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COMPUTATIONAL MECHANICSNew Trends and Applications

S. Idelsohn, E. Onate and E. Dvorkin (Eds.)cCIMNE, Barcelona, Spain 1998

A SPECTRAL ELEMENT SEMI-LAGRANGIAN METHODFOR THE SHALLOW WATER EQUATIONS ON

UNSTRUCTURED GRIDS

Francis X. Giraldo

Naval Research Laboratory

Monterey, CA 93943 USA

e-mail: giraldo@nrlmry.navy.mil, web page: http://www.nrlmry.navy.mil/giraldo

Key Words: Semi-Lagrangian Method, Shallow Water equations, Spectral Element

Method, Unstructured Grid

Abstract. The purpose of this paper is twofold: to give a brief yet comprehensive descrip-tion of the spectral element and semi-Lagrangian methods, and to introduce a new methodarrived at by fusing both of these impressive methods. The practical aspects of both meth-ods are described in detail by their implementation on the 2D shallow water equations.These equations have been used customarily to develop new numerical methods for weatherprediction models because they exhibit the same wave behavior as the more complex 3Datmosphere and ocean equations. The spectral element method is essentially a higher or-der finite element method that exhibits spectral convergence provided that the solution isa smooth function. The semi-Lagrangian method traces the characteristic curves of the

solution and, consequently, is very well suited for resolving the non-linearities introducedby the advection operator of the fluid dynamics equations. By using the basis functionsof the spectral element method as the interpolation functions for the semi-Lagrangianmethod, one can create a truly local method that can be used in conjunction with un-structured/adaptive grid generation and can be ported quite naturally to parallel computerarchitectures. Results for the fully non-linear 2D shallow water equations are presentedthus showing the benefits of this new scheme.

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1 INTRODUCTION

The spectral element method combines the benefits extracted from both the spectralmethod and the finite element method. The spectral element method can be describedas a method that can automatically generate any order polynomial basis function, as in

the spectral method, while allowing for the geometrical flexibility enjoyed by the finiteelement method. The prosperity of the spectral element method can be attributed tothe fact that any order polynomial can be generated automatically, concurrently withtheir numerical integration rules. In fact, the collocation points are the Gauss quadraturepoints for the integration rules. In addition, the basis functions are orthogonal whichmeans that we can obtain diagonal mass matrices. There is also no need to define thebasis functions explicitly because we can define implicit relations a priori for the innerproducts of the functions and their derivatives. It is also worth mentioning that since thecollocation points are not equi-spaced, one can automatically generate staggered grids byusing varying order polynomials for the different variables as is done in 6, which satisfiesthe Babuska-Brezzi condition.

The advective terms in the governing equations of fluid motion present formidablechallenges to many discretization methods including Galerkin methods. These terms makethe operator non-self-adjoint and as a result, prevents the optimization of these methods.By combining the time derivative and the advective terms into the Lagrangian derivativeand then discretizing the resulting operator, we can circumvent many of the difficulties;this is the semi-Lagrangian method. This method not only increases the solution accuracybut also allows for much larger time steps therefore making it potentially more efficientthan Eulerian methods.

2 SHALLOW WATER EQUATIONS

2.1 Spectral Element Method

The 2D shallow water equations are defined as

t+ (u) = 0

u

t+ u u +

x f v = 2u

v

t+ u v +

y+ f u = 2v

where they represent the conservation of mass, and horizontal and vertical momentum,respectively. The weak formulation of these equations in physical coordinates x and y iswritten as

ij d

j

t

ijk d ukj +

(N u)i d = 0

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ij duj

t+

ijk d ukuj +

ij

xd j

i jk d fkvj = v

i j d uj +

(N u)i d

ij dvj

t+

ijk d ukvj +

ijy

d j

+

i jk d fkuj = v

i j d vj +

(N v)i d

where N is the unit normal vector. The mappings from physical to computational spaceyields the following matrices

