Parallel adaptive discontinuous Galerkin approximation for thin layer avalanche modeling

ARTICLE IN PRESS

0098-3004/$ - se

doi:10.1016/j.ca

$Research s

ACI-0121254.�Correspond

fax: +1716 645

E-mail addr

Computers & Geosciences 32 (2006) 912–926

www.elsevier.com/locate/cageo

Parallel adaptive discontinuous Galerkin approximationfor thin layer avalanche modeling$

A.K. Patraa,�, C.C. Nichitac, A.C. Bauera, E.B. Pitmanc,M. Bursikb, M.F. Sheridanb

aDepartment of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260, USAbDepartment of Geology, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA

cDepartment of Mathematics, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA

Accepted 26 October 2005

Abstract

This paper describes the development of highly accurate adaptive discontinuous Galerkin schemes for the solution of the

equations arising from a thin layer type model of debris flows. Such flows have wide applicability in the analysis of

avalanches induced by many natural calamities, e.g. volcanoes, earthquakes, etc. These schemes are coupled with special

parallel solution methodologies to produce a simulation tool capable of very high-order numerical accuracy. The

methodology successfully replicates cold rock avalanches at Mount Rainier, Washington and hot volcanic particulate

flows at Colima Volcano, Mexico.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Debris flow; Numerical simulation; Parallel computing; Discontinuous Galerkin methods; Depth average avalanche models

1. Introduction

In recent years a set of depth-averaged modelshave been developed for describing a class ofpotentially hazardous geophysical mass flows (seefor, e.g. Hutter et al., 1993; Gray, 1997; Iverson andDenlinger, 2001; Pitman et al., 2003). Such flowsmay arise in the aftermath of volcanic activity,earthquakes, floods, etc.

e front matter r 2006 Elsevier Ltd. All rights reserved

geo.2005.10.023

upported by National Science Foundation Grant

ing author. Tel.: +1 716 645 2593;

3875.

ess: [email protected] (A.K. Patra).

These models constitute a set of non-linearhyperbolic equations (strictly hyperbolic when theflow depth h40) and have been used to constructsimulations of flows on realistic terrains. In earlierpapers (Pitman et al., 2003; Patra et al., 2005) wedescribed finite volume schemes for solving thissystem using first and second order Godunovschemes. Numerical tests with those schemesindicated that our solutions were quite dependenton the choice of grid size and fairly large computa-tions were necessary to resolve even the simplest oftest problems.

Motivated by these results, we have developedand implemented a set of numerical schemes basedon an adaptive discontinuous Galerkin (DG)formulation, that promise very high resolution at

.

ARTICLE IN PRESSA.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926 913

minimal extra cost. Such schemes based on thepioneering work of Cockburn et al. (2000), havebeen remarkably successful in producing computa-tionally efficient solutions of several linear and non-linear hyperbolic systems (see for, e.g. Remacleet al., 2003 or Cockburn, 2002). In these methodsthe field variables are approximated by piecewisepolynomials whose order and support can be locallydefined. Thus, the cell sizes and approximatingpolynomials can be chosen to best capture theevolving flow. Unlike traditional finite elementapproximations, the approximation is allowed tobe discontinuous at inter-element boundaries. Thishas particular advantages when using distributedmemory parallel computers, since less data syn-chronization is required. Our development of theadaptive DG schemes closely follows the work ofHartmann and Houston (2002) on the Eulerequations of gas dynamics and Aizinger andDawson (2002) on classical shallow water equa-tions. We note also that a major benefit of using thediscontinuous Galerkin formulation is the avail-ability of well developed methodology for a poster-iori error estimation in both the field variables andalso in more specific quantities of interest.

In this paper, we will outline the basic develop-ment of the discontinuous Galerkin-type schemesfor the debris flow equations, develop a simpleadaptive strategy using a residual-based errorindicator and parallel solution techniques. While,the basic DG methodology has been available for afew years, its application to the thin layer models ofavalanche flow using extensions to the basicmethodology as explained below, grid adaptivitywith residual-based error indicators, integrationwith geographical information systems and parallelsolution methodology are among the new contribu-tions. The code successfully mimics dynamics anddeposition of natural cases including cold rockavalanches at Mount Rainier, Washington and hotvolcanic particulate flows at Colima Volcano,Mexico.

2. Mathematical modeling

We begin with the equations modeling mass andmomentum conservation for an incompressiblecontinuum in O � R3:

r � u ¼ 0, (1)

qðr0uÞ þ r � ðr0u� uÞ ¼ �r � Tþ r0g, (2)

where r0 is the density of the medium, g is thegravitational acceleration, T is the stress and u is thevelocity. Kinematic boundary conditions are im-posed at the free surface interface, of equationFsðx; tÞ ¼ sðx; tÞ � z ¼ 0, and at the basal surfaceinterface, with equation Fbðx; tÞ ¼ bðx; tÞ � z ¼ 0.

qtFs þ ðu � rÞFs ¼ 0 at Fsðx; tÞ ¼ 0, (3)

qtFb þ ðu � rÞFb ¼ 0 at Fbðx; tÞ ¼ 0. (4)

