Parallel Adaptive Discontinuous Galerkin

Approximation For Thin Layer Avalanche

Modeling 1

A. K. Patra a,∗ , C. C. Nichita c, A. C. Bauer a,E. B. Pitman c, M. Bursik b and M.F. Sheridan b

a Department of Mechanical and Aerospace Engineering,University at Buffalo, TheState University of New York, Buffalo NY 14260

b Department of Geology, University at Buffalo, The State University of NewYork, Buffalo NY 14260

c Department of Mathematics, University at Buffalo, The State University of NewYork, Buffalo NY 14260

Abstract

This paper describes the development of highly accurate adaptive discontinuousGalerkin schemes for the solution of the equations arising from a thin layer typemodel of debris flows. Such flows have wide applicability in the analysis of avalanchesinduced by many natural calamities e.g. volcanoes, earthquakes etc. These schemesare coupled with special parallel solution methodologies to produce a simulationtool capable of very high order numerical accuracy.The methodology successfullyreplicates cold rock avalanches at Mount Rainier, Washington and hot volcanicparticulate flows at Colima Volcano, Mexico.

1 Introduction

In recent years a set of depth averaged models have been developed for describing a class ofpotentially hazardous geophysical mass flows (see for e.g. Hutter et. al. (1993); Gray (1997);Iverson and Denlinger (2001); Pitman et. al. (2003)). Such flows may arise in the aftermathof volcanic activity, earthquakes, floods etc.

∗ Corresponding author: 605 Furnas Hall, Department of Mechanical and Aerospace Engineering,University at Buffalo, Buffalo, NY 14260, Tel/FAX 716-645-2593/3875.

Email address: abani@eng.buffalo.edu (A. K. Patra).1 Research supported by Natioanl Science Foundation Grant ACI-0121254

Preprint submitted to Computers and Geosciences 11 November 2004

These models constitute a set of non-linear hyperbolic equations (strictly hyperbolic when theflow depth h > 0) and have been used to construct simulations of flows on realistic terrains.In earlier papers (Pitman et. al. (2003); Patra et. al. (2004)) we described finite volumeschemes for solving this system using first and second order Godunov schemes. Numericaltests with those schemes indicated that our solutions were quite dependent on the choice ofgrid size and fairly large computations were necessary to resolve even the simplest of testproblems.

Motivated by these results, we have developed and implemented a set of numerical schemesbased on an adaptive discontinuous Galerkin (DG) formulation, that promise very high res-olution at minimal extra cost. Such schemes based on the pioneering work of Cockburn et.al. (2000), have been remarkably successful in producing computationally efficient solu-tions of several linear and non-linear hyperbolic systems (see for e.g. Remacle et. al. (2003)or Cockburn (2002)). In these methods the field variables are approximated by piecewisepolynomials whose order and support can be locally defined. Thus, the cell sizes and ap-proximating polynomials can be chosen to best capture the evolving flow. Unlike traditionalfinite element approximations, the approximation is allowed to be discontinuous at inter-element boundaries. This has particular advantages when using distributed memory parallelcomputers, since less data synchronization is required. Our development of the adaptive DGschemes closely follows the work of Hartmann and Houston (2002) on the Euler equations ofgas dynamics and Aizinger and Dawson (2002) on classical shallow water equations. We notealso that a major benefit of using the discontinuous Galerkin formulation is the availabilityof well developed methodology for a posteriori error estimation in both the field variablesand also in more specific quantities of interest.

In this paper, we will outline the basic development of the discontinuous Galerkin typeschemes for the debris flow equations, develop a simple adaptive strategy using a residualbased error indicator and parallel solution techniques. While, the basic DG methodology hasbeen available for a few years, its application to the thin layer models of avalanche flow usingextensions to the basic methodology as explained below, grid adaptivity with residual basederror indicators, integration with geographical information systems and parallel solutionmethodology are among the new contributions. The code successfully mimics dynamics anddeposition of natural cases including cold rock avalanches at Mount Rainier, Washingtonand hot volcanic particulate flows at Colima Volcano, Mexico.

2 Mathematical Modeling

We begin with the equations modeling mass and momentum conservation for an incompress-ible continuum in Ω ⊂ R3:

∇ · u = 0 (1)

∂(ρ0u) +∇ · (ρ0u⊗ u) = −∇ ·T + ρ0g (2)

2

where ρ0 is the density of the medium, g is the gravitational acceleration,T is the stress andu is the velocity.Kinematic boundary conditions are imposed at the free surface interface,of equation Fs(x, t) = s(x, t) − z = 0, and at the basal surface interface, with equationFb(x, t) = b(x, t)− z = 0.

