The Hysea Project: a web-based platform for the simulation ... · The Hysea Project: a web-based...

IntroductionMathematical models

Numerical schemeApplications

The Hysea Project: a web-based platform for the simulation ofgeophysical flows

Carlos ParesGrupo EDANYA. Universidad de Malaga. Espana.

BCAM Workshop Environmental Mathematics Day. February 26, 2013.

Carlos Pares Bcam. February 2013



Outline

1 IntroductionHySEA: High Performance Cloud Computing Software

2 Mathematical models

3 Numerical scheme

4 ApplicationsHigh Performance Computing (HPC)Real applications




HySEA: High Performance Cloud Computing Software

Edanya Team

Team leader: Carlos Pares Madronal.

Marc de la Asuncion.

Manuel J. Castro Dıaz.

Jose Marıa Gallardo Molina.

Jose Manuel Gonzalez Vida.

Jorge Macıas Sanchez.

Tomas Morales de Luna.

Marıa de la Luz Munoz Ruiz.

Sergio Ortega Acosta.

Carlos Sanchez Linares





Collaborations

Enrique Fernandez Nieto, Gladys Narbona Reina (Univ. Sevilla)

Jose Antonio Garcıa Rodrıguez, Ana Ferreiro Ferreiro (Univ. a Coruna)

E. Toro (Univ. Trento), M. Dumbser (Univ. Trento), S. Mishra (ETH Zurich), P.Lefloch (Univ. Pierre et Marie Curie), Francois Bouchut (Univ. Paris-Est), G.Russo (Univ. Catania), S. Noelle (Univ. Aachen), R. LeVeque (Univ.Washington), ...

Jose Miguel Mantas (Univ. Granada), B. Fraguela (Univ. a Coruna)

Victor Dıaz del Rio (I.E.O), Anne Mangeney (Institut de Physique du Globe deParis )

M. Bruno (Univ. Cadiz - CASEM Andalusian Center for Marine Studies.)

D. Arcas, V. Titov (NOAA Center for Tsunami Reserch)

CGS Ingenierıa, Medio Ambiente Canarias, CONSULTEC S.L., SENERIngenierıa.





Goals

Edanya: Main Goals

Development of robust, reliable and low computational cost numerical toolsfor the simulation of geophysical flows and the prediction of emergencysituations such as river floodings or oil spills, tsunamis, debris avalanches . . .





Goals

Edanya: Main Goals

Development of robust, reliable and low computational cost numerical toolsfor the simulation of geophysical flows and the prediction of emergencysituations such as river floodings or oil spills, tsunamis, debris avalanches . . .

Ingredients

Mathematical models: based in Shallow-water equations.

Numerical methods: High order path-conservative finite volume schemes.

HySEA: High Performance Cloud Computing software to simulategeophysical flows.





HySEA: interdisciplinary platform

Models based on geophysical flows with applications in:

Physical Oceanography

Marine Geology and Ecology

Tsunami Research

Civil and Hydraulic Engineering





HySEA: how to make our software more accessible?

Hardware:

Users don’t need to have a supercomputer.

Codes have to be accessible from any architecture.

Software:

Users don’t need to install specific libraries, compilers, etc.

Models have to be independent from the operative system.

Easy update process for code changes.

Updated documentation.





HySEA: making more accessible our codes.

Hardware:

The simulations are run on a supercomputer (CPU’s and GPU’s) located at theNumerical Methods Laboratory of the University of Malaga.

Software:

The interface between the researcher and Cires is a web browser.





platformHySEA

EDANYA BetaHyperbolic Systems and Efficient Algorithms




River floodings: one-layer shallow water system

∂h∂t

+∂qx

∂x+∂qy

∂y= 0

∂qx

∂t+

∂

∂x

(q2

x

h+

g2

h2

)+

∂

∂y

(qxqy

h

)= gh

∂H∂x

+ Sf1 (w)

∂qy

∂t+

∂

∂x

(qxqy

h

)+

∂

∂y

(q2

y

h+

g2

h2

)= gh

∂H∂y

+ Sf2 (w),

h

H

η

x

0

yh is the water depth,

q = (qx, qy), u = (ux, uy) =qh

,

w = (h, qx, qy)T ,

H is the bathymetry,

g is the acceleration of gravity,

Sfi (w), i = 1, 2, parametrize the frictionterms.