After defining appropriate rheology to relate thestresses to strain rates and velocities the abovesystems can be solved for appropriate initial andboundary conditions. Recognizing that the depth inthe z direction is much smaller than that in the x

and y directions (Oð1Þ compared to Oð103Þ), Hutteret al. (1993) greatly simplified the complexity of thesystem by a process of depth averaging and scalingto obtain a system of equations much like thosegoverning ‘‘shallow water’’. The shallowness as-sumption gives a ‘‘hydrostatic’’ equation for thenormal stresses in the z direction,

Tzz ¼ ðh� zÞrgz, (5)

which after depth averaging becomes a relation forthe depth averaged normal stress in the z direction,Tzz ¼

12rgzh. Using the Mohr–Coulomb theory, the

depth-averaged normal stresses Txx;Tyy can berelated to the normal stress Tzz, by using a lateralstress coefficient kap, so that

Txx ¼ Tyy ¼ kapTzz. (6)

The active or passive state of stress is developed ifan element of material is elongated or compressed,and the formula for the corresponding states can bederived from the Mohr diagram. It may be shownthat

kap ¼ 21� ½1� cos2 jintð1þ tan2 jbedÞ�

1=2

cos2 jint

� 1 (7)

in which ‘‘�’’ corresponds to an active state(qvx=qxþ qvy=qy40), respectively ‘‘þ’’ to the pas-sive state (qvx=qxþ qvy=qyo0).

The shear stresses Tyx;Txy can also be related tothe normal stresses Txx;Tyy, using a simplificationof the Coulomb (nonlinear) model to assume aconstant proportionality simplification, based on along history of such a practice in soil mechanicsRankine (1857), and an alignment of the stress axis.The equation for the lateral shear stresses can nowbe written as (Fig. 1)

Tyx ¼ Txy ¼ �sgnðqvx=qyÞkap12rgzh sinjint. (8)

ARTICLE IN PRESS

Fig. 1. Domain and partitionings.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926914

Finally the formula for the shear stress at thebasal surface Tzx can be derived from the basalsliding law. For curving beds this relation is

Tzx ¼ �vxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2x þ v2y

q rgzh 1þvx

rxgz

� �� �tanjbed , (9)

where rx is the radius of local bed curvature, and the‘‘�’’ indicates that basal Coulomb stresses opposebasal sliding. The relationship above is slightlymodified from the original in Iverson and Denlinger(2001) where sgnðvxÞ was used instead of

vx=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2x þ v2y

q. With this modification the friction

mobilized is in proportion to the velocity in thatdirection. This is particularly important when theflows differ significantly in the x and y dimensions,e.g. flow in a channel.

Now using the different boundary conditions anddepth averaging we obtain the system of equationsgoverning the flow of dry avalanches on arbitrarytopography in terms of conservative variables, invectorial form as

Ut þ FðUÞx þGðUÞy ¼ SðUÞ, (10)

where U¼ ðh; hvx; hvyÞt¼ ðu1; u2; u3Þ

t, F ¼ ðhvx; hv2xþ

0:5kapgzh2; hvxvyÞt;G¼ ðhvy; hvxvy; hv2yþ 0:5kapgzh

2Þt,

and S ¼ ð0;Sx;SyÞt and where

Sx ¼ gxh� hkap sgnqvx

qy

� �qyðgzhÞ sinjint

�vxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2x þ v2y

q gzh 1þvx

rxgz

� �� �tanjbed ,

Sy ¼ gyh� hkap sgnqvy

qx

� �qxðgzhÞ sinjint

�vyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2x þ v2y

q gzh 1þvy

rygz

� �� �tanjbed .

The components of the unknown vector U

represent pile height and two components for thedepth-averaged momentum. The above system ofequations is strictly hyperbolic for h40 and can besolved numerically by using standard techniques.

3. Runge–Kutta discontinuous Galerkin

approximations in space and time

3.1. DG formulation

We introduce now a sequence M0;M1; . . . ofpartitionings of the domain O such that each Mi ¼

fOKg whereS

KOK ¼ O and each OK is the image ofO ¼ ½�1; 1� � ½�1; 1� under a set of bijective map-pings F K defined as is customary in the finiteelement method. This partitioning can be used todefine an approximation space for the componentsof the field variables U

W K ¼ wjwjOK� PpðOK Þ;

[K

OK ¼ O

( ),

where PpðOK Þ is the set of polynomials of order pp

defined on OK . Thus hðx; tÞ ¼P

K

Pi hiK ðtÞwiK ðxÞ

for all x 2 O, wiK 2W K and t 2 ½0;TÞ. We note thatW K can be composed of arbitrary orders ofpolynomials, e.g. for x; z 2 ½�1; 1� W K ¼ ½1; x; z�leads to a linear approximation whileW K ¼ ½1; x; z; x

2; z2; xz� will lead to a quadraticapproximation.