∂tFs + (u · ∇)Fs = 0 at Fs(x, t) = 0 (3)

∂tFb + (u · ∇)Fb = 0 at Fb(x, t) = 0 (4)

After defining appropriate rheology to relate the stresses to strain rates and velocities theabove systems can be solved for appropriate initial and boundary conditions. Recognizingthat the depth in the z direction is much smaller than that in the x,and y directions (O(1)compared to O(103) ), Hutter et. al. (1993) greatly simplified the complexity of the systemby a process of depth averaging and scaling to obtain a system of equations much like thosegoverning “shallow water”. The shallowness assumption gives a “hydrostatic” equation forthe normal stresses in the z direction,

Tzz = (h− z)ρgz (5)

which after depth averaging becomes a relation for the depth averaged normal stress in the zdirection, T zz = 1

2ρgzh. Using the Mohr-Coulomb theory, the depth averaged normal stresses

T xx, T yy can be related to the normal stress T zz, by using a lateral stress coefficient kap, sothat

T xx = T yy = kapT zz (6)

The active or passive state of stress is developed if an element of material is elongated orcompressed, and the formula for the corresponding states can be derived from the Mohrdiagram. It may be shown that

kap = 21± [1− cos2 ϕint(1 + tan2 ϕbed)]

1/2

cos2 ϕint

− 1 (7)

in which “-” corresponds to an active state (∂vx/∂x + ∂vy/∂y > 0), respectively “+” to thepassive state(∂vx/∂x + ∂vy/∂y < 0).

The shear stresses T yx, T xy can also be related to the normal stresses T xx, T yy, using asimplification of the Couloumb (nonlinear) model to assume a constant proportionality sim-plification, based on a long history of such a practice in soil mechanics Rankine (1857),and an alignment of the stress axis. The equation for the lateral shear stresses can now bewritten as:

T yx = T xy = −sgn(∂vx/∂y)kap1

2ρgzh sin ϕint (8)

3

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

Tzx = − vx√v2

x + v2y

[ρgzh

(1 +

vx

rxgz

)]tan ϕbed (9)

where rx is the radius of local bed curvature, and the “-” indicates that basal Coulombstresses oppose basal sliding. The relationship above is slightly modified from the originalin Iverson and Denlinger (2001) where sgn(vx) was used instead of vx√

v2x+v2

y

. With this

modification the friction mobilized is in proportion to the velocity in that direction. This isparticularly important when the flows differ significantly in the x and y dimensions e.g. flowin a channel.

Now using the different boundary conditions and depth averaging we obtain the system ofequations governing the flow of dry avalanches on arbitrary topography in terms of conser-vative variables, in vectorial form as:

Ut + F(U)x + G(U)y = S(U) (10)

where U = (h, hvx, hvy)t = (u1, u2, u3)

t, F = (hvx, hv2x + 0.5kapgzh

2, hvxvy)t,

G = (hvy, hvxvy, hv2y + 0.5kapgzh

2)t , and S = (0, Sx, Sy)t and where

Sx = gxh− hkapsgn(∂vx

∂y)∂y(gzh) sin ϕint −

vx√v2

x + v2y

[gzh

(1 +

vx

rxgz

)]tan ϕbed

Sy = gyh− hkapsgn(∂vy

∂x)∂x(gzh) sin ϕint −

vy√v2

x + v2y

[gzh

(1 +

vy

rygz

)]tan ϕbed

The components of the unknown vector U represent pile height and two components for thedepth averaged momentum. The above system of equations is stictly hyperbolic for h > 0and can be solved 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... of partitionings of the domain Ω such that eachMi = ΩK where ∪KΩK = Ω and each ΩK is the image of Ω = [−1, 1] × [−1, 1] undera set of bijective mappings FK defined as is customary in the finite element method. This

4

Fig. 1. Domain and partitionings

partitioning can be used to define an approximation space for the components of the fieldvariables U

WK = w|w|ΩK⊂ Pp(ΩK),∪KΩK = Ω

where Pp(ΩK) is the set of polynomials of order ≤ p defined on ΩK . Thus h(x, t) =∑K

∑i hiK(t)wiK(x) for all x ∈ Ω, wiK ∈ WK and t ∈ [0, T ). We note that WK can be

composed of arbitrary orders of polynomials e.g. for ξ, ζ ∈ [−1, 1] WK = [1, ξ, ζ] leads to alinear approximation while WK = [1, ξ, ζ, ξ2, ζ2, ξζ] will lead to a quadratic approximation.