Landslide + tsunami simulations: 2D (Simplified) Two-layer Savage-Huttershallow-water model

Hypothesis

We consider an stratified media composed by a non viscous and homogeneous fluid with constant density ρ1 (water)and a granular material with density ρs and porosity ψ0 . We suppose the mean density of the granular material is givenby: ρ2 = (1− ψ0)ρs + ψ0ρ1 .

Fluid and granular material are immiscible.

Layer 2

Layer 1

Reference Level

h2(x,y,t)

h1(x,y,t)

H(x,y)





∂h1

∂t+∂q1,x

∂x+∂q1,y

∂y= 0

∂q1,x

∂t+∂

∂x

q21,x

h1+

g

2h21

+∂

∂y

(q1,xq1,y

h1

)= −gh1

∂h2

∂x+ gh1

∂H

∂x+ Sf1

(w)

∂q1,y

∂t+∂

∂x

(q1,xq1,y

h1

)+∂

∂y

q21,y

h1+

g

2h21

= −gh1∂h2

∂y+ gh1

∂H

∂y+ Sf2

(w)

∂h2

∂t+∂q2,x

∂x+∂q2,y

∂y= 0

∂q2,x

∂t+∂

∂x

q22,x

h2+

g

2h22

+∂

∂y

(q2,xq2,y

h2

)= −grh2

∂h1

∂x+ gh2

∂H

∂x+ Sf3

(w) + τ3(w)

∂q2,y

∂t+∂

∂x

(q2,xq2,y

h2

)+∂

∂y

q22,y

h2+

g

2h22

= −grh2∂h1

∂y+ gh2

∂H

∂y+ Sf4

(w) + τ4(w),

hi, is the layer depth, i = 1, 2,

qi = (qi,x, qi,y), is the mass flow at each layer, i = 1, 2,

w = (h1, q1,x, q1,y, h2, q2,x, q2,y)T ,

H is the bathymetry, g is the acceleration of gravity,

r = ρ1/ρ2 is the ratio of the constant densities of the layers (ρ1 < ρ2),

Sfi (w), i = 1, . . . , 4, parametrize the friction,

τ3(w), τ4(w) parametrize the Coulomb/Pouliquen friction term.





∂h1

∂t+∂q1,x

∂x+∂q1,y

∂y= 0

∂q1,x

∂t+∂

∂x

q21,x

h1+

g

2h21

+∂

∂y

(q1,xq1,y

h1

)= −gh1

∂h2

∂x+ gh1

∂H

∂x+ Sf1

(w)

∂q1,y

∂t+∂

∂x

(q1,xq1,y

h1

)+∂

∂y

q21,y

h1+

g

2h21

= −gh1∂h2

∂y+ gh1

∂H

∂y+ Sf2

(w)

∂h2

∂t+∂q2,x

∂x+∂q2,y

∂y= 0

∂q2,x

∂t+∂

∂x

q22,x

h2+

g

2h22

+∂

∂y

(q2,xq2,y

h2

)= −grh2

∂h1

∂x+ gh2

∂H

∂x+ Sf3

(w) + τ3(w)

∂q2,y

∂t+∂

∂x

(q2,xq2,y

h2

)+∂

∂y

q22,y

h2+

g

2h22

= −grh2∂h1

∂y+ gh2

∂H

∂y+ Sf4

(w) + τ4(w),

E. Fernandez Nieto, F. Bouchut, D. Bresch, M.J. Castro, A. Mangeney.

A new Savage-Hutter type model for submarine avalanches and generated tsunami. J. Comp. Phys.,227: 7720-7754, 2008.




