 h0thth g(h h0)h dx dt

=

t1t0

Dh

(th + x[hx]) h

t +1

2(x)

2+ g(hh0)+ 13

h02t h

dx dt,

(2.48)

where we integrated by parts with respect to time and space from (2.4) to (2.48) and we usedthe arbitrariness of and h. Taking the first variations we obtain the following governingequations:

: th + x[hx] = 0, (2.49a)

h : t +1

2(x)

2 +1

3h0

2t h + g(h h0) = 0. (2.49b)


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Introducing the horizontal velocity u x we rewrite the equations of motion in the form:th + x[hu] = 0, (2.50a)

tu + x[1

2u2 + g(h h0) + 1

3h0

2t h] = 0. (2.50b)

Note that expression (2.50a) is a kinematic condition concerning conservation of mass andexpression (2.50b) is a dynamic condition, which denotes conservation of momentum. Withoutthe term 13

2t h we have the non-linear shallow water equations:

th + x[hu] = 0, (2.51a)

tu + x[1

2u2 + g(h h0)] = 0. (2.51b)

2.4.1 Whitham variational Boussinesq model including surface tension

Analogous to the derivation in section 2.3.1 we obtain for the Whitham Boussinesq modelthe following equations:

th + x[hu] = 0, (2.52a)

tu + x[1

2u2 + g(h h0) + 1

3h0

2t h

hx

1 + (hx)2

] = 0. (2.52b)

2.4.2 Linear dispersion for the original Whitham Boussinesq-model

We linearize the governing equations of the Whitham Boussinesq-model, in the same way asconsidered in section 2.4:

th + h0xu = 0, (2.53a)

tu + gxh +13

h02t xh 3xh = 0. (2.53b)

We introduce solutions h(x, t) = hei(xt) and u(x, t) = uei(xt). Substituting these intothe linear system (2.53) we observe that they satisfy the dispersion relation:

h02

g=

1 +

g2 (h0)2

1 + 13(h0)2

. (2.54)

We get the following Taylor expansion in , around = 0:

2 = gh02

(

1

3

gh30

h0)

4 + (1

9

gh50

1

3

h30)6

( 127

gh70 1

9h50)

8 + +O(10). (2.55)

2.5 Comparison of dispersion relations

In this subsection we compare the three models derived for 2D. The Taylor expansions around = 0 of the linear dispersion relations of the three models that we studied, are the following:

1. Full potential flow5, as derived from Lukes variational principle:

2Pot =g

h0(h0)

2 13

(h0)4 +

2

15(h0)

6 17315

(h0)8

+ h30(h0)4 1

3(h0)6 + 2

15(h0)8+ O(10), (2.56)

5Taylor approximation of the exact linear dispersion relation.


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2. Klopman variational Boussinesq Model (KVBM):

2K =g

h0

(h0)

2 13

(h0)4 +

2

15(h0)

6 17315

(h0)8

+

h3

0(h0)4

1

3

(h0)6 +

2

15

(h0)8

+ O(10) and (2.57)

3. Whitham variational Boussinesq Model (WVBM):

2W =g

h0

(h0)

2 13

(h0)4 +

1

9(h0)

6 127

(h0)8

+

h30

(h0)

4 13

(h0)6 +

1

9(h0)

8 + O(10). (2.58)

2.5.1 Comparison of dispersion relations without surface tension effects

We neglect surface tension effects, = 0. When comparing the Taylor expansions of the dis-persion relations, we see that the Klopman variational Boussinesq model is of order

O(10)

and the original Whitham Boussinesq-model is of order O(4) compared with the exact dis-persion relation. In fact, the linear dispersion relation of the KVBM is the same relation asfor the Boussinesq equations with improved linear dispersion of Madsen et al. [25].We will first consider the dispersion relations without surface tension effects, = 0. In figure2.5.1 we plot the dispersion relations6 of the three models for small h0, where = 2/.Long waves are represented by h0 0 and for h0 >> 1 we have short waves relative tothe water depth. For long waves the three dispersion relation denoted by defining () =

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

kh0

(

h0

/g)1/2

ExactKlopman VBM

Whitham VBM

Figure 2.5.1: Dispersion relations:

h/g as a function of h0, for the Klopman andWhitham Boussinesq-models and the exact dispersion relation. water depth h0 = 10m.

of equations (1)-(3) yield a group velocity Vgroup() equal to ()/. For infinitesimalwaves it is shown by Klopman et al. [19] that for waves up to h0 = the phase speedK()/

(gh0) of the Klopman variational Boussinesq model has a relative error,

K()/exact()/

1, (2.59)6Partly reconstruction of figure 5.1a in [20].


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less than 3%7 compared to the exact phase speed. For the original Whitham variationalBoussinesq model we have that the relative error,

W()/

exact/ 1, (2.60)

is equal to 15%8 for h0 = . The equality h0 = is particularly important if we liketo investigate breaking waves. For smaller water depths and around this it is where weconsider waves propagating from deep water into shallow water. For the nonlinear shallowwater-wave model propagating waves will break eventually. Note that Vinje and Brevig [36]investigated the breaking of nonlinear shallow water waves with an initial sinusoidal waveprofile, at a ratio of h0 = 0.53 (2) [36]. Due to the accurate approximation of the exactphase speed we have that the linear Klopman variational Boussinesq model holds for wavespropagating up to the edge of the deep-water zone. Moreover, it is particularly interesting toinvestigate within the non-linear Klopman variational Boussinesq model whether waves willeventually start to break in the presence of dispersion.