We also define

wint(x) = lims→0+

w(x + snK) (11)

wext(x) = lims→0−

w(x + snK) (12)

〈w(x)〉 =1

2(wint(x) + wext(x)) (13)

[w(x)] = (wint(x)− wext(x)) (14)

where nK is the outward pointing normal on the element boundary ∂ΩK at x.

Multiplying (10) by w = (w1, w2, w3) and integrating over each element ΩK we have

∫ΩK

∂U

∂twdΩK +

∫ΩK

∂F

∂xwdΩK +

∫ΩK

∂G

∂ywdΩK =

∫ΩK

SwdΩK (15)

where, to simplify notations we used the convention (Sw)i = Siwi without the usual Einsteinsummation.

5

Using Green’s formulae

∫ΩK

∂U

∂twdΩK −

∫ΩK

F∂w

∂xdΩK −

∫ΩK

G∂w

∂ydΩK

+∮

∂ΩK

(Fnx + Gny)wds =∫

ΩK

SwdΩK (16)

Assembling over all the elements and defining Γ = ∪K∂ΩK\∂Ω

∑K

∫

ΩK

∂U

∂twdΩK −

∫ΩK

F∂w

∂xdΩK −

∫ΩK

G∂w

∂ydΩK −

∫ΩK

SwdΩK

+∮

∂Ω

(Fnx + Gny)wds +∮Γ

[(Fnx + Gny)w]ds = 0 (17)

To describe the solution scheme we now introduce the following notations. Let A denote the3×2 matrix (F G), or on components Ai,1 = Fi and Ai,2 = Gi. To simplify notations wewrite the equations resulting from (16) on components.

∫ΩK

∂uj

∂twdΩK −

∫ΩK

∑j

A(j,.)(U) · ∇wdΩK +

+∮

∂ΩK

∑j

A(j,.)(U) · nwds =∫

ΩK

Sj(U)wdΩK (18)

N (uj, w) = (Sj, w) (19)

We are looking for an approximation Uh ∈ WK ,Uh = (uhj ), j = 1..3, for the state variables

U ∈ (L∞[0, T ))3 × L2(Ω) so that

∫ΩK

∂uhj

∂twhdΩK −

∫ΩK

∑j

A(j,.)(Uh) · ∇whdΩK + (20)

+∮

∂ΩK

∑j

A(j,.)(Uh) · nwhds =

∫ΩK

Sj(Uh)whdΩK , ∀wh ∈ WK .

When we assemble these equations over all elements and use the notations from 16

∑K

∫

ΩK

∂uhj

∂twhdΩK −

∫ΩK

∑j

A(j,.)(Uh) · ∇whdΩK+

6

+∮

∂Ω

∑j

A(j,.)(Uh) · nwhds +

∮Γ

∑j

[A(j,.)(Uh) · nwh]ds =

∫ΩK

Sj(Uh)whdΩK , ∀wh ∈ WK . (21)

3.2 Fluxes and Slope Limiters

3.2.1 Fluxes

The approximate solution Uh may be discontinuous across the element interface and thereforewe need to approximate the integral containing the physical flux

∮Γ

∑j[A(j,.)(U

h) ·nwh]dsby a numerical flux times the average of the test function values across the element interface∮Γ η(Uhint

(x),Uhext(x)) 〈wh〉ds where, Uhint

(x) and Uhext(x) are defined as in (11) and (12)

and where Uhext(x) is replaced by the appropriate boundary value on ∂Ω ∩ ∂ΩK .

The numerical flux function η must be a two-point monotone function (nondecreasing withrespect to the second argument, nonincreasing with respect to the first) which is consistentand conservative. We use here the HLL fluxes described in Toro (1997) and tested in ourfinite differences numerical code (Patra et. al. (2004)). We briefly describe these fluxes below.

The Riemann problem at the element interfaces with the left and the right states given byUh(int), and Uh(ext) respectively, has characteristic speeds which are given by the eigen-values of the Jacobian matrix of F for the x-direction and by the eigenvalues of G for the

y direction. For the x-direction these are given by (u + c, u, u− c), where c =√

kapgzh, andwhere gz is the component of the gravitational acceleration normal to the basal surface andkap is the active/passive coefficient of the depth averaged theory by Hutter et. al. (1993).To propagate information in the x-direction we estimate the signal velocities in the solutionof the Riemann problem by the following choice proposed by Davis (1998) where uint, uext

are the “left” and “right” values of the x components of the velocity, hint, hext are the “left”and “right” values of height. These values are obtained by interpolation to the appropriatelocation on the element boundary from the approximate solution element and it’s neighborrespectively.