General framework

General formulation

∂w∂t

+∂F1

∂x(w)+

∂F2

∂y(w)+ B1(w)

∂w∂x

+ B2(w)∂w∂y

= S1(w)∂H∂x

+ S2(w)∂H∂y

+ SF(w),




General framework

General formulation

∂w∂t

+∂F1

∂x(w)+

∂F2

∂y(w)+ B1(w)

∂w∂x

+ B2(w)∂w∂y

= S1(w)∂H∂x

+ S2(w)∂H∂y

+ SF(w),

w(x, t) : D× (0, T) 7→ ω ⊂ Rn, the vector of unknowns,

D bounded domain of R2; ω convex subset of Rn,

Fi : ω 7→ Rn, i = 1, 2, regular and locally bounded functions,

Bi : Ω 7→ MN×N(R), i = 1, 2 regular and locally bounded matrix-valuedfunctions,

Si, SF : ω 7→ Rn, i = 1, 2, regular and locally bounded functions,

H : D ⊂ R2 : 7→ R known function.




General framework

General formulation

∂w∂t

+∂F1

∂x(w)+

∂F2

∂y(w)+ B1(w)

∂w∂x

+ B2(w)∂w∂y

= S1(w)∂H∂x

+ S2(w)∂H∂y

+ SF(w),

Nonconservative products

The nonconservative products Bi(w) ∂w∂α

, Si(w) ∂H∂α

, i = 1, 2, α = x, y do not makesense in general within the framework of distributions. Here, we follow the theorydeveloped by [Dal Maso, LeFloch and Murat] to give a sense to these products asBorel measures. This theory is based on the choice of a family of paths.




General Formulation: Nonconservative form

General formulation

∂w∂t

+∂F1

∂x(w) +

∂F2

∂y(w) + B1(w)

∂w∂x

+ B2(w)∂w∂y

= S1(w)∂H∂x

+ S2(w)∂H∂y

+ SF(w)





General formulation

∂w∂t

+∂F1

∂x(w) +

∂F2

∂y(w) + B1(w)

∂w∂x

+ B2(w)∂w∂y

= S1(w)∂H∂x

+ S2(w)∂H∂y

+ SF(w)

[LeFloch] Introducing the trivial equation ∂tH = 0 and taking H as a new unknown

W := [w H]> ∈ RN , N = n + 1, Jk(w) =∂Fk

∂wand

Ak =

[Jk(w) + Bk(w) −Sk(w)

0 0

], k = 1, 2 ∈MN×N ,

the system can be written as follows:





General Formulation

∂W∂t

+A1(W)∂W∂x

+A2(W)∂W∂y

= SF(W),





General Formulation

∂W∂t

+A1(W)∂W∂x

+A2(W)∂W∂y

= SF(W),

Given a unitary vector η = (η1, η2) ∈ R2:

A(W,η) = A1(W)η1 +A2(W)η2.

We assume that the system is strictly hyperbolic, i.e. ∀W ∈ Ω ⊂ RN and∀ η ∈ S1, the matrix A(W,η) has N real eigenvalues:λ1(W,η) < . . . < λN(W,η), being Rj(W,η), j = 1, . . . ,N the associatedeigenvectors.





A family of paths (Dal Maso, LeFloch, Murat) in Ω ⊂ RN is a locally Lipschitz map

Φ: [0, 1]× Ω× Ω× S1 → Ω,

where S1 ⊂ R2 denotes the unit sphere, that satisfies some regularity conditions and1 Φ(0; WL,WR,η) = WL and Φ(1; WL,WR,η) = WR, for any WL,WR ∈ Ω,

η ∈ S1.2 Φ(s; WL,WR,η) = Φ(1− s; WR,WL,−η), for any WL,WR ∈ Ω, s ∈ [0, 1],

η ∈ S1.3 Φ(s; W,W,η) = W, for any s ∈ [0, 1], η ∈ S1.






Φ: [0, 1]× Ω× Ω× S1 → Ω,




A piecewise regular function W is a weak solution if and only if the two followingconditions are satisfied:

(i) W is a classical solution where it is smooth.