We also define

wintðxÞ ¼ lims!0þ

wðxþ snK Þ, (11)

wextðxÞ ¼ lims!0�

wðxþ snK Þ, (12)

hwðxÞi ¼ 12ðwintðxÞ þ wextðxÞÞ, (13)

½wðxÞ� ¼ ðwintðxÞ � wextðxÞÞ, (14)

where nK is the outward pointing normal on theelement boundary qOK at x.

ARTICLE IN PRESSA.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926 915

Multiplying (10) by w ¼ ðw1;w2;w3Þ and integrat-ing over each element OK we haveZ

OK

qUqt

wdOK þ

ZOK

qFqx

wdOK þ

ZOK

qGqy

wdOK

¼

ZOK

SwdOK , ð15Þ

where, to simplify notations we used the conventionðSwÞi ¼ Siwi without the usual Einstein summation.

Using Green’s formulaeZOK

qUqt

wdOK �

ZOK

Fqwqx

dOK �

ZOK

Gqwqy

dOK

þ

IqOK

ðFnx þGnyÞwds ¼

ZOK

SwdOK . ð16Þ

Assembling over all the elements and definingG ¼

SK qOKnqOX

K

ZOK

qUqt

wdOK �

ZOK

Fqwqx

dOK

��

ZOK

Gqwqy

dOK

�

ZOK

SwdOK

�þ

IqOðFnx þGnyÞwds

þ

IG½ðFnx þGnyÞw�ds ¼ 0. ð17Þ

To describe the solution scheme we now intro-duce the following notations. Let A denote the 3� 2matrix (F G), or on components Ai;1 ¼ Fi andAi;2 ¼ Gi. To simplify notations we write theequations resulting from (16) on componentsZ

OK

quj

qtwdOK �

ZOK

Xj

Aðj;:ÞðUÞ � rwdOK

þ

IqOK

Xj

Aðj;:ÞðUÞ � nwds

¼

ZOK

SjðUÞwdOK , ð18Þ

Nðuj ;wÞ ¼ ðSj ;wÞ. (19)

We are looking for an approximation Uh 2

W K ;Uh ¼ ðuh

j Þ; j ¼ 1 . . . 3; for the state variablesU 2 ðL1½0;TÞÞ3 � L2ðOÞ so thatZ

OK

quhj

qtwh dOK �

ZOK

Xj

Aðj;:ÞðUhÞ � rwh dOK

þ

IqOK

Xj

Aðj;:ÞðUhÞ � nwh ds

¼

ZOK

SjðUhÞwh dOK ; 8w

h 2W K . ð20Þ

When we assemble these equations over allelements and use the notations from (16)

XK

ZOK

quhj

qtwh dOK �

ZOK

Xj

Aðj;:ÞðUhÞ � rwh dOK

( )

þ

IqO

Xj

Aðj;:ÞðUhÞ � nwh ds

þ

IG

Xj

½Aðj;:ÞðUhÞ � nwh�ds

¼

ZOK

SjðUhÞwh dOK 8wh 2W K . ð21Þ

3.2. Fluxes and slope limiters

3.2.1. Fluxes

The approximate solution Uh may be discontin-uous across the element interface and therefore weneed to approximate the integral containing thephysical flux

HG

Pj ½Aðj;:ÞðU

hÞ � nwh�ds by a numer-ical flux times the average of the test functionvalues across the element interface

HGZðU

hint

ðxÞ;Uhext

ðxÞÞhwhids where, Uhint

ðxÞ and Uhext

ðxÞ aredefined as in (11) and (12) and where Uhext

ðxÞ isreplaced by the appropriate boundary value onqO \ qOK .

The numerical flux function Z must be a two-point monotone function (nondecreasing withrespect to the second argument, nonincreasing withrespect to the first) which is consistent andconservative. We use here the HLL fluxes describedin Toro (1997) and tested in our finite differencesnumerical code (Patra et al., 2005). We brieflydescribe these fluxes below.

The Riemann problem at the element interfaceswith the left and the right states given by UhðintÞ,and UhðextÞ, respectively, has characteristic speedswhich are given by the eigenvalues of the Jacobianmatrix of F for the x-direction and by theeigenvalues of G for the y direction (see for e.g.Leveque, 1992). For the x-direction these are givenby ðuþ c; u; u� cÞ;where c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffikapgzh

p, and where

gz is the component of the gravitational accelerationnormal to the basal surface and kap is the active/passive coefficient of the depth-averaged theory byHutter et al. (1993). To propagate information inthe x-direction we estimate the signal velocities inthe solution of the Riemann problem by thefollowing choice proposed by Davis (1998) whereuint; uext are the ‘‘left’’ and ‘‘right’’ values of the x

ARTICLE IN PRESSA.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926916

components of the velocity, hint; hext are the ‘‘left’’and ‘‘right’’ values of height. These values areobtained by interpolation to the appropriate loca-tion on the element boundary from the approximatesolution element and it’s neighbor, respectively:

Cint1 ¼ minð0;minðuext � cext; uint � cintÞÞ, (22)

Cext1 ¼ maxð0;maxðuext þ cext; uint þ cintÞÞ. (23)

Similarly,

Cint2 ¼ minð0;minðvext � cext; vint � cintÞÞ, (24)