2.5.2 Comparison of dispersion relations with surface tension effects

Now we include surface tension effects in our dispersion relations. We investigate surfacetension effects since they affect the velocity of propagating waves. In figure 2.5.2 we plot thelinear dispersion relations for = 7.4 102 N m1 and = /, since we eliminated thewater density previously. This is an experimental value for water in contact with air at15.6C [29]. We take = 1.0 103. This figure is partly a reproduction of fig. 57 in [23].We observe that the Klopman variational Boussinesq model more closely resembles the exactdispersion relation including surface tension. In table 2.5.1 and figure 2.5.2 this has been

made more explicitely. In table 2.5.1 the relative error of the phase speeds of the KVBM C()k

and WVBM C

()

w with the exact phase speed C

()

e are given:Observe that in the KVBM with surface tension the error is bigger, as compared with theWhitham formulation for longer waves.Now the question arises if it is always appropriate to include surface tension effects. Denotingthe phase speed when neglecting surface tension effects by C(0) and the phase speed withsurface tension effects by C(), we can calculate the relative error by

C()e

C(0)e

1 = g + 2

g 1 (2.61)

since C(0)e g and C() g + 2 for a certain waterdepth h0 and wavenumber and

again =

/ with

= 7.4 102

N m1

and the water density, here 1000 kg m3

. Forwavelengths equal to 2/ the relative error is : We added the third column (h0 = 5 cm)in table to compare the results with figure 2.5.1. Lighthill [23] states that in general surfacetension is negligible for water waves > 7 cm; from (2.5.2) we see that the phase speed hasa relative error of 0.061 %. Longer waves propagate as pure gravity waves.When including surface tension effects we may conclude that for the region where surfacetension effects are important, the KVBM including surface tension shows smaller relativeerrors than the WVBM.

7We found a relative error of 2.79%.8The actual error was 14.7%, which implicitly yields that propagating waves will be dissipated and will

not be dispersed.


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0 1 2 3 40

0.5

1

1.5

2

/h0

Phase

speed

/(k2gh0

)1/2

With st. h0= 0.01 m

With st. h0= 0.02 m

With st. h0= 0.04 m

With st. h0= 0.08 m

No st. (all depths h0)

(a) Exact dispersion relation and pure gravity waves

0 1 2 3 40

0.5

1

1.5

2

/h0

Phase

speed

/(k2gh0

)1/2

Klopman VBM with st.Exact with st.Whitham VBM with st.Exact, no st.

(b) Water depth 1 cm.

0 1 2 3 40

0.5

1

1.5

2

/h0

Phase

speed

/(k2gh0

)1/2


(c) Water depth 5 cm.

0 1 2 3 40

0.5

1

1.5

2

/h0

Phase

speed

/(k2gh0

)1/2


(d) Water depth 10 cm.

Figure 2.5.2: Phase speed /(

gh) plotted against the ratio 2/(h0) = /h0 for small

water depths h0. Here: Surface tension (st.) parameter = 7.4 102N m1 for water incontact with air at 15.6C. For reference the pure gravity wave at this water depth is plotted.(a) the exact dispersion relation including surface tension effects for various water dephts.Presented are the dispersion relations for the KVBM, WVBM and the exact model includingsurface tension effects at depth b) h0 = 1 cm, c) h0 = 5 cm and d) h0 = 10 cm.


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2.6 Moving boundaries in the KVBM

The three variational models that we investigated previously were all considered on a horizon-tal domain with a fixed horizontal extent. When we consider a fluid layer which is floodingand drying on a seabed hb(x) with a slope, the fluid boundaries may not be fixed in time.

Moreover, a part of the seabed may fall dry when the fluid is moving away creating distinctpatches of fluid. On the other hand, distinct fluid patches may merge and flood a land patchin between. The horizontal domain now becomes Dh = [xL(t), xR(t)] for example and theaction functional becomes

0 = L(,h,,xL, xR) = t1t0

L(,h,,xL , xR) dt. (2.62)

We illustrate the moving boundary condition for the Klopman Boussinesq-model [19]. Wechoose the Lagrangian density L as

L(,h,,xL, xR) = xR(t)

xL(t) th H dx, (2.63)where H was derived in equation (2.35c):

H(,h) = 12

h

x 2

3h2x

2+

2

45h5(x)

2 +2

3h32 +

1

2g(h + hb)

2 12

gh2b .

(2.64)

Taking variations with respect to , h, , xL and xR we have

0 =

t1

t0

xR(t)

xL(t)

th

x

hx 23 h3x (2.65a)+t(h)

E

h

1

2(x)

2 2h2xx + 43

h4(x)2 + 2h22 + g(h + hb)

(2.65b)

+x

2

3h3x 8

15h5x

4

3h3 dx (2.65c)

+lim0

1

xR+xRxR

( + )t(h + h) Hb( + ,h + h, + )dx (2.65d)

+lim0

1

xL

xL+xL

( + )t(h + h) Hb( + ,h + h, + )dx dt. (2.65e)

For the integrals terms at the boundaries xL(t) and xR(t), terms (2.65d) and (2.65e) we have:

t1t0

lim0

1

xR+xRxR

( + )t(h + h) Hb( + ,h + h, + )dx

+lim0

1

xLxL+xL

( + )t(h + h) Hb( + ,h + h, + )dx dt

=

t1

t0 th

x=xR(t)

xR

th

x=xL(t)

xL dt. (2.66)

since the water levels h(xR(t), t), h(xL(t), t) are equal to zero at the fluid boundary. Theterm E, t(h) requires special treatment, since the boundaries are now time dependent.


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We have the following relation to rewrite term t(h):

t1t0

d

dt

xR(t)xL(t)

h dx =

t1t0

xR(t)xL(t)

ht

F+ t(h)

Edx

+h

x=xR(t)

dxR(t)

dt h

x=xL(t)

dxL(t)

dt

dt, (2.67)

where F is the term that completes the terms (2.65a)-(2.65c) to regain the variational deriva-tive resulting in the governing equations in the fluid domain, as collected in (2.38). We express

E explicitely using the above relation and we substitute E in (2.65b).We combine the terms at the boundaries with the boundary terms (2.65d) and (2.65e). Atthe boundary xR(t), we have for xR and h the following relation:

hxR(t)

= xhx=xR(t)

xR(t), (2.68)

and similarly for x = xL(t). Combining the remaining terms with respect to (2.38), we have:

0 =

xR(t)xL(t)

h dxt1t0

+

t1t0

xR

th + x(h)

dxR(t)

dt

x=xR(t)

(2.69a)

xL

th + x(h)dxL(t)

dt

x=xL(t)

dt. (2.69b)

Observe that everywhere in the fluid we have conservation of mass

th + x(hx) = 0 (2.70)

and we have that h(xR(t), t) = h(xL(t), t) = 0 at the boundaries. Substitution of the aboverelations and using the arbitrariness of the variables xL, xR we obtain conditions for themoving boundaries

xR : (x)(xh) (xh)dxR(t)dt

= 0 at x = xR(t), (2.71)

xL : (x)(xh) + (xh) dxL(t)dt

= 0 at x = xL(t), (2.72)

which is simplified to

xR : x dxR(t)dt = 0 at x = xR(t), (2.73)

xL : x dxL(t)dt

= 0 at x = xL(t). (2.74)

Together with equations (2.39), (2.40) and (2.41) these conditions govern the Klopman vari-ational Boussinesq model for fluid boundaries that can move in time.