Mijkl =

ij d =

+11

+11

|J|hi() hj() hk() hl() dd

Auijklmn =

ijxk d = +1

1+11

|J|hi() hj() hk

() hl() hm() hn() x

dd

Avijklmn =

i

j

yk d =

+11

+11

|J| hi() hj() hk()hl

() hm() hn()

ydd

Puijkl =

i

jx

d =+11

+11

|J| hi() hj()hk

() hl()

xdd

Pvijkl =

i

j

yd =

+11

+11

|J| hi() hj() hk()hl

()

ydd

Cijklmn = ijk d = +1

1 +1

1

|J| hi() hj() hk() hl() hm() hn() dd

Dijkl =

ix

j

x+

iy

j

yd =

+11

+11

|J|hi

() hj()

hk

() hl()

x

2dd

++11

+11

|J| hi()hj()

hk()

hl

()

y

2dd

Guijklmn =

ix

jk d =+11

+11

|J|hi

() hj() hk() hl() hm() hn()

xdd

Gvijklmn = iy jk d =

+1

1 +1

1 |J| hi()

hj

() hk() hl() hm() hn()

y dd

where M is the mass, Au and Av the advection, Pu and Pv the pressure, C the Coriolis,D the diffusion, and Gu and Gv the terms which arise from Greens theorem, and R the

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boundary terms. The 2D shallow water equations can now be written in the followingmatrix form

Mijklkl

t Guijklmn umnkl G

vijklmn vmnkl = 0

Mijkl

ukl

t + Au

ijklmn umnukl + Av

ijklmn vmnukl + Pu

ijkl kl Cijklmn fmnvkl = Dijkl ukl

Mijklvlk

t+ Auijklmn umnvkl + A

vijklmn vmnvkl + P

vijkl kl + Cijklmn fmnukl = Dijkl vkl

for i ,k,m = 0,...,N and j,l ,n = 0,...,M. The shallow water equations can now bewritten as a vector system of coupled ordinary differential equations as

dU

dt= H(U)

where

U =

uv

and

H(U) =M1ijkl

Guijklmn umnkl + G

vijklmn vmnkl

Auijklmn umnukl Avijklmn vmnukl P

uijkl kl + Cijklmn fmnvkl Dijkl ukl

Auijklmn umnvkl Avijklmn vmnvkl P

vijkl kl Cijklmn fmnukl Dijkl vkl

.The spatial accuracy is determined by the order of the basis functions and will be of

order O(N + 1). As an example, for N = 1 we have linear elements, but second orderspatial accuracy. The element basis functions are the Legendre cardinal functions

hi() = (1 2)LN()

N(N + 1)LN(i)( i)(1)

where LN is the Nth order Legendre polynomial and LN is its derivative, and the mapping

from physical space to computational space is achieved by the transformation

=2

x2 x1(x x1) 1, x [x1, x2] (2)

where x1 and x2 are the physical coordinates defining the spectral element. The Legendrecardinal functions have the following properties

hi(j) =

1 if i = j0 if i = j

(3)

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and

hi

(j) =

LN(i)

Ln(j)(ij)if i = j

0 if i = j = 0, N

N(N+1)4

if i = j = 0

+

N(N+1)

4 ifi

=j

=N

(4)

which simplifies the implementation of the spectral element method because we do nothave to construct the basis functions explicitly.

The coordinates within an element are approximated by the basis functions as

x =Ni=0

Mj=0

xijhi()hj()

and so its derivative can be approximated by

x

=

N

i=0M

j=0 xijhi

()hj()

where N and M represent the number of grid points in the and directions. Theremainder of the derivatives are obtained following the same procedure. Note that theextension to three dimensions is immediately obvious from the two dimensional case.

To keep the algorithm as general and as automatic as possible, we evaluate the integralsnumerically. Therefore, the mass matrix can be evaluated as

Mijkl =Qq=0

Rr=0

|J(q, r)| wqr hi(q)hj(r)hk(q)hl(r)

where Q and R represent the number of Legendre-Gauss-Lobatto quadrature points inthe and directions. Note that the mass matrix only contains polynomials of the order2N. However this is not the maximum polynomial order contained in the equations. TheCoriolis matrix Cijklmn contains polynomials of the order 3N and so we require at leastQ = 3N+1

2quadrature points. These quadrature rules guarantee the exact integration of

all of the matrices in the equations.The matrix J(, ) is the Jacobian of the transformation from physical to computational

space and relates the ratio of areas of the element between the two spaces. In order todescribe the Jacobian, we begin with the chain rule

dx =x

d+x

d

dy =y

d+

y

d

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where the Jacobian is then defined as

|J| =

xx

yy

.For general grids in 2D, we have x = x(, ) , y = y(, ) and

|J| =x

y

x

y

where the derivatives are obtained numerically from the basis functions.