Cext2 ¼ maxð0;maxðvext þ cext; vint þ cintÞÞ (25)

are the signal velocities in the y-direction.The function Z giving the numerical flux will have

the form

ZðUhint

ðxÞ;Uhext

ðxÞ; nÞ

¼X

j

fCextj Aðj;:ÞðU

hext

Þ � n� CljAðj;:ÞðU

hint

Þ � n

þ Cintj Cext

j ðUhext

�Uhint

Þg=Cextj � Cint

j . ð26Þ

Flow fronts occur when zero flow depth existsadjacent to a cell with nonzero flow depth. Theerrors in front propagation speeds at flow fronts canbe very large, and more accurate estimates forspeeds are needed in such cases. For a front movingin the positive x direction cext ¼ hext

¼ 0, and thecorrect solution consists of a single rarefaction waveassociated with the left eigenvalue. The front itselfcorresponds to the tail of the rarefaction moving tothe ‘‘left’’ and has exact propagation speeduext ¼ uint þ 2cint. This problem is similar to theproblem involving vacuum states in shock tubes,and the rationale for this approach is discussed inToro (1997).

Finally, the system of equations becomes

XK

ZOK

quhj

qtwh dOK �

ZOK

Xj

Aðj;:ÞðUhÞ � rwh dOK

( )

þ

IqO

Xj

Aðj;:ÞðUhÞ � nwh ds

þ

IGZðUhint

ðxÞ;Uhext

ðxÞ; nÞhwhids

¼

ZOK

SjðUhÞwh dOK 8wh 2W K . ð27Þ

The integrals may be evaluated using quadrature,and the equations may be written as a system of

differential equations in time, which has the form

d

dtUh ¼ LðUhÞ. (28)

This system can be solved using a total variationdiminishing in the means (TVDM) RK timediscretization. However, as we describe next, theapproximate solution at every stage must bemodified by a process of slope limiting to eliminatespurious oscillations.

3.2.2. Slope limiting

The numerical scheme derived by directly inte-grating 28 does not provide an approximatesolution that satisfies the TVDM property. Wediscuss our strategy for slope limiting in the case ofpiecewise linear approximations, since the samealgorithm applies to higher-order approximations.The slope limiter must maintain the conservation ofmass, satisfy the sign conditions that prevent totalvariation from increasing, and must not degrade theaccuracy of the method.

In the global coordinate system, the piecewiselinear approximation of the solution is

Uh ¼ Uh þ ðx� x0ÞðUxÞ

hþ ðy� y0ÞðU

yÞh, (29)

where Uh are the cell averages of the systemvariables, ðUxÞ

h and ðUyÞh are the coefficients of

linear shape functions used to construct the localapproximation of the exact solution and ðx0; y0Þ arethe coordinates of the center of the current elementO, i.e. it is the map of the centroid of OK andFK ð0; 0Þ ¼ ðx0; y0Þ. We use the generalized slopelimiter of the MUSCL schemes by Van Leer, whichis described in detail in Cockburn (2002).

Slope limiting for systems must be performed inthe local characteristic variables (Cockburn, 2002).To achieve this we need the characteristic speedsand direction for wave propagation, that is theeigenvalues and the eigenvectors of the compositematrix Q ¼Mnx þNny where M and N are theJacobian matrices of F and G. For our choice ofrectangular shaped elements, the grid is aligned withthe x and y coordinates. For the x-component of theflux the composite matrix reduces to M, and similartreatment applies to y-fluxes with the Jacobianmatrix M replaced by N. Here we describe thex-component case in detail.

The essence of the limiting process is to comparethe slopes of the approximate solution computed

ARTICLE IN PRESSA.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926 917

directly from the piecewise linear approximationwith finite difference approximations of the slopesobtained by comparing cell averages of neighboring

cells. Hence the quantities of interest are ðUxÞh;

ððUhÞright�UhÞ=ðxright

0 � x0Þ, and ðUh � ðUhÞleftÞ=

ðx0 � xleft0 Þ, where the left, right suffixes denote

neighboring elements to the negative x direction andthe positive x direction of the current element. Let Rbe the matrix having the eigenvectors of M ascolumns. These vectors can be transformed to thecoordinate system of the characteristic variables bymultiplying them with the inverse of R. Let

R�1Uxh¼ mxh. The limiter is then applied to each

component, i.e.

ðmxj Þ

hlim ¼ m ðmx

j Þh;ðmh

j Þright� ðmh

j Þ

xright0 � x0

;mh

j � ððmhj ÞÞ

left

x0 � xleft0

0@

1A,

(30)

where the mð:; :; :Þ is the usual minmod functiondefined by

mða1; a2; . . . ; anÞ

¼

smin1pnpnjanj if s ¼ signða1Þ

¼ � � � ¼ signðanÞ;

0 otherwise:

8>><>>:

The limited coefficients ðmxÞhlim are transformed back

to the original coordinate system by multiplication

with the matrix R, and we denote them by ðUxÞhlim.

The procedure for the y-component follows thesame procedure with appropriate neighbors.