Cint1 = min(0, min(uext − cext, uint − cint)) (22)

Cext1 = max(0, max(uext + cext, uint + cint)) (23)

Similarly,

Cint2 = min(0, min(vext − cext, vint − cint)) (24)

Cext2 = max(0, max(vext + cext, vint + cint)) (25)

are the signal velocities in the y-direction.

7

The function η giving the numerical flux will have the form

η(Uhint

(x),Uhext

(x),n) =∑j

Cextj A(j,.)(U

hext

) · n− C ljA(j,.)(U

hint

) · n +

Cintj Cext

j (Uhext −Uhint

)/Cextj − Cint

j (26)

Flow fronts occur when zero flow depth exists adjacent to a cell with nonzero flow depth.The errors in front propagation speeds at flow fronts can be very large, and more accurateestimates for speeds are needed in such cases. For a front moving in the positive x directioncext = hext = 0, and the correct solution consists of a single rarefaction wave associatedwith the left eigenvalue. The front itself corresponds to the tail of the rarefaction moving tothe “left” and has exact propagation speed uext = uint + 2cint. This problem is similar tothe problem involving vacuum states in shock tubes, and the rationale for this approach isdiscussed in Toro (1997).

Finally, the system of equations becomes

∑K

∫

ΩK

∂uhj

∂twhdΩK −

∫ΩK

∑j

A(j,.)(Uh) · ∇whdΩK+

+∮

∂Ω

∑j

A(j,.)(Uh) · nwhds +

∮Γ

η(Uhint

(x),Uhext

(x),n)〈wh〉ds =

=∫

ΩK

Sj(Uh)whdΩK , ∀wh ∈ WK . (27)

The integrals may be evaluated using quadrature, and the equations may be written as asystem of differential equations in time, which has the form

d

dtUh = L(Uh) (28)

This system can be solved using a total variation diminishing in the means (TVDM) RKtime discretization. However, as we describe next, the approximate solution at every stagemust be modified by a process of slope limiting to eliminate spurious oscillations.

3.2.2 Slope Limiting

The numerical scheme derived by directly integrating 28 does not provide an approximatesolution that satisfies the TVDM (Total Variation Diminishing in the Means) property. Wediscuss our strategy for slope limiting in the case of piecewise linear approximations, since thesame algorithm applies to higher order approximations. The slope limiter must maintain the

8

conservation of mass, satisfy the sign conditions that prevent total variation from increasing,and must not degrade the accuracy of the method.

In the global coordinate system, the piecewise linear approximation of the solution is

Uh = Uh + (x− x0)(Ux)h + (y − y0)(U

y)h (29)

where Uh are the cell averages of the system variables, (Ux)hand(Uy)h) are the coefficientsof linear shape functions used to construct the local approximation of the exact solution and(x0, y0) are the coordinates of the center of the current element Ω i.e. it is the map of thecentroid of ΩK and FK(0, 0) = (x0, y0). We use the generalized slope limiter of the MUSCLschemes by Van Leer, which is described in detail in Cockburn (2002).

Slope limiting for systems must be performed in the local characteristic variables (Cockburn(2002)). To achieve this we need the characteristic speeds and direction for wave propagation,that is the eigenvalues and the eigenvectors of the composite matrix Q = Mnx + Nny

where M and N are the Jacobian matrices of F and G. For our choice of rectangular shapedelements, the grid is aligned with the x and y coordinates. For the x-component of the fluxthe composite matrix reduces to M, and similar treatment applies to y-fluxes with theJacobian matrix M replaced by N. Here we describe the x-component case in detail.

The essence of the limiting process is to compare the slopes of the approximate solution com-puted directly from the piecewise linear approximation with finite difference approximationsof the slopes obtained by comparing cell averages of neighboring cells. Hence the quanti-

ties of interest are (Ux)h, (Uh)right−Uh

xright0 −x0

, and Uh−(Uh)left

x0−xleft0

, where the left, right suffixes denote

neighboring elements to the negative x direction and the positive x direction of the currentelement. Let R be the matrix having the eigenvectors of M as columns. These vectors canbe transformed to the coordinate system of the characteristic variables by multiplying themwith the inverse of R. Let R−1Uxh = µxh. The limiter is then applied to each componenti.e.

(µxj )

hlim

= m

(µxj )

h,(µh

j )right

− (µhj )

xright0 − x0

,µh

j − ((µhj ))

left

x0 − xleft0

(30)

where the m(.,.,.) is the usual minmod function defined by

m(a1, a2, · · · , aν) =

s min1≤n≤ν |an|, if s = sign(a1) = · · · = sign(aν)

0, otherwise

The limited coefficients (µx)hlim are transformed back to the original coordinate system by

multiplication with the the matrix R, and we denote them by (Ux)hlim. The procedure for

the y-component follows the same procedure with appropriate neighbors.