2.2 Spectral Element Semi-Lagrangian Method

Semi-Lagrangian methods belong to the general class of upwinding methods. Thesemethods incorporate characteristic information into the numerical scheme. The La-grangian form of the 2D shallow water equations is defined as

ddt

+ u = 0 (5)

du

dt+

x f v = 2u

dv

dt+

y+ f u = 2v

x

t= u,

y

t= v (6)

where ddt

denotes the total derivative and is defined as

d

dt =

t + u

x + v

y .

The weak formulation of these equations is given as

ij ddjdt

ijk d ukj +

(N u)i d = 0

ij dduj

dt+

i

j

xd j

i jk d fkvj = v

i j d uj

+

(N u)i d

ij ddvj

dt+

i

jy

d j +

i jk d fkuj = v

i j d vj

+

(N v)i d

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in terms of the physical coordinates x and y, where N is the unit normal vector. Aftermappings from physical to computational space we can write the Lagrangian form of the2D shallow water equations in the following matrix form

Mijkldkl

dtGuijklmn

umnkl

Gvijklmn

vmnkl

= 0

Mijkldukl

dt+ Puijkl kl Cijklmn fmnvkl = Dijkl ukl

Mijkldvlk

dt+ Pvijkl kl + Cijklmn fmnukl = Dijkl vkl

for i ,k,m = 0,...,N and j,l ,n = 0,...,M. Note that the matrices are defined in a similarmanner to the Eulerian case. The Lagrangian form of the shallow water equations cannow be written as the vector system of coupled ordinary differential equations

dU

dt= H(U)

where once again

U =

uv

and now the right hand side vector is given by

H(U) =M1ijkl

Guijklmn umnkl + Gvijklmn vmnkl

Puijkl kl + Cijklmn fmnvkl Dijkl uklPvijkl kl Cijklmn fmnukl Dijkl vkl

where H(U) = H(U(x ,t t)) are the interpolations at the departure points(x ,t t). Integrating (6) by the mid-point rule we get

= t u

x

2, t +

t

2

(7)

which defines a recursive relation for the trajectories. There are other options but thisapproach works extremely well and is quite efficient. There are also some important issuesinvolving the implementation of the semi-Lagrangian that ought to be mentioned.

We still need to consider how we are going to interpolate the values of H at departure

points. Interpolation is required because in general, the departure points will not fallon grid points but rather between them. Typically, the interpolations are constructedone of three ways: Lagrange, Hermite or spline polynomials. The difficulty with thesemethods is that they require some structure in the grid (i.e., curvilinear coordinates)which would be quite limiting in 2D where the elements may not have a true structure.

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(A caveat is in order here: cubic splines can be constructed from unstructured data, butthese methods are generally impractical for CFD type computations because they areprohibitively expensive). The incentive for combining the spectral element method withthe semi-Lagrangian method is due to the high order polynomials of the spectral elementmethod that are also locally defined within each element. We can determine in which

spectral element a given departure point lies, and then use the basis functions within theelement to construct the interpolation required by the semi-Lagrangian method. Sincethe order of the basis functions are typically high order, the interpolation will be of asufficiently high order to ensure the high order spatial and temporal accuracy of thenumerical scheme.

Since the Legendre cardinal basis functions have to be able to interpolate any departurepoint, they have to be constructed explicitly. The Legendre cardinal basis functions canbe written using the definition for Lagrange polynomials

hi() =N

j = 0j = i

ji

j (8)

and its derivatives are

hi

() =

Nk = 0k = i

Nj = 0j = i

1

i k

ji j

(9)

where the i, j , k are the permutations of the Legendre-Gauss-Lobatto (collocation)points. These two relations are very general and valid for any order Legendre cardinal basisfunction and can be used quite easily to generate the desired interpolating polynomial forthe semi-Lagrangian method.

The boundary condition requires that the departure point of the semi-Lagrangianmethod remain within the domain. On a sphere, the conditions used are those of period-icity. Periodicity can also be used here but the exact solution requires that the functionvanish at infinity. However, the velocity fields do not guarantee that the departure pointswill lie within the domain and so we must devise some method by which to check whetherthe departure point is in the interior of the domain.