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2.7 Conclusion

In this section we have studied variational principles for free surface water waves, wherethe total energy H is the basis of an action functional and where H varies per model. Thefollowing three variational principles have been studied:

1. Lukes variational principle for (3D) non-linear potential flow [27].

2. Klopmans variational Boussinesq model (1D) [19].

3. Whithams variational Boussinesq model (1D) [37]

We have included surface tension effects by adding an extra term in the total energy H.For all three principles the Euler-Lagrange equations were derived which govern the fluid sur-face motion. Comparing the linear dispersion relations we conclude that the Klopman varia-tional Boussinesq model approximates the exact dispersion relation better than the Whithamvariational Boussinesq model.In the region where initial sinusoidal waves start to break [36], we find that the relative errorof the KVBM is small compared to the exact linear dispersion relation.Furthermore, surface tension effects can be approximated more accurately with the KVBMin water depths where surface tension effects may not be neglected, i.e. for short waves with < 10 cm.We conclude that for longer waves propagating over deeper water the Klopman Boussinesqvariational is equivalent to the Boussinesq formulations of Madsen [25]. The advantage ofthe KVBM is that energy is conserved and guarantueed to be positive-definite. Furthermore,moving boundaries of the fluid domain can be nicely incorporated, such that the physics offlooding and drying is described easily.


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3 Numerical modeling: continuous Galerkin 19

3 Numerical modeling: continuous Galerkin

3.1 Introduction

To approximate the solutions of the governing equations derived in section 2 we use the

continuous Galerkin finite element method, which is also know as Ritz-Galerkin method.In this method we formulate a weak formulation of the Boundary Value Problems that weobserve. Discretizing the equations offers the possibility to obtain the approximated solutionnumerically.First we discretize Whithams variational Boussinesq formulation by using the arbitrarinessof the variational derivatives with respect to each variable. Note that they are elements of thetest function-space C0 (Dh) on the domain, which can be restricted to the test functions oneach element Ki with the test function space C

0 (Ki). Hereby we then formulate the finite

element weak formulations.In section 3.3 the assembly of the discretized system is considered. First the linearized shallowwater model is considered, then the nonlinear shallow water model and finally the Boussinesq

term is included, where we used an auxiliary variable to treat the second-order time derivative.Then a non-flat seabed is introduced, which enables us to test if waves will start to break.In section 3.6 time-integration methods are proposed, namely the total variation diminishingRunge-Kutta 3 and 4 method and the symplectic Stormer-Verlet method.In section 3.7 some conclusion are drawn.

3.2 Whithams Boussinesq variational principle

For the Whitham Boussinesq-model including surface tension (including the -term, section2.4) the variational formulation Ls was

0 =

t1t0

Dh

Ls(h, ) dxdt (3.1)

=

t1t0

Dh

th 12

h(x)2 +

1

6h0(th)

2 12

g(hh0)2 (

1 + (xh)2 1) dx dt. (3.2)

Solving the system with a finite element method requires that we partition the horizontaldomain Dh = x (xL, xR) with elements to approximate the solution. We introduce thepartitioning Th existing of N open elements Kj = {x|x (xj , xj+1)} where xi and xj+1 arethe so-called nodes. The result is a tessalation

Th = {Ki Ni=1 Ki = Dh and Ki Kj = if i = j, 1 i, j N} (3.3)with Ki the closure of element Ki. This discretization is illustrated by figure 3.2.1.

x1

xL K1

x2

K2

x3 xj

Kj

xj+1 xN

KNxR

xN+1

Figure 3.2.1: Subdivision into N elements Kj of the horizontal domain Dh.

We introduce global Galerkin basis-functions j(x)

Pn(Dh), which means that they arecontinuous, have compact support on Dh and are polynomials of degree n [6]. We chooselinear basis-functions j(x) = 1 on global node j and zero at the other nodes, see figure 3.2.2.For convenience we consider periodic boundaries, such that node x1 = xN+1 and we consider


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20 3 Numerical modeling: continuous Galerkin

Kj1xj1 xj xj+1

Kj

j j+1

Kj2 Kj+1

j1

0 1

Figure 3.2.2: Linear basis functions for continuous Galerkin finite element methods; thick lineis j(x).

the surrounding elements KN and K1 for node x1. We approximate the unknown functionsof the system on the nodes xj, j = 1, . . ,N+ 1 with trial-functions h(t) and (t) for h(x, t)and (x, t) respectively in terms of the basis-functions j [5]

h(x, t) =h(x, t) = N+1j=1

j(x)Hj(t), (x, t) = (x, t) = N+1j=1

, j(x)j(t) (3.4)

where j is only non-zero in the elements Kj1 and Kj . In element Kk the basis-functions kand k+1 are considered, since the other basis-functions are zero. It is convenient to introducethe element basis functions m (m = 0, 1), such that 0(x) = k(x) and 1(x) = k+1(x) forx Kk. The representation of the trial functions (x, t) and h(x, t) can now be written as

(x, t) = k(t)0(x) + k+1(t)1(x) x Kk, (3.5a)

h(x, t) = Hk(t)0(x) + Hk+1(t)1(x) x Kk. (3.5b)

When we consider a non-flat seabed, we may represent the still water level h0(x) with:

h0(x) = H0,k0(x) + H0,k+11(x) x Kk. (3.6)3.2.1 Discretized variational formulation

Neglecting surface tension, the discretized variational formulation is the substitution of (3.5)and (3.6) in (3.1):

0 =

t1

t0

N

k=1xk+1

xk

(k

0

+ k+1

1

)(dHk

dt0

+dHk+1

dt1

)

12

(Hk0 + Hk+11)(kd0dx

+ k+1d1dx

)2 +1

6h0(

dHkdt

0 +dHk+1

dt1)