9

The approximate solution after slope limiting process is complete will then be

Uh(x, y) =Uhj + (x− x0)(U

x)hlim + (y − y0)(U

y)hlim (31)

The approximate solution obtained from the equation (31) is said to be limited and it isdenoted by ΛUh. Now this slope limited approximation can be used in a time integrationscheme.

3.3 Second Order Runge Kutta Discretization in Time

We implemented a second order Runge-Kutta (RK)algorithm and we investigated piecewiselinear solutions. Cockburn et. al. (2000) has established that if pth order basis functions areused in space then we require p + 1th order RK schemes in time to maintain a balance in therrors in time and space discretization.

The algorithm for second order(p = 1) can be written as follows

Uh(1) =Uh

(n) + ∆tL(ΛUh(n)) (32)

Wh =Uh(1) + ∆tL(ΛUh

(1)) (33)

Uh(n+1) =

1

2(Uh

(n) + Wh) (34)

Higher order versions involve additional stages with different coefficients as documented inCockburn et. al. (2000).

4 Computational Issues

4.1 Adaptive Strategies

The adaptivity in this simulation has three goals:

• to control the approximation error,• to capture the flow from more accurately, and,• to approximate terrain better.

A survey of the literature on a posteriori error estimation (see for e.g.Hartmann and Houston(2002)) reveals that the numerical approximation error may be well controlled by controllingappropriate norms of the residuals defined below. If u is the exact solution of (18) and uh

10

is the approximate numerical solution computed from (20) then the element-wise residual isdefined by:

RK(w) = (N (u, w)− Nuh, w)− ((S(u), w)− (S(uh), w)) (35)

Clearly RK(w) cannot be computed and must be estimated. Techniques for such estimates(in a multitude of norms) are well developed (see for e.g.Hartmann and Houston (2002)).We note that a significant contribution to the residual in this case will be related to the jumpin the physical fluxes. Thus we can define a primary indicator of numerical approximationerror βK as

β2K =

∮∂ΩK

[F (U) · w]2 + [G(U) · w]2ds (36)

To accomplish the other two goals of adaptivity though, we need more information than iscontained in this indicator. Measures of change in topography and techniques that refinethe grid at the flow front are also necessary. For capturing the front we have implemented ascheme in which cells at the front are explicitly tracked (by monitoring the change in flowdepth among neighboring cells) and refined. While, we have not implemented adaptivitybased on terrain features, we update terrain information at finer resolutions as the mesh isupdated.

Similarly unrefinement schemes have been implemented to remove cells that are not activein the computation. When the flow has proceeded through a region and the flow depths andmomentums have been reduced below a threshold we unrefine the cells.

4.2 Parallel Computing

We will use parallel processing to enable us to solve the very large systems necessary for highfidelity computations on realistic terrain. Our approach to parallel processing is to use dataparallel computations. The particular challenges due to mesh adaptivity and consequentlyterrain adaptivity are surmounted using the ideas discussed in Laszloffy et. al. (2000); Patraet. al. (2003) and more recently adapted for thin layer granular flow models using adaptivefinite difference schemes in Patra et. al. (2004).

The central idea is to organize data and computations using a space filling curve based order-ing of the cells. Parallel decomposition for p processors is achieved by a p-way partitioningof this ordering with work associated with the cells in a partition being undertaken by asingle processor (see Fig. 2 for an illustration). Data from a layer of cells at each partitionboundary needs to be available to the processor computing the neighboring partition. Hencewe create a layer of “ghost” cells along each partition boundary which is also made availableto the neighboring processor. Upon completion of each time step of computation data as-sociated with these cells must be exchanged among processors. As the flow evolves and the

11

for parallel computingPartition Lines for

space fillingcurve ordering

Ω

Ω2

4Ω Ω3

Ω1

Ω

Possible

1

Ω2

4Ω Ω3

to partition linesLayers of ghost cells adjacent

Fig. 2. Partitioning of sample domain into 4 partitions for parallel computing. Note samplespace-filling curve ordering and it’s 4 − way dissection to obtain the partitioning and a layer ofghost cell along partition lines.

adaptation pattern changes the new cells are are introduced in the ordering and the partitionboundaries 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 have integrated our simulation codes with appropriategeographical information system tools. The tool automatically extracts the required eleva-tion, slope and curvature data from standard digital elevation models. Details of the manyissues involved in making this linkage are described in our earlier work (Patra et. al. (2004)).The highlight of our methodology is that the interpolation used to generate the elevationsis matched to the size of the computational grid to avoid spurious artifacts. Secondly, as thegrid is locally refined finer topographical details are obtained from the database resulting inclear definition of channels and other sharp features resulting smaller modeling errors.