As is done in typical unstructured mesh generation, the boundary array is stored ina counterclockwise direction. For a multiply-connected domain, the interior boundaries

are stored clockwise. This strategy ensures that the interior of the domain always liesto the left of each boundary segment. Therefore, the first step requires taking the crossproduct of the vector defining the boundary segment x1x2 = (x2 x1) with the vectorx1xd = (xd x1). Thus, if

x1x2 x1xd < 0

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then the orthogonal vector is pointing in the negative z direction. In other words, theresulting vector points into the paper, meaning that the departure point xd is to theright of the boundary segment and hence outside of the domain. When this condition isencountered, the intersection point of the boundary vector x1x2 and the departure pointvector xd is found. The vectors can be written parametrically as

x = x1 + s(x2 x1) and x = o + t(xd o)

where o is the origin, and s and t are the parametric variables. Thus the linear systemto be solved is the following

(x2 x1) xd(y2 y1) yd

s

t

=

x1y1

where a valid intersection occurs if and only if (s, t) [0, 1].The element containing the departure point must also be found. For general grids, the

best approach is to use a quadtree data structure. For the icosahedral grids used as onetest case in this paper, the data structure described in 4 should be used. But once weisolate the element claiming the departure point, we still need to determine its coordinatesin terms of the computational space. Equation (7) will only give the coordinates of thedeparture point in terms of the physical space. We can write the coordinates in physicalspace of the departure point using the basis functions in the form

xd =Ni=0

Mj=0

xijhi(d)hj(d)

and by virtue of Newtons method, we can write the iterative scheme for the roots ( d, d)

asFk+1 = Fk +Fk(kd,

kd) (d,d) = 0

where

F =Ni=0

Mj=0

xijhi(d)hj(d) xd .

This leads to the solutions

d =

Fk1 Fk1Fk2

Fk2

Fk1 Fk1Fk2

Fk2

, d = Fk1 Fk1Fk2

Fk2

Fk1 Fk1Fk2

Fk2

where

k+1d = kd + d,

k+1d =

kd + d

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which only requires 5 iterations at most. Thus if (k+1d , k+1d ) [1, 1], then we can be sure

that the departure point is contained within this element. Upon obtaining the departurepoint in terms of computational space coordinates the interpolation can be obtained usingthe basis functions of the element.

3 TIME DISCRETIZATION

A general family of explicit and implicit, one-step and multi-step time discretizationschemes for the ordinary differential equation

dU

dt

U

t= H(U)

where U is a vector quantity, can be given by the relation

(1 + ) Un Un1 = t

Hn+1(U) + (1 + ) Hn(U)

(10)

t

Hn1(U) + Hn2(U)

where Un = Un+1 Un and Un1 = Un Un1. For = 0, we get explicit methodsand implicit for 0 < 1. For , = 0 we get one-step methods, and multi-step methodsotherwise. The multi-step 3rd order explicit Adams-Bashforth (AB) scheme is obtainedby the parameters = = 0, = 16

12, = 5

12and the one-step 2nd order implicit Crank-

Nicholson (CN) by = 12

, = = = 0. Note that for the multi-step methods, anotherscheme must be used until we have a sufficient number of time levels required by themulti-step method. This other scheme should probably be a one-step method and mustbe at least of the same order of accuracy as the multi-step method. A popular choice isto use the following general family of Runge-Kutta schemes

Uk+1 = tK k + 2 H(Uk) for k = 0,...,K 1

where Uk+1 = Uk+1 Un and U0 = Un. For the explicit methods, we need not worryabout the non-linearity of the problem, but for the implicit methods we have to considerthis case. A straightforward approach is to consider a local linearization of the non-linearquantities. Therefore, taking a Taylor series expansion, we get

Hn+1(U) = Hn(U) +Hn(U) Un

which yields a second order approximation to the non-linear terms. Of course we cango to higher approximations but in this paper only 3rd order explicit and 2nd order

implicit schemes are considered. Applying this expansion to (10) we arrive at the followinglinearized general time discretization scheme

[(1 + ) t Hn(U) ] Un = Un1 + t [(1 + ) Hn(U)] (11)

t

Hn1(U) + Hn2(U)

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1 2

3

1 2

3

4

56

Figure 1: The subdivision of a triangle into 3 quadrilaterals. In this case, since the triangle isequilateral, the 3 quads will also be exactly the same size.