2

12

g(Hk0 + Hk+11 h0(xk)0 h0(xk+1)1)2 dxdt. (3.7)

We may now vary with respect to the variables Hk(t), Hk+1(t), k(t) and k+1(t) over theelements Kk. The first variation with respect to k(t) and k+1(t) is:

0 =N

k=1 Kk (k0 + k+11)dHk

dt

0 +dHk+1

dt

1)

Hk0 + Hk+11k d0dx

+ k+1d1dx

k

d0dx

+ k+1d1dx

dx (3.8)


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and with respect to Hk(t) and Hk+1(t) is:

0 =Nk=1

Kk

(Hk0 + Hk+11)(dk

dt0 +

dk+1dt

1)

+ 12

(Hk0 + Hk+11)(k d0dx

+ k+1d1dx

)2

+1

3h0(Hk0 + Hk+11)(

d2Hkdt2

0 +d2Hk+1

dt21)

+g(Hk0 + Hk+11)(Hk0 + Hk+11 H0,k0 H0,k+11) dx. (3.9)

3.3 Assembly of global nodes

In this section an assembly framework is created that we can use to calculate the fully dis-cretized variational formulation. The assembly framework results in an algebraic system.Using the arbitrariness of the variational derivatives, Hk, k C0 (Kk1 Kk), we are

able to collect the function values in element Kk1 and Kk that contribute to the functionvalues at node xk.We start with the linearized shallow water model. Then we include nonlinear terms thatplay a role in the shallow water formulations. Thereafter we consider the discrete variationalformulation for the Boussinesq model as derived in the previous section. For each step thetotal energy of the system is considered.

3.3.1 Discrete linearized shallow water variational formulation

When we linearize the system (3.8, 3.9) around the still water position z = h0 and neglectthe higher order time derivatives, we obtain the following discrete system principle for thelinearized shallow water equations;

0 =Nk=1

Kk

(0 + 1)dHk

dt0 +

dHk+1dt

1)

h0

k(d0dx

)2 + k+1(d1dx

)2 + (k + k+1)d0dx

d1dx

)

dx (3.10a)

0 =Nk=1

Kk

(0 + 1)(dk

dt0 +

dk+1dt

1)

+g(0 + 1)(Hk0 + Hk+11 H0,k0 H0,k+11) dx, (3.10b)where we used the arbitrariness of the variational derivatives. These relationships can besimplified as follows. We transform the integrals over the element Kk into integrals over thereference element K = [0, 1] with local basis-functions 0 = 1 , 1 = , K. We are nowable to reformulate the integrals over elements Kk with respect to the reference element K:

Akmn = |Kk|1

0

mn d, Bkmn = |Kk|

10

dmdx

dndx

d, (3.11)

with |Kk| = xk+1 xk the length of the element Kk and {m, n} = {0, 1}. These integralsover the reference element can be computed with Gauss two-point formula, which is exactfor linear polynomials [24]. The representations

and

h in (3.5) evaluated on the elements

Kk, together with the evaluation of the inner-products on the reference element (3.11), givesus a linear system per node k:

Ak110dHk1

dt+ (Ak111 + A

k00)

dHkdt

+ Ak01dHk+1

dt


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= h0

Bk110 k1 + (B

k111 + B

k00)k + B

k01k+1

, (3.12a)

Ak110dk1

dt+ (Ak111 + A

k00)

dkdt

+ Ak01dHk+1

dt(3.12b)

=

g A

k110 (Hk1

H0,k1) + (A

k111 + A

k00)(Hk

H0,k) + A

k101 (Hk+1

H0,k+1)

whit the stillwater height h0 constant in time. We can write this system using mass matrixMA and stiffness matrix MB as follows:

MAdH

dt= h0MB (3.13a)

MAd

dt= gMA (H h0), (3.13b)

where MA and MB are tri-diagonal matrices with the exception of the first and last row,where boundary condition are playing a role. Since MA is invertible we can write the systemas follows:

dH

dt= h0M

1A MB (3.14a)

d

dt= g(H h0), (3.14b)

Integration over time can be done using the third-order Runge-Kutta method, see 3.6. Thespatial averaged energy H(t) for the linearized shallow water variational formulation is givenby:

H(t) = 1 k Kk

1

2h0(k

d0dx

+ k+1d1dx

)2

+1

2g(Hk0 + Hk+11 H0,k0 H0,k+11)2dx. (3.15)

3.3.2 Discrete shallow water variational formulation

The discrete nonlinear shallow water system can be obtained from the discretized variationalformulation described by system (3.8, 3.9). The discretized shallow water system reads:

0 =Nk=1

Kk

(k0 + k+11)dHk

dt0 +

dHk+1dt

1)

Hk0 + Hk+11k d0dx

+ k+1d1dxk d0

dx+ k+1

d1dx dx (3.16a)

0 =Nk=1

Kk

(Hk0 + Hk+11)(dk

dt0 +

dk+1dt

1)

+1

2(Hk0 + Hk+11)(k

d0dx

+ k+1d1dx

)2

+g(Hk0 + Hk+11)(Hk0 + Hk+11 H0,k0 H0,k+11) dx. (3.16b)

For convenience we introduce the element integration

Dkmnp = |Kk|1

0

mdndx

dpdx

d. (3.17)


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Assembly of the elements Kk1 and Kk around xk Dh using the arbitrariness of the variables and H results in the following system:

Ak110dHk1

dt+ (Ak111 + A

k00)

dHkdt

+ Ak01dHk+1

dt(3.18a)

= h0 Bk110 k1 + (Bk111 + Bk00)k + Bk01k+1+Dk1010 Hk1k1 + D

k1011 )Hk1k + D

k1110 Hkk1 + D

k1111 Hkk,

+Dk000Hkk + Dk001Hkk+1 + D

k100Hk+1k + D

k101Hk+1k+1,

Ak110dk1

dt+ (Ak111 + A

k00)

dkdt

+ Ak01dHk+1

dt(3.18b)

= g

Ak110 (Hk1 H0,k1) + (Ak111 + Ak00)(Hk H0,k) + Ak101 (Hk+1 H0,k+1)

12

Dk1100 (k1)2 1

2Dk1101 k1k

1

2Dk1110 kk1

1

2Dk1111 (k)