12

5 Numerical Tests and Validation

We will now present a series of numerical tests that 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 of sand down simple ramps (a standardtable-top experiment). We will simulate the flow of approximately 425 g of sand (volumeapproximately 2.7 × 10−03m3 sliding down a ramp at 44.3 degrees (see Fig 3 a for details).The results of this experiment and its use in validating the TITAN2D tool are documentedin our earlier work Patra et. al. (2004). An interior friction angle of 37.3 degrees and bedfriction angle of 32.48 degrees are used in the simulations and are in line with experimentalmeasurements. We will now compare the results from our new schemes with those from usingthe finite volume schemes (Patra et. al. (2004)). Figure 4 shows a series of frames in whichthis comparison is illustrated.In both cases we start the computation with 100 grid cells andallow adaptivity (refinement and unrefinement based on the adaptive strategy) to controlthe number of cells at any time step.

The adaptive strategy for the finite volume is based largely on heuristics (as explained inPatra et. al. (2004)) while the residual based βK defined in equation (36) provides a moresystematic and mathematically consistent basis for refinement. The solutions are similarin many aspects. However, the frames clearly show the higher resolution of the DG basedscheme. Hence, our confidence in the numerical correctness of this new solution schemeis greatly reinforced by these results. The next figure 5 shows comparisons of the extents(spreads) to some simple table top experiments described in Patra et. al. (2004). Thesimulations are quite good early on for time less than 0.6 seconds when the flow is on theinclined part of the ramp. After reaching the flat part of the ramp the correlations are notgood. The experimental observations of x extents(distance between head and tail) seems toincrease to a much higher value before finally collapsing to a value close to that reachedby the simulation. Upon careful examination of the images of the experiment we noticethat a very thin layer on the inclined portion of the ramp causes the tail of the flow in theexperiment to be located further back than in the simulation. In the simulation also we seea very thin O(10−5 m) thick layer which matches this layer. However, this thickness is toosmall and usually neglected as non-physical. Similarly for the y extents the experimentalvalues increase quite rapidly as it reaches the bottom of the ramp; the simulated extentsare initially smaller but over time seem to be trending towards these higher numbers. Wehypothesize that this discrepancy between simulation and experiment is either due to thesharp change in terrain slope and curvature as the flow moves from the inclined part to theflat part or due to the numerical error introduced in computing flows with very thin layersince estimates of flow speed u, v used in computing the transport are derived from the depthaveraged momentum hu, hv by dividing with h.

13

In the next figure we make a detailed comparison of the maximum pile height versus timefor both the new and finite volume schemes. The maximum pile heights versus time areremarkably similar until we reach the flat part of the ramp where the heights are muchhigher for the new DG schemes. The greater resolution of the DG scheme and/or the differentcomputations for the source terms involving friction are possible reasons for the differences.We note that the solution scheme in TITAN2D is substantially similar to those in Denlingerand Iverson (2001).

In the next set of tests down a curvilinear ramp (shown in Fig. 7 and Fig. 8), we plot the flowdepth, adaptive mesh and error indicator βK . The evolving mesh designed to capture theflow accurately is clearly seen. The mesh is highly refined at the front (cells containing theinterface of zero flow depth and non-zero flow depth and in the interior where the indicatorβK is high. Note that this does not necessarily coincide with areas of high flow depth as isseen in frames labeled e) and f) in Fig. 8 where the stable center area is not highly refined butthe rapidly moving outside of the pile is. βK is a measure of the local numerical error in thecomputation. Thus, high values of βK are expected in areas where there is more rapid flowand smaller grid cells are required to resolve the flows. In the next figure Fig. 9 we displaythe evolution of a sample parallel adaptive mesh and its partitioning for parallel computingon four processors. Proper partitioning of the cells is required for efficiently computing onmulti-processor machines.

5.2 Flows on Natural Terrain

In the next two tests we test the new methodology on sample digital elevation models oflittle Tahoma peak and Colima volcano.

In 1963 a series of 7 avalanches occurred at Little Tahoma Peak on Mount Rainier, Washing-ton (Fahnestock (1963); Norris (1994)). The avalanches, totaling approximately 1.1×107m3

of broken lava blocks and other debris, traveled 6.8 km horizontally and fell 1.8 km verti-cally (H/L = 0.246). Velocities calculated from run-up range from 24 to 42 m/s and mayhave been as high as 130 m/s while the avalanches passed over Emmons Glacier Crandell andFahnestock (1965). The avalanches formed a total deposit thickness of 30 m near their distalterminus where they ponded against a terminal moraine. Because topographic surveys weremade both before (by Fahnestock (1963)) and after (by Crandell and Fahnestock (1965))the event, various aspects of the flowing avalanche and its deposits are well documented. Forthis reason we have used the Little Tahoma Peak avalanches to calibrate the model here andin earlier work for similar types medium-sized rock avalanches (Sheridan et. al. (2004)).