Figure 2: The grid and contours for the rotating Gaussian hill for the spectral element methodon an icosahedral grid with N = 1, Nref = 3, = 2.11, and the CN scheme.

In figure 3, one can see the traces of the background or parent triangular elements. The

blue lines denote the spectral elements and the red lines denote the collocation points. Inaddition, the quadrilateral decomposition of the triangles forms the Voronoi polygons ofthe grid points. This occurs because the triangulation of this collection of grid points is infact the Delaunay triangulation which is the well known dual of the Voronoi (or Dirichlet)tessellation. This can be seen much more easily in figure 2, where the grid tiles representthe Voronoi polygons with the grid points as their centroids.

This icosahedral grid has a very efficient data structure associated with it that can beused for searching. The other beneficial property of this grid is that all the quadrilateralelements are exactly the same size because the parent triangles are equilateral. This is animportant property if an uniform representation throughout the domain is desired whichis usually the case for geophysical flows on the surface of the sphere where grid biasing is

typically undesirable. However, if totally unstructured grids are desired as is the case withadaptive grids in computational fluid dynamics, will this strategy work? The followingsection addresses this case.

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Figure 3: The grid and contours for the rotating Gaussian hill for the spectral element semi-Lagrangian method on an icosahedral grid with N = 4, Nref = 1, = 2.84, and the CNscheme.

4.1.3 Unstructured Grid

Figure 4 and 5 show the grid and contours for both methods using an unstructuredgrid with N = 4. The unstructured grid generator is described in 3. However, thisgrid generator creates only triangles and so the triangles have to be subdivided intoquadrilaterals as illustrated by figure 1. After the quadrilaterals have been constructed,the collocation points are then generated along the boundaries of the elements first, andthen in the interior. This approach ensures that the edge collocation points are continuousacross the neighboring elements.

Figure 4: The grid and contours for the rotating Gaussian hill for the spectral elementmethod on an unstructured grid with N = 4, = 4.54, and the CN scheme.

The results for this example show that the spectral element method does indeed workeven on such an irregular grid as the one presented here but that it does not work too

well for this Courant number ( = 12.86) as the stability analysis illustrates in5

.However, the combination of spectral elements with the semi-Lagrangian method yields

an extremely accurate result even when large Courant numbers are used (see figure 5).This example truly shows the power and flexibility of the proposed strategy of combiningthe spectral element method with the semi-Lagrangian method.

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Figure 5: The grid and contours for the rotating Gaussian hill for the spectral element semi-Lagrangian method on an unstructured grid with N = 4, = 4.54, and the CN scheme.

4.2 Rossby Soliton Wave

4.2.1 Problem Statement

This problem describes an equatorially trapped Rossby soliton wave 2. The exactsolution is given by

(x , y , t) = (0) + (1)

u(x , y , t) = u(0) + u(1)

v(x , y , t) = v(0) + v(1)

where the superscripts (0) and (1) denote the zeroth and first order asymptotic solutionsof the shallow water equations, respectively. They are given by

(0) = 9 + 6y24

e y22u(0) =

(2y) e

y2

2

v(0) =

3 + 6y2

4

e

y2

2

and

(1) = c(1)9

16 5 + 2y2

e

y2

2 + 2(1)(y)

u(1) = c(1) 9163 + 2y2 ey22 + 2U(1)(y)

v(1) =

V(1)(y)

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where (, t) = A sech2B, = x c t, A = 0.771 B2, B = 0.394, and c = c(0) + c(1)

where c(0) = 13

and c(1) = 0.395 B2 as given in 7. The variable is the solution to theequation

+ n

+ n

3

3= 0

which is the famous Korteweg-de Vries equation that yields soliton solutions. The shallowwater equations can be simplified into this equation using the method of multiple scalesas presented in 1. Finally, the remaining terms are given by

(1)(y)U(1)(y)V(1)(y)

= e y22 n=0

nunvn

Hn(y)where Hn(y) are the Hermite polynomials and n, un, vn are the Hermite series coefficientsgiven in 2.