(ii) At every point of a discontinuity W satisfies the jump condition∫ 1

0A(Φ(s; W−,W+,η),η)

∂Φ

∂s(s; W−,W+,η) ds = σ

(W+ −W−

),

where I is the identity matrix; σ, the speed of propagation of the discontinuity; η aunit vector normal to the discontinuity at the considered point; and W−, W+, thelateral limits of the solution at the discontinuity.






Φ: [0, 1]× Ω× Ω× S1 → Ω,




For conservative problems, the jump conditions reduce to the standardRankine-Hugoniot condition regardless of the chosen family of paths:

Fη(W+)− Fη(W−) = σ(W+ −W−

),

where, given η = (η1, η2) ∈ S1:

Fη(W) = η1F1(W) + η2F2(W).






Φ: [0, 1]× Ω× Ω× S1 → Ω,




The choice of the family of paths is important because it determines the speed ofpropagation of discontinuities.

It has to be based on the physical background of the problem: limit of viscousprofiles . . . , but in practice can be very difficult.

From the mathematical point of view, some hypotheses concerning the relation ofthe paths with the integral curves of the characteristic fields can be imposed.




High order numerical schemes

Goal

Develop high order finite volume schemes for problems that can be written under theform:

∂W∂t

+A1(W)∂W∂x

+A2(W)∂W∂y

= 0,




High order numerical schemes

Goal

Develop high order finite volume schemes for problems that can be written under theform:

∂W∂t

+A1(W)∂W∂x

+A2(W)∂W∂y

= 0,

Basic idea

Split the computational domain into subsets of simple geometry called cells orfinite volumes.

Define a reconstruction operator of the unknowns W on each cell.

Combine with a first order finite volume scheme for nonconservative systems(path-conservative schemes).




Notation

The computationa domain is decomposed into subsets (closed polygons) calledcells of finite volumes, Vi ⊂ R2;

Ni is the set of indexes j such that Vj is a neighbor of Vi;

Eij is the common edge to two neighbor cells Vi and Vj, and |Eij| represents itslength;

ηij = (ηij,x, ηij,y) is the normal unit vector of the edge Eij pointing towards thecell Vj;

∆x is the maximum of the diameters of the cells;

Wni will represent the constant approximation of the averaged solution in the cell

Vi at time tn provided by the numerical scheme:

Wni∼=

1|Vi|

∫Vi

W(x, tn)dx.




High order state reconstruction operators: examples

X.D. Liu, S. Osher, T. Chan.

Weighted essentially nonoscillatory schemes, J. Comput. Phys. 115 (1994) 200-212.

C.-W. Shu.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolicconservation laws. ICASE Report 97-65, 1997.

C.-W. Shu and S. Osher.

Efficient implementation of essentially non- oscillatory shock capturing schems. J. Comp. Phys., 77:439-471, 1998.

H. J. Schroll and F. Svensson.

A Bihyperbolic Finite Volume Method for Quadrilateral Meshes. SIAM: J. Sci. Comput.,26(2):237-260, 2006.

M. Dumbser, M. Kaser.

Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linearhyperbolic systems. J. Comput. Phys. 221, 693-723,2007.

Jose M. Gallardo, Sergio Ortega, Marc de la Asuncion and Jose Miguel Mantas

Two-dimensional compact third-order polynomial reconstructions. Solving nonconservative hyperbolicsystems using GPUs. J. Sci. Comput. 48, 141-163, 2011.




High Order 2D scheme: conservative systems

Let us consider the conservative system

∂W∂t

+∂F1

∂x(W) +

∂F2

∂y(W) = 0,

and let W i(t) denotes the cell average over the cell Vi at time t.

The equation satisfies by W i(t) is the following

W′

i (t) = − 1|Vi|

∑j∈Ni

∫Eij

F(W(γ, t)) · ηij dγ

. (1)





A first order method and a reconstruction operator is used to approach thevalues of fluxes at the edges:

W′i (t) = − 1

|Vi|

∑j∈Ni

∫Eij

G(W−ij (γ, t),W+ij (γ, t),ηij) dγ

, (1)

being Wi(t) the approximation to W i(t) and W±ij (γ, t) the reconstruction atγ ∈ Eij.

limx→ γx ∈ Vi

Pti(x) = W−ij (γ, t), lim

x→ γx ∈ Vj

Ptj(x) = W+

ij (γ, t).

where Pti is a smooth approximation of the solution at the cell Vi computed

from the values at some neighbor cells of Wj(t)j∈Bi , where Bi define thestencil of Vi.