2

1

2

Dk000(k)2

1

2

Dk001kk+1

1

2

Dk010k+1k

1

2

Dk011(k+1)2

We rewrite this assembly in terms of solution vectors as follows

MAdH

dt= h0MB + MD(H) (3.19a)

MAd

dt= gMA(H h0) 1

2ME(), (3.19b)

where MD(H(t)) is denoted for periodic boundaries (x1 xN+1) by

MD = (3.20)

M D12 M D13 0 . . . . . . 0 M DN1M D21 M D

22 M D

23 . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . M Dk1 M Dk2 M Dk3 . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . M DN11 M DN12 M DN13M D13 0 . . . . . . 0 M D

N1 M D

N2

with

M Dk1(H(t)) = Dk1010 Hk1(t) + D

k1110 Hk(t) (3.21a)

M Dk2(H(t)) = Dk1011 Hk1(t) + Dk1111 Hk(t) + Dk000Hk(t) + Dk100Hk+1(t) (3.21b)

M Dk3(H(t)) = Dk001Hk(t) + D

k101Hk+1(t). (3.21c)

Similarly, matrix ME((t)) is denoted by M Ei,j() = M Di,j(), i , j [1, . . ,N] with coeffi-cients computed as follows:

M Ek1 ((t)) = Dk1100 k1(t)D

k1110 k(t) (3.22a)

M Ek2 ((t)) = Dk1101 k1(t) + D

k1111 k(t) + D

k000k(t) + D

k010k+1(t) (3.22b)

M Ek3 ((t)) = Dk001k(t) + D

k011k+1(t). (3.22c)

Note that in the assembly of the nonlinear terms in (3.19a) and (3.19b), the matrix coefficientsM Dk1 (3.21a) and M E

k1 (3.22a) are multiplied by k1. Similarly, the coefficients M D

k2

(3.21b) and M Ek2 (3.22b) are multiplied by k and finally M Dk3 (3.21c) and M E

k3 (3.22c)

are multiplied by k+1.


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The spatial averaged energy H(t) for the shallow water variational formulation is given by:

H(t) = 1

k

Kk

1

2(Hk0 + Hk+11)(k

d0dx

+ kd0dx

)2

+ 12

g(Hk0 + Hk+11 H0,k0 H0,k+11)2 dx. (3.23)

3.3.3 Discrete Whitham variational Boussinesq formulation

Equivalently to the previous formulations for the shallow water model the Whitham Boussi-nesq model can be discretized. The Boussinesq model includes the second-order time deriva-tive of the wave elevation, 2t (x, t) in (3.9). The system to be solved now reads:

MAdH

dt= h0MB + MD(H) (3.24a)

MA

d

dt +

1

3 h0MA

d2H

dt2 = gMA(H h0) 1

2 ME(), (3.24b)

For convenience, write d2Hdt2

= ddt(dHdt ) and introducing F = +

13h0

dHdt yields the following

system:

F = +1

3h0M

1A (h0MB + MD(H))

=

IN +

1

3h0M

1A (h0MB + MD(H)

(3.25a)

MAdH

dt= h0MB + MD(H) (3.25b)

MA dFdt

= gMA(H h0) 12 ME(). (3.25c)

where IN is the [N N] unity matrix. When H(tn), F(tn) known, we can solve (3.25a) andobtain (tn). We then can compute (3.25b, 3.25c) in time using an explicit time steppingmethod.The spatial averaged energy H(t) for the Whitham Boussinesq variational formulation is givenby:

H(t) = 1

k

Kk

1

2(Hk0 + Hk+11)(k

d0dx

+ kd0dx

)2

+ 12

g(Hk0 + Hk+11)2 dx 16

( dHkdt

0 + dHk+1dt

1)2 (3.26)

3.4 Klopmans variational Boussinesq model

In order to solve the weakly nonlinear Klopman variational Boussinesq model numerically weintroduce the following expansions for the trial and test functions

h(x, t) =N+1

j=1 j(x)Hj(t), h(x, t) =N+1

j=1 j(x)Vj ,

(x, t) =N+1

j=1 j(x)j(t), (x, t) =N+1

j=1 j(x)Wj ,

(x, t) =

N+1j=1 j(x)j(t), (x, t) =

N+1j=1 j(x)Zj .

(3.27)

where j are linear basisfunctions of degree dp = 1 that are only non-zero in the elementsKj1 and Kj as introduced in section 3.2. The seabed slope h0(x) is approximated by H0,which we will assume constant in the rest of this subsection. We integrate the variational


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formulation of the Klopman Boussinesq model (2.37) by parts with respect to time and wehave

0 =

t1

t0 Dhth x

hx 2

3h3x

(3.28a)

ht h

1

2(x)

2 2h2xx + 43

h4(x)2 + 2h22 + g(h h0)

(3.28b)

+x

2

3h3x 8

15h5x

4

3h3 dxdt. (3.28c)

We will now substitute the expansions for our trial and test functions to discretize our weakformulation for each varied variable. Evaluation of the global basisfunction is done over thereference element as described in section 3.2.

3.5 Assembly of global nodes

We assemble our system per varied variable at each elements belonging to the tesselation ofa periodic domain. We introduce convenient integrals over the elements and collect thesecontributions per element to solve our system in time.

3.5.1 Assembly for the first part

We first consider part (3.28a), the variation with respect to . We haveDh

th x

hx 23

h3x

dx = 0 (3.29a)

N

j=1 Kj th x hx 2

3h3x dx = 0. (3.29b)

We substitution the expansions (3.27). For convenience we introduce the following integralsat elements Kj of our tessalation Th,

A(j, i, m) =|Kj|

2

11

im d (3.30a)

B1(j, i, HL, HR, L, R) =|Kj|

2

11

didx

HL1 + HR0

L

d1dx

+ Rd0dx

d (3.30b)

C1(j, i, HL, HR, L, R) = |Kj

|2

1

1

didx HL1+HR03Ld1dx +Rd0dx d (3.30c)

where i represents the expansion of our arbitrary testfunction and indices L and Rrepresent respectively the left (xj) and right (xj+1) node of the element concerned. Notethat, for fast computation of the integrals, didx = (1)1i 2|Kj| for linear polynomials. Wecollect the integrals above by summation over all elements and add the contributions to theleft and right node. We obtain for equation (3.29b)

M AdH

dt M B1 + 2

3M C1 = 0 (3.31)

where the matrices are found by summation of all contributions from elements Kj, j =

1, . . ,Ne to both nodal values at the left and right boundaries, xj and xj+1, of elements Kj.They are indicated by xj+1i, i = 0, 1.