The approximate solution after slope limitingprocess is complete will then be

Uhðx; yÞ ¼ Uhj þ ðx� x0ÞðU

xÞhlim þ ðy� y0ÞðU

yÞhlim.

(31)

The approximate solution obtained from Eq. (31) issaid to be limited and it is denoted by LUh. Nowthis slope limited approximation can be used in atime integration scheme.

3.3. Second-order Runge– Kutta discretization in time

We implemented a second-order Runge–Kutta(RK) algorithm and we investigated piecewise linearsolutions. Cockburn et al. (2000) has establishedthat if pth order basis functions are used in spacethen we require ðpþ 1Þth order RK schemes in time

to maintain a balance in the errors in time and spacediscretization.

The algorithm for second order (p ¼ 1) can bewritten as follows:

Uhð1Þ ¼ Uh

ðnÞ þ Dt LðLUhðnÞÞ, (32)

Wh ¼ Uhð1Þ þ Dt LðLUh

ð1ÞÞ, (33)

Uhðnþ1Þ ¼

12ðUhðnÞ þWhÞ. (34)

Higher-order versions involve additional stages withdifferent coefficients as documented in Cockburn etal. (2000).

4. Computational issues

4.1. Adaptive strategies

The adaptivity in this simulation has three goals:

�

A survey of the literature on a posteriori errorestimation (see for, e.g. Hartmann and Houston,2002) reveals that the numerical approximationerror may be well controlled by controlling appro-priate norms of the residuals defined below. If u isthe exact solution of (18) and uh is the approximatenumerical solution computed from (20) then theelement-wise residual is defined by

RK ðwÞ ¼ ðNðu;wÞ �Nuh;wÞ � ððSðuÞ;wÞ � ðSðuhÞ;wÞÞ.

(35)

Clearly RK ðwÞ cannot be computed and must beestimated. Techniques for such estimates (in amultitude of norms) are well developed (see for,e.g. Hartmann and Houston, 2002). We note that asignificant contribution to the residual in this casewill be related to the jump in the physical fluxes.Thus we can define a primary indicator of numericalapproximation error bK as

b2K ¼I

qOK

½F ðUÞ � w�2 þ ½GðUÞ � w�2 ds. (36)

To accomplish the other two goals of adaptivitythough, we need more information than is con-tained in this indicator. Measures of change intopography and techniques that refine the grid atthe flow front are also necessary. For capturing the

ARTICLE IN PRESS

Fig. 2. Partitioning of sample domain into 4 partitions for

parallel computing. Note sample space-filling curve ordering and

it’s 4-way dissection to obtain the partitioning and a layer of

ghost cell along partition lines.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926918

front we have implemented a scheme in which cellsat the front are explicitly tracked (by monitoring thechange in flow depth among neighboring cells) andrefined. While, we have not implemented adaptivitybased on terrain features, we update terraininformation at finer resolutions as the mesh isupdated.

Similarly unrefinement schemes have been im-plemented to remove cells that are not active in thecomputation. When the flow has proceeded througha region and the flow depths and momentums havebeen reduced below a threshold we unrefine thecells.

4.2. Parallel computing

We will use parallel processing to enable us tosolve the very large systems necessary for highfidelity computations on realistic terrain. Ourapproach to parallel processing is to use dataparallel computations. The particular challengesdue to mesh adaptivity and consequently terrainadaptivity are surmounted using the ideas discussedin Laszloffy et al. (2000); Patra et al. (2003) andmore recently adapted for thin layer granular flowmodels using adaptive finite difference schemes inPatra et al. (2005).

The central idea is to organize data and computa-tions using a space filling curve based ordering ofthe cells. Parallel decomposition for p processors isachieved by a p-way partitioning of this orderingwith work associated with the cells in a partitionbeing undertaken by a single processor (see Fig. 2for an illustration). Data from a layer of cells ateach partition boundary needs to be available to theprocessor computing the neighboring partition.Hence we create a layer of ‘‘ghost’’ cells along eachpartition boundary which is also made available tothe neighboring processor. Upon completion ofeach time step of computation data associated withthese cells must be exchanged among processors. Asthe flow evolves and the adaptation pattern changesthe new cells are introduced in the ordering and thepartition boundaries are adjusted to reflect this.Cells are then migrated to new processors to reflectthe new partitioning.

4.3. Integration with geographical information

systems

To model flows on natural terrain we haveintegrated our simulation codes with appropriate

geographical information system tools. The toolautomatically extracts the required elevation, slopeand curvature data from standard digital elevationmodels. Details of the many issues involved inmaking this linkage are described in our earlierwork (Patra et al., 2005). The highlight of ourmethodology is that the interpolation used togenerate the elevations is matched to the size ofthe computational grid to avoid spurious artifacts.Secondly, as the grid is locally refined finertopographical details are obtained from the data-base resulting in clear definition of channels andother sharp features resulting smaller modelingerrors.

5. Numerical tests and validation

We will now present a series of numerical teststhat were used to validate the new schemes.