Using the divergence theorem, the previous semi-discrete numerical schemecan be rewritten as follows:

W′i (t) = − 1|Vi|

∑j∈Ni

∫Eij

(G(W−ij (γ, t),W+

ij (γ, t),ηij)− F(W−ij (γ, t)) · ηij

)dγ

− 1|Vi|

∑j∈Ni

∫Eij

(F(W−ij (γ, t)) · ηij

)dγ

= − 1|Vi|

∑j∈Ni

∫Eij



)dγ

− 1|Vi|

∫Vi

∇ · (F Pti)(x)) dx

= − 1|Vi|

∑j∈Ni

∫Eij



)dγ

− 1|Vi|

∫Vi

(J1(Pt

i(x))∂Pt

i

∂x(x) + J2(Pt

i(x))∂Pt

i

∂y(x)

)dx,




High Order 2D scheme for nonconservative systems

W′i (t) = − 1

|Vi|

∑j∈Ni

∫Eij

D−ij (W−ij (γ, t),W+ij (γ, t),ηij)dγ

+

∫Vi

(A1(Pt

i(x))∂Pt

i

∂x(x) +A2(Pt

i(x))∂Pt

i

∂y(x)

)dx]

where D±ij (W−ij (γ, t),W+ij (γ, t),ηij) is a path-conservative first order finite volume

scheme.In order to obtain a fully discrete numerical scheme:

Use a quadrature formulae to approximate the integrals.

Use a high order TVD Runge-Kutta numerical scheme for time integration.




Path-conservative schemes

We consider Path-conservative schemes in the sense introduced by Pares, 2006

D±(WL,WR,η

),

are two Lipschitz continuous functions from Ω× Ω× S1 to Ω such that:

D±(W,W,η) = 0, ∀W ∈ Ω, ∀η ∈ S1,

D−(WL,WR,η) +D+(WL,WR,η) =∫ 1

0

(A(Φ(s; WL,WR,η),η)

)∂Φ

∂s(s; WL,WR,η) ds.

D+(WL,WR,η) = D−(WR,WL,−η),

for every WL,WR ∈ Ω, and η ∈ S1;




Roe based method for two-dimensional nonconservative systems

Roe Linearization (Toumi 1992)

Given a family of paths Φ, a function AΦ : Ω× Ω× S1 →MN×N(R) iscalled a Roe linearization, if it verifies:

1 For each WL,WR ∈ Ω and η ∈ S1, AΦ(WL,WR,η) has N distinct realeigenvalues:

λ1(WL,WR,η) < λ2(WL,WR,η) < · · · < λN(WL,WR,η).

2 AΦ(W,W,η) = A(W,η), for every W ∈ Ω, η ∈ S1.3 For any WL,WR ∈ Ω, η ∈ S1:

AΦ(WL,WR,η)(WR−WL) =

∫ 1

0A(Φ(s; WL,WR,η),η)

∂Φ

∂s(s; WL,WR,η)ds.





Remark

If Ak(W), k = 1, 2 are the Jacobian matrices of two smooth flux functionsFk(W), k = 1, 2, Prop. 3 is independent of the family of paths and it reduces tothe usual Roe property:

AΦ(WL,WR,η) · (WR −WL) = Fη(WR)− Fη(WL),

for any η = (η1, η2) ∈ S1, where

Fη(W) = η1F1(W) + η2F2(W),

represents the flux along the η direction.





Roe method corresponds to the choice

D±(WL,WR,η) = A±Φ(WL,WR,η) · (WR −WL),

where A±Φ(WL,WR,η) is the diagonalizable matrix whose eigenvalues are:

λ±1 (WL,WR,η), λ±2 (WL,WR,η), . . . , λ±N (WL,WR,η),

and whose eigenvectors coincide with those of AΦ(WL,WR,η).