4.2.2 Structured Grid

Table 4 shows the results on a structured grid with the AB scheme. These resultsshow that the SEM-SLM works better than the h-type SEM and only slightly better thanthe p-type SEM. In the rotating Gaussian hill, the SEM-SLM AB scheme also showedlittle or no gain over the p-type SEM. The power of the SEM-SLM is evident at largeCourant numbers (or time steps). However, the AB scheme is explicit and even whenused with the SLM, the method must obey ODE rather than PDE stability restrictions.ODE stability restrictions are more lenient than the PDE restrictions, but they must beadhered to nonetheless. The SEM-SLM with the CN scheme in time is currently underdevelopment. This approach should prove to be far better than the SEM particularly at

very large Courant numbers as the analysis in 5 suggests.

4.2.3 Unstructured Grid

Table 5 shows the results on an unstructured grid with the AB scheme. These resultsillustrate that the SEM works rather well on this unstructured/adaptive grid. The SEM-SLM also gives good results but the real advantages of the SEM-SLM are not evident inthis example because the AB scheme is a fully explicit method. Because it is explicit,small time steps must be used in order to maintain the stability of the scheme. The SEM-SLM CN scheme ought to give far better results. At the very least we hope to achievethe same level of accuracy of the SEM while using much larger time steps.

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Figure 7: The grid and contours for the Rossby soliton wave at t=40 for the spectral elementsemi-Lagrangian method on an unstructured grid with N = 2, NE = 2100,

tN

= 110 , and theAB scheme.

Figure 8: The grid and contours for the Rossby soliton wave at t=40 for the spectral elementsemi-Lagrangian method on an unstructured grid with N = 4, NE = 2100,

tN

= 110 , and theAB scheme.

5 CONCLUSIONS

The spectral element method and the semi-Lagrangian method are described in detail.They are applied to the 2D shallow water equations in order to show the practical aspects

of the implementation of the methods. A new method is introduced whereby the spectralelement method and the semi-Lagrangian method are combined. The main attractionof this method (SEM-SLM) is that it uses the basis functions of the spectral elementmethod as the interpolating polynomial for the semi-Lagrangian calculation of the depar-ture points. This makes the method quite local in that the interpolation and calculationsare all performed in an element per element basis. This property is very important par-ticularly if we have an interest in implementing the method with unstructured/adaptivegrids and on parallel computers.

The results for the rotating Gaussian hill show that the spectral element semi-Lagrangianmethod (SEM-SLM) yields extremely accurate solutions even while using large Courantnumbers and on different types of grids, that is, as long as an implicit method (the CNscheme) is used. The icosahedral grid results are quite encouraging particularly becausethis grid has proven to be quite promising for applications on the sphere 4. The resultsfor the unstructured grid show that the spectral element semi-Lagrangian method yieldsvery good results regardless of the size of the Courant number.

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The results for the Rossby soliton wave show that the spectral element method stillworks rather well even with such unstructured grids but the semi-Lagrangian version ofthe method is disappointing, at least when used in conjunction with the AB scheme.However, this idea needs to be considered further and the development of the SEM-SLMwith the CN scheme is currently underway. At the very least, we hope to achieve the same

level of accuracy as the SEM but while using much larger time steps thereby increasingthe efficiency of the numerical method.

ACKNOWLEDGMENTS

I would like to thank the sponsor, the Office of Naval Research (ONR), and the programmanager, the Naval Research Laboratory (NRL), for supporting this work through pro-gram element PE-0602435N. I would also like to thank Tim Hogan of NRL for allowingme the opportunity to explore these new methods.

REFERENCES

[1 ]John P. Boyd, Equatorial Solitary Waves. Part 1: Rossby Solitons, Journal ofPhysical Oceanography, 10, 1699-1717, (1980).

[2 ] John P. Boyd, Equatorial Solitary Waves. Part 3: Westward-Traveling Modons,Journal of Physical Oceanography, 15, 46-54, (1985).

[3 ] Francis X. Giraldo, Efficiency and Accuracy of Lagrange-Galerkin Methods onUnstructured Adaptive Grids, Mathematical Modelling and Scientific Computing,8, (1997).

[4 ] Francis X. Giraldo, Lagrange-Galerkin Methods on Spherical Geodesic Grids, Jour-nal of Computational Physics, 136, 197-213, (1997).

[5 ] Francis X. Giraldo, The Lagrange-Galerkin Spectral Element Method on Unstruc-tured Quadrilateral Grids, Journal of Computational Physics, submitted (December,1997).

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