A series of tests were made with simulated flows of 9.4× 106m3 volume. This value approxi-mates the size of individual avalanches at Little Tahoma Peak. The length of the simulatedflow runout was calibrated to 6.8 km, matching the actual avalanches, by adjusting the basalfriction angle and internal friction angle. Best results were obtained with values of 10 and30, respectively. Fig. 10 shows results from a run of the code as a series of time steps. The

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a) Ramp with sharp turn to flatzone

0.603

1.2.42

44.3 deg

b)Ramp with gradualcurvature

0.5

1.0

Fig. 3. Geometry of two ramps used in testing the code. Ramp on the right gradually curves intoa flat plane while ramp on left has a sharp turn into a flat surface.

simulation results compare well with the actual 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) flowvelocity, and 5) maximum thickness of the deposit.

The flow boundaries of the moving mass from the simulations fit reasonably well with themapped extent of the avalanches Sheridan et. al. (2004). The area of the mapped depositsis 1.3km2 compared with 0.6km2 for the simulation. The run-up heights for the avalancheswas 40 to 90 m whereas the simulation run-up was 60 m. The maximum velocity of theavalanches calculated from super elevation at bends is 140 m/s whereas the simulation gavea value of maximum velocity ranging from 80-150 m/s. The total deposit thickness of theseven actual avalanches is 30 m ( 4.4 m average thickness) where the model results gave 3.6m for a single flow thickness.

A second set of simulations is conducted on terrain data from Colima volcano. Results ofsimulations for a parabolic pile of volume 9.384405×106m3 centered at the UTM coordinates(644935,2171380) and of extent 200m and maximum height 150m flowing down the volcanoare shown in Fig. 11. The flows appear to channelize appropriately and splits among themultiple channels. We predict a maximum velocity in the range 121-174m/s.

6 Conclusions and Future Work

In this paper we have described the development of highy accurate adaptive discontinuousGalerkin schemes for the solution of the equations arising from a thin layer type model ofdebris flows. These schemes are then coupled with special solution methodologies to producea simulation tool capable of very high order numerical accuracy. The tool is then applied to

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Fig. 4. Flow of a parabolic pile of sand down a 44.3 degree ramp onto a flat surface simulatedusing both finite volume schemes (bottom pictures) and the new discontinuous Galerkin schemes.Finite volume results are from the TITAN2D code (Patra et. al. 2004) and have been comparedto experiments. Flows are simulated with adaptive grids and choices of interior friction angle of37.3 degrees and basal friction angle of 32.47 degrees. Flow depths are substantially similar in bothflows but the new schemes show better resolution of flow depth contours.

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

- Tai

l

Time

DG SimulationX-extent FD50 adaptive grid

X-extent Expt.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Y-E

xten

t

Time

DG SimulationExperimental Data

FD Simulation

Fig. 5. Flow of a parabolic pile of sand down a 44.3 degree ramp onto a flat surface simulatedusing both finite volume schemes (bottom pictures) and the new discontinuous Galerkin schemescompared 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 degrees andbasal friction angle of 32.47 degrees.

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Fig. 6. Maximum pile height versus time for the flow of a parabolic pile of sand down a ramp of44.30 and choices of interior friction angle of 37.30 and basal friction angle of 32.470. Flow depthsare substantially similar in both flows until the flow hits the flat part of the ramp where the DGcomputation yields a flow that is much less spread out and hence has a higher pile height.

several test problems to illustrate the use of these schemes.

The new schemes outlined here will enable several lines of future work. Most prominentamong these will be the development of classes of a posteriori error estimates for the nu-merical approximation error and control thereof leading to robust and reliable simulations.Strategies for exploiting the local adaptivity features of this tool will also need to be betterdeveloped.

References

Aizinger, V. and Dawson, C., A discontinuous Galerkin method for two dimensional flowand transport in shallow water, Advances in Water Resources,25, 2002, pp.67-84.

Crandell D.R. and R.K. Fahnestock, 1965. Rockfalls and Avalanches from Little TahomaPeak on Mount Rainier, Washington. US Geological Survey Bull. 1221-A, A1-A30.

Cockburn, B., Karniadakis, G., Shu, C., The Development of Discontinuous Galerkin Meth-ods, in Discontinuous Galerkin Methods Ed. Cockburn, B., Karniadakis, G., Shu, C,Springer, Heidelberg, 2000.