5.1. Flows down ramps

In the first set of tests we will use flow of a pile ofsand down simple ramps (a standard table-topexperiment). We will simulate the flow of approxi-mately 425 g of sand (volume approximately 2:7�10�03 m3 sliding down a ramp at 44:3 (see Fig. 3 fordetails). The results of this experiment and its use invalidating the TITAN2D tool are documented inour earlier work of Patra et al. (2005). An interiorfriction angle of 37:3 and bed friction angle of

ARTICLE IN PRESSA.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926 919

32:48 are used in the simulations and are in linewith experimental measurements. We will nowcompare the results from our new schemes withthose from using the finite volume schemes (Patraet al., 2005). Fig. 4 shows a series of frames in whichthis comparison is illustrated. In both cases we startthe computation with 100 grid cells and allowadaptivity (refinement and unrefinement based onthe adaptive strategy) to control the number of cellsat any time step.

The adaptive strategy for the finite volume isbased largely on heuristics (as explained in Patra

0.603

1.20.42

44.3 deg

0.5

1.0

(a) (b)

Fig. 3. Geometry of two ramps used in testing the code. Ramp

on the right gradually curves into a flat plane (b) while ramp on

left has a sharp turn into a flat surface (a).

Fig. 4. Flow of a parabolic pile of sand down a 44:3 ramp onto a fl

pictures) and the new discontinuous Galerkin schemes. Finite volume r

been compared to experiments. Flows are simulated with adaptive grid

angle of 32:47. Flow depths are substantially similar in both flows but

et al., 2005) while the residual-based bK defined inEq. (36) provides a more systematic and mathema-tically consistent basis for refinement. The solutionsare similar in many aspects. However, the framesclearly show the higher resolution of the DG-basedscheme. Hence, our confidence in the numericalcorrectness of this new solution scheme is greatlyreinforced by these results. Next, Fig. 5 showscomparisons of the extents (spreads) to some simpletable top experiments described in Patra et al.(2005). The simulations are quite good early on fortime less than 0.6 s when the flow is on the inclinedpart of the ramp. After reaching the flat part of theramp the correlations are not good. The experi-mental observations of x extents (distance betweenhead and tail) seems to increase to a much highervalue before finally collapsing to a value close tothat reached by the simulation. Upon carefulexamination of the images of the experiment wenotice that a very thin layer on the inclined portionof the ramp causes the tail of the flow in theexperiment to be located further back than in thesimulation. In the simulation also we see a very thinOð10�5 mÞ thick layer which matches this layer.However, this thickness is too small and usuallyneglected as nonphysical. Similarly for the y extentsthe experimental values increase quite rapidly as it

at surface simulated using both finite volume schemes (bottom

esults are from the TITAN2D code (Patra et al., 2004) and have

s and choices of interior friction angle of 37:3 and basal friction

the new schemes show better resolution of flow depth contours.

ARTICLE IN PRESS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

X-E

xten

t=H

ead

- T

ail

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time

DG Simulation

X-extent FD50 adaptive grid

X-extent Expt.

0

0.2

0.4

0.6

0.8

1

Y-E

xten

t

DG Simulation

Experimental Data

FD Simulation

Fig. 5. Flow of a parabolic pile of sand down a 44:3 ramp onto a flat surface simulated using both finite volume schemes (bottom

pictures) and the new discontinuous Galerkin schemes compared to experiments. Finite volume results are from the TITAN2D code

(Patra et al., 2004). Flows are simulated with adaptive grids and choices of interior friction angle of 37:3 and basal friction angle of 32:47.

Fig. 6. Maximum pile height versus time for flow of a parabolic

pile of sand down a ramp of 44:3 and choices of interior friction

angle of 37:3 and basal friction angle of 32:47. Flow depths are

substantially similar in both flows until flow hits flat part of ramp

where DG computation yields a flow that is much less spread out

and hence has a higher pile height.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926920

reaches the bottom of the ramp; the simulatedextents are initially smaller but over time seem to betrending towards these higher numbers. We hy-pothesize that this discrepancy between simulationand experiment is either due to the sharp change interrain slope and curvature as the flow moves fromthe inclined part to the flat part or due to thenumerical error introduced in computing flowswith very thin layer since estimates of flow speedu; v used in computing the transport are derivedfrom the depth-averaged momentum hu; hv bydividing with h.

In the next figure we make a detailed comparisonof the maximum pile height versus time for both thenew and finite volume schemes. The maximum pileheights versus time are remarkably similar until wereach the flat part of the ramp where the heights aremuch higher for the new DG schemes. The greaterresolution of the DG scheme and/or the differentcomputations for the source terms involving frictionare possible reasons for the differences. We notethat the solution scheme in TITAN2D is substan-tially similar to those in Denlinger and Iverson(2001).

In the next set of tests down a curvilinear ramp(shown in Figs. 6–8), we plot the flow depth,adaptive mesh and error indicator bK . The evolvingmesh designed to capture the flow accurately isclearly seen. The mesh is highly refined at the front(cells containing the interface of zero flow depth andnonzero flow depth and in the interior where theindicator bK is high. Note that this does notnecessarily coincide with areas of high flow depthas is seen in frames labeled (e) and (f) in Fig. 8 where

the stable center area is not highly refined but therapidly moving outside of the pile is. bK is a measureof the local numerical error in the computation.Thus, high values of bK are expected in areas wherethere is more rapid flow and smaller grid cellsare required to resolve the flows. In the next figureFig. 9 we display the evolution of a sample paralleladaptive mesh and its partitioning for parallelcomputing on four processors. Proper partitioningof the cells is required for efficiently computing onmulti-processor machines.