Roe methods require the explicit calculation of the eigenstructure of theintermediate matrix what can be costly. They also require the use of anentropy-fix technique.

The following equality holds:

A±Φ(WL,WR,η) =12

(AΦ(WL,WR,η)± |AΦ(WL,WR,η)|) .





A different decomposition of the Roe matrix can be considered:

D±(WL,WR,η

)=

12(AΦ(WL,WR,η)± QΦ(WL,WR,η)

)· (WR −WL)

where QΦ(WL,WR,η) is a certain viscosity matrix.

Different numerical schemes can be obtained by considering different viscositymatrices QΦ(WR,WL,η).




PVM methods

Castro, Fernandez-Nieto (2013) have introduced a family of finite volumemethods defined by

QΦ(WL,WR,η) = Pl,LR(AΦ(WL,WR,η)),

being Pl,LR(x) a polinomial of degree l,

Pl,LR(x) =

l∑j=0

αLR,j xj.

QΦ(WL,WR,η) can be seen as a Polynomial Viscosity Matrix (PVM).

QΦ(WL,WR,η) has the same eigenvectors than AΦ(WL,WR,η) and if λ is aneigenvalue of AΦ(WL,WR,η), then Pl,LR(λ) is an eigenvalue of QΦ(WL,WR,η).

This strategy was first applied to systems of conservation laws in: P. Degond,P.F. Peyrard, G. Russo, Ph. Villedieu. Polynomial upwind schemes forhyperbolic systems. C. R. Acad. Sci. Paris 1 328, 479-483, 1999.




PVM methods

Some classical methods for systems conservation laws can be interpreted andgeneralized to nonconservativ hyperbolic systems as PVM-methodscorresponding to some particular choices of polynomials:

Polynomial of degree 0: Rusanov, Lax-Friedrichs.Polynomial of degee 1: HLL.Polynomial of degree 2: Lax-Wendroff, FORCE, GFORCE.

New methods with good properties can be obtained. The computation of theeigenstructure of the matrix and entropy-fix techniques are not required.Examples: PVM-2U, IFCP.

More recently, Castro, Gallardo, Marquina have derived a further extensionbased on Rational Viscosity Matrices.




PVM-2U(SM, Sm) method

P2,LR(x) = αLR,0 + αLR,1x + αLR,2x2,

such asP2,LR(Sm) = |Sm|, P2,LR(SM) = |SM|, P′2,LR(SM) = sgn(SM),

where

SM =

λ1,LR if |λ1,LR| ≥ |λN,LR|,λN,LR if |λ1,LR| < |λN,LR|.

Sm =

λN,LR if |λ1,LR| ≥ |λN,LR|,λ1,LR if |λ1,LR| < |λN,LR|.

SL !1 !2 !j...

!N SR

PVM−2U(SL,SR)




PVM-2U(SL, SR, Sint) method or IFCP method

P2,LR(x) = αLR,0 + αLR,1x + αLR,2x2, 1 SL S2L

1 SR S2R

1 Sint S2int

αLR,0

αLR,1

αLR,2

=

|SL||SR||Sint|

,

SL (respectively SR) is an approximation of the minimum (respectively maximum)wave speed and

Sint = Sext max(|λ2,LR|, . . . , |λN−1,LR|),

Sext =

sgn(SL + SR), if (SL + SR) 6= 0,1, otherwise.




PVM-2U(SL, SR, Sint) method or IFCP method

!1 !N"int... ...




References

References

C. Pares.Numerical methods for nonconservative hyperbolic systems: a theoretical framework.SIAM J. Numer. Anal., 44(1):300-321, 2006.

M. Castro, E.D. Fernandez, A. Ferreiro, J.A. Garcıa and C. Pares.High order extensions of Roe schemes for two dimensional nonconservative hyperbolicsystems. J. Sci. Comput., 39: 67-114, 2009.