M Aj+1i,j+1m = A(j, i, m) m = 0, 1 (3.32a)


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M B1j+1i = B1(j, i, Hj , Hj+1, j , j+1) (3.32b)

M C1j+1i = C1(j, i, Hj , Hj+1, j, j+1) (3.32c)

Observe that the parameter j + 1 i in combination with element Kj represents the globalnode, nameley i = 1 refers to the left and i = 0 to the right node of an element.

Note that we still have to account for boundary conditions; eg. in case of periodic boundaryconditions we concern nodal value x1 if xj+1i = xNe+1 and we add this contribution to theglobal matrix accordingly.

3.5.2 Assembly for the second part

We now consider part (3.28b), the variation with respect to . We haveDh

h

t +1

2(x)

2 2h2xx + 43

h4(x)2+ 2h22+ g(hh0))

dx = 0 (3.33a)

N

j=1 Kj ht +1

2 (x)2

2h2

xx +

4

3 h4

(x)2

+ 2h2

2

+ g(hh0)dx =0 (3.33b)and adopt the convenient integrals for our tessellation Th

A(j, i, m) =|Kj|

2

11

in d (3.34a)

B2(j, i, L, R) =|Kj|

2

11

id1

dxL +

d1dx

R2

d (3.34b)

C2(j, i, HL, HR, L, R, L, R) =

|Kj

|2

1

1

iHL1 + HR02L d1dx + Rd0dx Ld1dx + R d0dx d (3.34c)D2(j, i, HL, HR, L, R) =

|Kj|2

11

i

HL1 + HR04

Ld1dx

+ Rd0dx

2d (3.34d)

E2(j, i, HL, HR, L, R) =|Kj |

2

11

i

HL1 + HR02

L1 + R02

d. (3.34e)

where i represents the expansion of our arbitrary testfunction h. We obtain for equation(3.33b)

M AdH

dt

+1

2

M B2

2M C2 +

3

4

M D2 + 2M E2 + gM A (H

H0) = 0 (3.35)

where

M Aj+1i,j+1m = A(j, i, m) m = 0, 1 (3.36a)M B2j+1i = B2(j, i, j , j+1) (3.36b)

M C2j+1i = C2(j, i, Hj , Hj+1, j , j+1, j , j+1) (3.36c)

M D2j+1i = D2(j, i, Hj , Hh+1, j, j+1) (3.36d)

M E2j+1i = E2(j, i, Hj , Hj+1, j , j+1) (3.36e)

3.5.3 Assembly for the third part

We now consider part (3.28c), the variation with respect to . We haveDh

x

2

3h3x 8

15h5x

4

3h3 dx = 0 (3.37)


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Nj=1

Kj

x

2

3h3x 8

15h5x

4

3h3 dx = 0. (3.38)

and use the integrals for our tessellation Th

C3(j, i, HL, HR, L, R) =|Kj |

2

11

didx

HL1+HR0

3L

d1dx

+Rd0dx

d (3.39a)

D3(j, i, HL, HR, m) =|Kj |

2

11

didx

HL1+HR0

5dmdx

d (3.39b)

E3(j, i, HL, HR, m) =|Kj|

2

11

i

HL1+HR03

m d (3.39c)

where i represents the expansion of our arbitrary testfunction and with the parameterm we represent the nodel value of j+1m at xj+1m. We obtain for equation (3.38)

2

3M C3 8

15M D3 +

4

3M E3

= 0 (3.40)

where

M C3j+1i = C3(j, i, Hj , Hj+1, j, j+1) (3.41a)

M D3j+1i,j+1m = D3(j, i, Hj , Hj+1, m) (3.41b)

M E3j+1i,j+1m = E3(j, i, HL, HR, m) (3.41c)

where for periodic boundaries if j + 1

m = N e + 1 we have to put j + 1

m = 1.

3.5.4 Solving the system in time

Since we represent each trial function in terms of polynomial basisfunctions of degree dp = 1we have to make sure that our spatial integration routine preserves accuracy. For exampleintegral D3; we numerically integrate h5 over an element. Observe that as a result we obtainpolynomials of order at most dp = 5.xj+1

xj

h5 dx |Kj|2

11

Hj(

1 2

) + Hj+1(1 +

2)5

d. (3.42)

Therefore an integration routine should be chosen that is at least exact for polynomials ofdegree 5. We choose the three point Gauss quadrature rule, which is exact for polynomialsof degree dp = 2 3 1.Note that we have the following invertible matrices; M A, M D3 and M E3 of size [N e N e].We have the following vectors of size [N e 1]: the trial function vectos H, and , and theassembly vectors M B1, M B2, M C1, M C2, M C3, M D2, M D3, M E2 and M E3.When solving the system in time, we first have to solve the third part to obtain . Then weobtain our residuals for a time integration method.

=4

5M D3 + 2M E3

1M C3 (3.43)

dH

dt = M A1M B1 23 M C1 (3.44)

dH

dt= M A1

gM A (H H0) 12

M B2 + 2M C2 34

M D2 2M E2 (3.45)


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3.6 Time integration

Untill now the time was considered to be continuous. We now discretize our systems withrespect to time, to integrate numerically in time from one time step to another. In literaturemany time integration methods are proposed. Since we consider energy conserved systems,

we would particularly like to conserve energy in our numerical scheme. Therefore we considerthe Stormer-Verlet method. The two Runge-Kutta methods are considered, since they arehigh order in time (third- and fourth-order, respectively). However, they are total variation(T V) diminishing [14] under the CFL coefficient. For our systems this results in numericaldissipation. The TV diminishing is expressed as follows:

T V(un+1) :=iTh

|un+1i+1 Un+1i | iTh

|uni+1 Uni | =: T V(un+1), (3.46)

where u(x, t) = u(xi, tn) is the solution at the ith node xi and at time step tn [14]. The CFL

conditions states:

t maxjTh xjmaxjTh

|unj |. (3.47)

Another result is that a solution of the systems observed only is oscillatory when the numericalsolution is representing a discontinuous jump [14], in our case representing the breaking waves.