ARTICLE IN PRESS

Fig. 7. Initial stages of grid for flow of a parabolic pile of sand down a curved ramp. Colors used to indicate level of field variable in cell.

The contours on left on each panel shows flow depth (pile height) and contours on right labeled flux show error indicator bK used in

adapting the grid. Zoom on top left shows details of adaptive grid.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926 921

ARTICLE IN PRESS

Fig. 8. Final stages of grid for flow of a parabolic pile of sand down a curved ramp. Colors used to indicate level of field variable in cell.

The contours on the left on each panel shows flow depth (pile height) and contours on right labeled flux show error indicator bK used in

adapting grid. Zoom on top left shows details of adaptive grid.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926922

ARTICLE IN PRESS

Fig. 9. Evolution of sample adaptive mesh and partitioning for parallel computing using four processors. Grid line colors indicate

processor and contours indicate flow depth.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926 923

5.2. Flows on natural terrain

In the next two tests we test the new methodologyon sample digital elevation models of little Tahomapeak and Colima volcano.

In 1963 a series of 7 avalanches occurred at LittleTahoma Peak on Mount Rainier, Washington(Fahnestock, 1963; Norris, 1994). The avalanches,totaling approximately 1:1� 107 m3 of broken lavablocks and other debris, traveled 6.8 km horizon-tally and fell 1.8 km vertically ðH=L ¼ 0:246Þ.Velocities calculated from run-up range from 24 to42m/s and may have been as high as 130m/s whilethe avalanches passed over Emmons Glacier Cran-dell and Fahnestock (1965). The avalanches formeda total deposit thickness of 30m near their distalterminus where they ponded against a terminalmoraine. Because topographic surveys were madeboth before (by Fahnestock, 1963) and after (byCrandell and Fahnestock, 1965) the event, variousaspects of the flowing avalanche and its deposits arewell documented. For this reason we have used theLittle Tahoma Peak avalanches to calibrate themodel here and in earlier work for similar typesmedium-sized rock avalanches (Sheridan et al.,2005).

A series of tests were made with simulated flowsof 9:4� 106 m3 volume. This value approximatesthe size of individual avalanches at Little TahomaPeak. The length of the simulated flow runout wascalibrated to 6.8 km, matching the actual ava-lanches, by adjusting the basal friction angle and

internal friction angle. Best results were obtainedwith values of 10 and 30, respectively. Fig. 10 showsresults from a run of the code as a series of timesteps. The simulation results compare well with theactual flows in terms of: (1) lateral extent of theflowing avalanche, (2) area of the actual deposit, (3)run-up at bends in the flow path, (4) flow velocity,and (5) maximum thickness of the deposit.

The flow boundaries of the moving mass from thesimulations fit reasonably well with the mappedextent of the avalanches Sheridan et al. (2005). Thearea of the mapped deposits is 1:3 km2 comparedwith 0:6 km2 for the simulation. The run-up heightsfor the avalanches was 40–90m whereas thesimulation run-up was 60m. The maximum velocityof the avalanches calculated from super elevation atbends is 140m/s whereas the simulation gave a valueof maximum velocity ranging from 80 to 150m/s.The total deposit thickness of the seven actualavalanches is 30m (4.4m average thickness) wherethe model results gave 3.6m for a single flowthickness.

A second set of simulations is conducted onterrain data from Colima volcano. Results ofsimulations for a parabolic pile of volume9:384405� 106 m3 centered at the UTM coordinatesð644935; 2171380Þ and of extent 200m and max-imum height 150m flowing down the volcano areshown in Fig. 11. The flows appear to channelizeappropriately and splits among the multiple chan-nels. We predict a maximum velocity in the range121–174m/s.

ARTICLE IN PRESS

Fig. 10. Simulated flow of a debris pile of volume 9:4� 106 m3 using digital elevation models of little Tahoma peak. Basal friction angle of

10 and internal friction angle of 30 are used in calculation. Contours indicate flow depth at different times. Comparisons with field

observations yield very good correlations with deposits after a series of avalanches in 1963.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926924

6. Conclusions and future work

In this paper, we have described the developmentof highly accurate adaptive discontinuous Galerkinschemes for the solution of the equations arisingfrom a thin layer-type model of debris flows. Theseschemes are then coupled with special solution

methodologies to produce a simulation tool capableof very high-order numerical accuracy. The tool isthen applied to several test problems to illustrate theuse of these schemes.

The new schemes outlined here will enable severallines of future work. Most prominent among thesewill be the development of classes of a posteriori

ARTICLE IN PRESS

Fig. 11. Simulated flow of a debris pile of volume 9:384405� 106 m3 using digital elevation models of Colima volcano. The superposed

grid shows adaptive grid with finer resolutions to capture flow features. The flow is initiated at UTM coordinate ð644935:1; 2171380:25Þand attains maximum velocities of 121–174m/s. Flow reaches the state in final frame after 2min and 18 s.