M.J. Castro, A. Pardo, C. Pares, E.F. Toro.On some fast well-balanced first order solvers for nonconservative systems. Math. Comp.79, 1427-1472, 2010.

E.D. Fernandez-Nieto, E.D. Fernandez Nieto, C.Pares.On an Intermediate Field Capturing Riemann solver based on a Parabolic viscosity matrixfor the two-layer shallow water system. J. Sci. Comp.48, 117-140, 2011.

M.J. Castro, E.D. Fernandez.A class of computationally fast first order finite volume solvers: PVM methods. SIAM J.Sci. Comput., 34(4), 21732196, 2013.




High Performance Computing (HPC)Real applications

HPC

Domain Decomposition: mesh partitioning + MPI.

GPUs as computing kernels.

GPUs: Graphics processing units are very efficient at manipulating computergraphics.GPUs have multiple parallel processors: Nvidia GTX Gforce 580 has 512 Cudacores.Modern GPUs provide both single- and double-precision computations but atdifferent efficiency.





Why GPUs?

CPUs

Northwood: 1(2 logical) Core(s), 2’2 GHz, 512 KB L2Cache

Harpertown: 4 Cores, 3’4 GHz, 2x6 MB L2 Cache

Westmere: 6 (12 logical) Cores, 6x256 KB L2 Cache,12 MB L3 Cache

GPUs

GeForce FX 5900: 450 MHz, 256 MB

GForce 8800 GTX: 128 Cores, 575 MHz, 728 MB

GeForce GTX 580: 512 Cores, 1544 MHz, 1546 MB





An example (Two-layer shallow-water system). First order





An example (Two-layer shallow-water system). Second and third order

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Dambreak problem: Limonero Dam (close to Malaga (Spain))

Resolution 5 m × 5 m.

Number of cells: 1052224.

Real simulated time: 20 min.

Used scheme: Second order HLL or PVM-1U(SL,SR).

Positivity of the water height is ensured.

Graphics card: GeForce GTX 570. Speedup: 230 (1 Intel Core i7 920).





Dambreak problem: Limonero Dam (close to Malaga (Spain))





Physical Model of Gilbert InletPicture from H. Fritz et. al., 2009

Based on the generalized Froude similarity a cross section Gilber Inlet was builtat 1:675 scale in a 2D physical laboratory model (L ×W × H: 11m, 0.5 m, 1m) by Fritz et al., 2001.We use these experimental data for our numerical experiment.





Settings

Settings

xmin: −1000 m.

xmax: 2000 m.

ymin: −170 m.

ymax: 170 m.

nx: 1200.

ny: 136.

Simulated time: 240 s.

CFL: 0.9

Ratio of densities: r = 0.50

Coulomb angle of repose: 25.

Coef. Friction between layers: 10−3.

Coef. Friction water-bottom: 2.5 · 10−2.





Tsunami generated by a landslide





Comparison with experimental data





Tsunami at the Alboran Sea I





Tsunami at the Alboran Sea II: Bathymetry reconstruction





Real applications: Tsunami in the continental margin of Alboran island

Settings

Dimensions: 180 km × 190 km.

∆x = ∆y = 50 m.

13.680.000 cells grid.

Simulated time: 3600 s.

CFL: 0.9

Ratio of densities: r = 0.55

Friction angle: 11.

Coef. Friction between layers: 10−5.

Coef. Friction water-bottom: 5 · 10−2.





Tsunami at the Alboran Sea III: simulation





Tsunami at the Alboran Sea IV: Comparison with actual bathymetry

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Batimetría actualBatimetría simulada






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Tsunami generated by a submarine landslide in Sumatra

Resolution 20 m × 20 m.

Number of cells: 939500.

Real simulated time: 12 min.

Used scheme: Second order PVM-2U(SL,SR,Sint).





Tsunami generated by a submarine landslide in Sumatra





Webpage:

http://edanya.uma.es

Youtube Channel:

http://youtube.com/grupoedanya


http://edanya.uma.es

http://youtube.com/grupoedanya

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