3.6.1 Stormer-Verlet

Stormer-Verlet is a symplectic method; it is reversible in time and it conserves energy forHamiltonian systems such as the (non)-linear shallow water equations. [15].For the linearized Shallow-Water equations:

Hn+1

2 = Hn +t2

MBn

n+1 = n t2

MAHn+ 1

2 (3.48)

Hn+1 = Hn+1

2 +t

2MB

n+1.

3.6.2 Third order Runge Kutta

We collect the discretized variational formulations in the following system:

tU = R(U), (3.49)

with U = [H, ] the state vector. We can use the third-order Runge Kutta to discretizethe system in time. We compute the new state vector Un+1(x) at time-step tn+1 = tn + texplicitely

U(1) = Un + tR(Un, tn)

U(2) =

3Un + U(1) + tR(U(1), tn + t)

/4 (3.50)

Un+1 =

Un + 2U(2) + 2tR(U(2), t2 + t/2)

/3.

A rough time step estimation is given by

t

CF L mink

(Kk)/c (3.51)

with CF L the Courant-Friedrichs-Lewy number; CF L 1 and c is a constant velocity, eg.c =

gh0 in the shallow water case 3.3.1.


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3.6.3 Fourth-order Runge-Kutta

The classical fourth-order Runge-Kutta formula is [31]:

U

(1)

= tR(U

n

, t

n

)U(2) =

1

2tR(Un +

1

2U(1), tn + t)

U(3) =1

2tR(Un +

1

2U(2), tn + t) (3.52)

U(4) = tR(Un + U(3), tn + t)

Un+1 = Un +U(1)

6+

U(2)

3+

U(3)

3+

U(4)

6

3.7 Conclusion

In this section we derived for fully nonlinear Whitham variational Boussinesq model a contin-uous Galerkin finite element method to approximate propagating wave solutions numerically.In the progress of this derivation we have derived a cG FEM for the linear and nonlinearshallow water equations.Furthermore, we have derived a cG FEM formulation for the weakly nonlinear Klopman vari-ational Boussinesq model for a horizontal seabed.Finally, we have presented two total variation diminishing in time integration methods, namelythe Runge-Kutta methods 3 and 4, and the St ormer-Verlet method, which is a symplecticmethod.


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4 Numerical Modeling: Discontinuous Galerkin 31

4 Numerical Modeling: Discontinuous Galerkin

4.1 Introduction

The first discontinuous Galerkin (DG) method was introduced in 1973 by Reed and Hill [32]

in the framework of neutron mass transport, i.e. for hyperbolic equations. Since then DGfinite element methods have been developed for a wide range of models and applications. Inour model besides hyperbolic terms also elliptic terms appear. The DG method has beendeveloped for purely elliptic problems by Bassi and Rebay [3]. In this report we will followthe approach of Arnold et al. [1] to compute the elliptic terms that appear in the Klopmanmodel. We will use the method of Brezzi et al. which is a local, consistent and conservativemethod [2]. The discontinuous Galerkin method is particularly useful when hydraulic jumpsoccur.The basic idea of the Galerkin method is to multiply the system of partial differential equa-tions by arbitrary test functions. In Discontinuous Galerkin methods we perform a partialintegration over each element resulting in a weak formulation. When we discretize the weak

formulation the discontinuous Galerkin methods specifically uses a numerical flux at theboundaries of the element that concerns the two neighboring elements. The discontinuitymay occur at the element boundaries. The advantage of this method that we can refine hplocally, which means that we may refine the element size h of one element or that we mayadjust the order of our polynomial expansion p in an element [38].In subsection 4.2 we will first introduce the discretization that we will use and we introducesome conventional notations and properties, such as the average {{}} and jump [[]] at a faceS and the global and local lifting operators R and Rs.The DG method is illustrated for the linear shallow water model9 in subsection 4.3. It is shownhow we can design the relations defining the numerical fluxes when energy is conserved, as isthe case in our variational models. These fluxes affect the stability and the accuracy of the

method. We will follow the approach of Yan and Shu (2002) [38] for KdV type equations toderive conditions for the numerical fluxes.In subsection 4.4 three basic properties of fluxes are formulated, namely consistency, conser-vation and stability in time. Furthermore the Rankine-Hugoniot relation is formulated withwhich we can calculate the propagation speed of the local discontinuity.Thereafter we present two choices for the fluxes, namely an alternating flux and the simpleLax-Friedrich flux [22].In subsection 4.5 we illustrate the local discontinuous Galerkin method to compute the ellip-tic terms on a Poisson equation. In subsection 4.6 we apply the local discontinuous Galerkinmethod to the linearized equations that were derived from the Klopman variational model.It is in these equations that the elliptic terms appear. It is discussed why the formulation

of the primal form as proposed by [38] was not followed and where discrepancies are withEskillsons work [12, 13].

4.2 Finite element discretization

In this section first the finite element discretization is introduced that we will use in theDG method. Second we discretize the trial functions which enables us to solve the systemnumerically. Third the jump and average over a face are defined and some algebraic propertiesare given. Fourth the global and local lifting operator are defined, which we will use to solve

our elliptic terms.

9The lsw is a system of hyperbolic equations when written in [h, u]


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32 4 Numerical Modeling: Discontinuous Galerkin

4.2.1 Tesselation of the horizontal domain

The spatial discontinuity in the DG method requires that we discretize the horizontal domainslightly different than in the continuous Galerkin case. Again we partition the horizontaldomain Dh with elements to approximate the solution. We introduce the partitioning Th

existing of N open elements Kj = {x|x (xj 12 , xj+ 12 )} where xj 12 and xj+ 12 are the so-called nodes. The result is a tesslation

Th = {Kj Nj=1 Kj = Dh and Ki Kj = if i = j, 1 i, j N}. (4.1)

with Kj the closure of element Kj. The discontinuity may occur at the internal faces xj 12

and xj+ 12

of element Kj. Values of u left and right of face xj 12

are therefore denoted by

respectively

uj 1

2

:= limxx

j 12

u(x, t) and u+j 1

2

:= limxx

j 12

u(x, t) (4.2)

as illustrated in figure 4.2.1

K1 K2 Kj KN

x1 12

x1+12

x2 12

xj+ 12

xN 12

xN+ 12

+++++ +

xj 12

Figure 4.2.1: Spatial discretization of the horizontal domain Dh for the discontinuous Galerkinfinite element method.