Cockburn, B., Discontinuous Galerkin Methods for Convection Dominated Problems,preprint.

Davis, S. F., Simplified second order Godunov type methods, SIAM J Sci Statist Comput 9,445-473, 1998

Denlinger, R.P and Iverson, R.M Flow of variably fluidized granular material across three-

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Fig. 7. Initial stages of grid for flow of a parabolic pile of sand down a curved ramp. Colors usedto indicate level of field variable in cell. The contours on the left on each panel shows the flowdepth(pile height) and the contours on the right labeled flux show the error indicator βK used inadapting the grid. Zoom on top left shows details of adaptive grid.

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Fig. 8. Final stages of grid for flow of a parabolic pile of sand down a curved ramp. Colors usedto indicate level of field variable in cell. The contours on the left on each panel shows the flowdepth(pile height) and the contours on the right labeled flux show the error indicator βK used inadapting the grid. Zoom on top left shows details of adaptive grid.

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Fig. 9. Evolution of sample adaptive mesh and partitioning for parallel computing using four pro-cessors. Grid line colors indicate processor and contours indicate flow depth.

dimensional terrain 2. Numerical predictions and experimental tests, J. Geophys. Res.106,2001,pp??.

Fahnestock, R.K., 1963. Morphology and hydrology of a glacial stream - White River, MountRainier, Washington. U.S. Geol. Soc. Prof. Paper 422-A, A1-A70.

Gray, J.N.M.T., Granular Avalanches on Complex Topography, in N.A. Fleck and A. C.F. Cocks (ed.) Procedings of IUTAM Symposium on Mechanics of Granular and PorousMaterials, pp. 275-286, Kluwer Academic Publishers, 1997.

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Iverson, R.M and Denlinger, R.P, Flow of variably fluidized granular material across three-dimensional terrain 1. Coulomb mixture theory, J. Geophys. Res. 106 2001, pp. 537-552

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LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhauser Verlag, 1992Long, J., Integrated Data Management and Dynamic Load Balancing for hp and Generalized

FEM, Ph.D. Dissertation, Mechanical and Aerospace Engineering Dept., University atBuffalo.

Norris R.D., 1994. Seismicity of rockfalls and avalanches at three Cascade Range volcanoes:Implications for seismic detection of hazardous mass movements. Bull. Seismol. Soc. Am.84,1994, 1925-1939

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., Parallel Adaptive Numeri-cal Simulation of Dry Avalanches over Natural Terrain, in press, J. Volc. GeothermalResearch.

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Fig. 10. Simulated flow of a debris pile of volume 9.4 × 106m3 using digital elevation models oflittle Tahoma peak. Basal friction angle of 100 and internal friction angle of 300 are used in thecalculation. Contours indicate flow depth at different times. Comparisons with field observationsyield very good correlations with the deposits after a series of avalanches in 1963.

Patra, A.,Laszloffy, A., and Long,J. Data Structures and Load Balancing for Parallel Adap-tive hp Finite Element Methods”, to appear in Computers and Mathematics with Appli-cations.

Pouliquen, O., Forterre, Y., Friction Laws for dense granular flows: application to the motion

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Fig. 11. Simulated flow of a debris pile of volume 9.384405× 106m3 using digital elevation modelsof Colima volcano.The superposed grid shows the adaptive grid with finer resolutions to captureflow features. The flow is initiated at UTM coordinate (644935.1,2171380.25) and attains maximumvelocities of 121-174 m/s. Flow reaches the state in the final frame after 2 minutes and 18 seconds.

of a mass down a rough inclined plane, J. Fluid Mech (2002) v. 453, pp 133-151.Pitman, E.B, Nichita, C. C., Patra, A. and Bauer, A., Sheridan, M.F., and Bursik,

M.,Computing granular avalanches and landslides, Physics of Fluids 15 no. 12, 3638-3647.

Rankine, W.J.M. On the stability of loose earth, Philos. Trans. Roy. Soc. London 147, 1857,9-27

Remacle, J.F., Flaherty, J., Shephard, M., An Adaptive Discontinuous Galerkin Techniquewith an Orthogonal Basis Applied to Compressible Flow Problems, SIAM Review, vol45,no.1, 2003 pp. 53-72.

Sheridan,M.F., A.J. Stinton, A. Patra, E.B. Pitman, A. Bauer, C.C. Nichita, in press, Eval-

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uating TITAN2D mass-flow model using the 1963 Little Tahoma Peak avalanches, MountRainier, Washington, Jour. Volcanology Geothermal Research.

Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag,1997

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