A.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926 925

error estimates for the numerical approximationerror and control thereof leading to robust andreliable simulations. Strategies for exploiting thelocal adaptivity features of this tool will also need tobe better developed.

References

Aizinger, V., Dawson, C., 2002. A discontinuous Galerkin

method for two dimensional flow and transport in shallow

water. Advances in Water Resources 25, 67–84.

Crandell, D.R., Fahnestock, R.K., 1965. Rockfalls and ava-

lanches from Little Tahoma Peak on Mount Rainier,

Washington. US Geological Survey Bulletin 1221-A,

A1–A30.

Cockburn, B., 2002. Discontinuous Galerkin methods for

convection dominated problems, preprint.

Cockburn, B., Karniadakis, G., Shu, C., 2000. The development

of discontinuous Galerkin methods. In: Cockburn, B.,

Karniadakis, G., Shu, C. (Eds.), Discontinuous Galerkin

Methods. Springer, Heidelberg, pp. 3–50.

Davis, S.F., 1998. Simplified second order Godunov type

methods. SIAM Journal Scientific and Statistical Computing

9, 445–473.

ARTICLE IN PRESSA.K. Patra et al. / Computers & Geosciences 32 (2006) 912–926926

Denlinger, R.P., Iverson, R.M., 2001. Flow of variably fluidized

granular material across three-dimensional terrain 2. Numer-

ical predictions and experimental tests. Journal of Geophy-

sical Research 106, 533–566.

Fahnestock, R.K., 1963. Morphology and hydrology of a glacial

stream—White River, Mount Rainier, Washington. US

Geological Survey Professional Paper 422-A, pp. A1–A70.

Gray, J.N.M.T., 1997. Granular avalanches on complex topo-

graphy. In: Fleck, N.A., Cocks, A.C.F. (Eds.), Proceedings

of IUTAM Symposium on Mechanics of Granular and

Porous Materials. Kluwer Academic Publishers, Dordrecht,

pp. 275–286.

Hartmann, P., Houston, P., 2002. Adaptive discontinuous

Galerkin finite element methods for nonlinear hyperbolic

conservation laws. SIAM Journal of Scientific Computing 24

(3), 979–1004.

Hutter, K., Siegel, M., Savage, S.B., Nohguchi, Y., 1993. Two

dimensional spreading of a granular avalanche down an

inclined plane; Part 1: Theory. Acta Mechanica 100, 37–68.

Iverson, R.M., Denlinger, R.P., 2001. Flow of variably fluidized

granular material across three-dimensional terrain 1. Cou-

lomb mixture theory. Journal of Geophysical Research 106,

537–552.

Laszloffy, A., Long, J., Patra, A.K., 2000. Simple data manage-

ment, scheduling and solution strategies for managing the

irregularities in parallel adaptive hp finite element simula-

tions. Parallel Computing 26, 1765–1788.

LeVeque, R.J., 1992. Numerical Methods for Conservation

Laws. Birkhauser Verlag, Berlin.

Norris, R.D., 1994. Seismicity of rockfalls and avalanches at

three Cascade Range volcanoes: implications for seismic

detection of hazardous mass movements. Bulletin of the

Seismological Society of America 84, 1925–1939.

Patra, A., Laszloffy, A., Long, J., 2003. Data structures and load

balancing for parallel adaptive hp finite element methods.

Computers and Mathematics with Applications 46, 105–123.

Patra, A.K., Bauer, A.C., Nichita, C.C., Pitman, E.B., Sheridan,

M.F., Bursik, M., Rupp, B., Webber, A., Stinton, A.J.,

Namikawa, L.M., Renschler, C.S., 2005. Parallel adaptive

numerical simulation of dry avalanches over natural terrain.

Journal of Volcanology and Geothermal Research 139 (1),

1–22.

Pitman, E.B., Nichita, C.C., Patra, A., Bauer, A., Sheridan,

M.F., Bursik, M., 2003. Computing granular avalanches and

landslides. Physics of Fluids 15 (12), 3638–3647.

Rankine, W.J.M., 1857. On the stability of loose earth.

Philosophical Transactions of the Royal Society of London

147, 9–27.

Remacle, J.F., Flaherty, J., Shephard, M., 2003. An adaptive

discontinuous Galerkin technique with an orthogonal basis

applied to compressible flow problems. SIAM Review 45 (1),

53–72.

Sheridan, M.F., Stinton, A.J., Patra, A., Pitman, E.B., Bauer, A.,

Nichita, C.C., 2005. Evaluating TITAN2D mass-flow model

using the 1963 Little Tahoma Peak avalanches, Mount

Rainier, Washington. Journal of Volcanology and Geother-

mal Research 139 (1), 89–102.

Toro, E.F., 1997. Riemann Solvers and Numerical Methods for

Fluid Dynamics. Springer, Berlin, 648pp.