4.2.2 DiscretizationTo reduce the partial differential equations of our model explicitely to ordinary differentialequations, we choose a finite number of polynomials to expand the variables in each elementKj. We hereby obtain the so-called trial functions uh(x, t) and vh(x, t) for the solution u(x, t)and for test function v(x, t), respectively:

uh(x, t) =

dPm=0

Um(xj, t)m(x), wh(x, t) =

dPm=0

Wm(xj, t)m(x), (4.3)

with polynomial basis functions m(x) PdP(Kk) in which PdP(Kj) denotes the space ofpolynomials in Kj of degree dP and uh, wh Vh, with Vh = {v v|Kj PdP(Kj), k = 1, . . ,N}.The so-called global basis functions m are defined on element Kj by:

m,j(x) =

1 if m = 0m,j(x) 1|Kj|

Kj

m,j(x) dx if m > 0.(4.4)

with compact support (equal to zero outside the element) and m = m on the local reference

element [1, 1]. With a mapping x = Fj() [xj 12

, xj+ 12

] with degree dp = 1 we map

from the reference element to the partitioning

Fj() =1

2

xj 12

+1 +

2

xj+ 12

(4.5)

and vise versa from the partitioning to the reference element with its inverse F1(x). Integralsover an element can now be computed with a two-point Gaussian quadrature rule which is


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exact for polynomials of degree 1 [28]. An integral over an element can be computed as followsfor a smooth function g(x) Vh

xj+1

2

xj 12

g(x) dx =|Kj |

2

1

1 g(Fj()) d =|Kj |

2 g(Fj(13

)) + g(Fj(1

3)). (4.6)

4.2.3 Jump and average on a face

The union of the internal faces is denoted by i and the boundary faces are united in the setb. All the boundaries are represented by = i b. At an internal face S we define the

jump [[]] and the average {{}}

[[v]] = vn + v+n+ for S i (4.7a){{v}} = 1

2(v + v+) for S i. (4.7b)

with n and n+ the outward normal with respect to the left and right element of face S. AtS b the jump is v+n+ or vn and the average equals the internal value v+ at the leftboundary or v at the right boundary of the domain. Note that in the 1D case n = 1 andn+ = +1.We state some algebraic properties that we will use in the following subsections. For vi , w

i

at the left and v+i , w+i at the right of face Si we have that

v(wn) + v+(w+n+) = {{v}}[[w]] + [[v]]{{w}} at S i (4.8)

and at the left or right boundary, we have that

v(wn) = {{v}}[[w]] = [[v]]{{w}} at S b. (4.9)with n the outward unit normal. Observe that the following relations hold for the jumpaverage

{{v + w}} = {{v}} + {{w}}, [[v + w]] = [[v]] + [[w]],{{{{v}}}} = {{v}} and [[{{v}}]] = 0. (4.10)

4.2.4 Lifting operator

To deal with elliptic terms in our systems first the global and local lifting operators are defined[3] in this subsection. Then they are discretized and rewritten as a linear system [34]. The

global lifting operator R(p) is defined with an arbitrary test function v Th as follows:Dh

vR(p) dx :={{v}}p ds. (4.11)

The local lifting operator at face S is Rs(p). It has compact support outside the elementsKj , Kj+1 adjacent to the face S = xj+ 1

2

= Kj Kj+1 is defined as:Dh

vRs(p) dx :=S{{v}}p ds. (4.12)

The local lifting operatorR

s(p) is related to the global lifting operatorR

(p) by:

Rs(p) =S

Rs(p). (4.13)


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34 4 Numerical Modeling: Discontinuous Galerkin

We discretize the local lifting operator Rs(p) at face s left or right of element Kj, by expansionin polynomials of degree dp as [34]

(Rs(p))j =dp

i=0 Rs,ji i(x) for x Kj (4.14)

in order to solve elliptic terms numerically. Likewise the expansions of our test-function vh(x)and trial function h(x) read

vh(x) =

dpl=0

Vj,ll(x) and h(x) =

dpi=0

j,ii(x) for x Kj. (4.15)

We substitute these expansions in the definition for the local lifting operator (4.12) and for aface S of element Kj

dpi=0

Rsr,jl

Kj

li dx = dpi=0

12

k,i S

i l ds + S

+i l ds for l = 0, . . ,dp(4.16)

where i and +i represent value at the left and at the right element of face S, respectively.

For convenience we define a matrix Aj R(dp+1)(dp+1) at element Kj and the matrix Ps,j R(dp+1)2 at face S adjacent to elements Kj and Kj+1 as

Ajli =

Kj

li dx and PsR,j(j,i, j+1,i)li =

1

2k,i

Sr

li ds. (4.17)

Now we can collect the integrals over the elements and over the faces to obtain the followinglinear system [34]

AjRsR,j = PsR,j(j , j+1). (4.18)

Similarly, we have on the left face of element Kj:

AjRsL,j = PsL,j(j1, j) (4.19)

Note that, since the basis functions i are orthogonal, automatically yields Aj is invertible.

Finally the lifting operator coefficients Rs,j are found for a left or right face s = {SL, SR} byinversion of Aj and the expansion of the trial function for the local lifting operator now reads

(Rs)j =dpi=0

Rs,ji i =

dpi=0

(Aj)1Ps,ji (4.20)

for face S left or right of element Kj.

4.3 Linear shallow water system

In this subsection we apply the discontinuous Galerkin method on the linear shallow waterequations over a flat seabed and illustrate how our numerical fluxes can be chosen such thatenergy is conserved. First we start with our system of equations, which are

th + h0xu = 0 (4.21a)

tu + gxh = 0. (4.21b)


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We multiply with arbitrary test functions v(x), w(x) C(Kj) on a certain element Kj andintegrate by parts over each element Kj Th. For convenience, we write vx = dvdx for the firstorder spatial derivative with respect to its only argument. At each element we have

Kj vth vxh0u dx + vh0u

xj+

12

vh0u+

xj

12

(4.22a)Kj

wtu wxgxh dx

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