A quadtree-adaptive multigrid solver for the Serre-Green ...A quadtree-adaptive multigrid solver for...

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Introduction Scale distribution for tsunamis The Tohoku tsunami

A quadtree-adaptive multigrid solver for theSerre-Green-Naghdi equations

Stephane Popinet

Institut Jean le Rond ∂’AlembertCNRS/Universite Pierre et Marie Curie

October 15, 2014

Outline

1 Wave equations, multigrid and adaptive meshes

2 Scale distribution for tsunamis

3 The Tohoku tsunami of 11th March 2011

Outline

Wave equations: Navier–Stokes and Euler

Navier–Stokes: two-phases, incompressible, 3D

Inviscid, irrotational fluid: potential flow solution

u = ∇φ

Incompressibility∇2φ = 0

Momentum equation

∂tφ+1

2∇ · φ2 + gη = 0

Free-surface boundary condition

∂yφ = ∂tη + ∂xφ∂xη

Wave equations: Navier–Stokes and Euler

Navier–Stokes: two-phases, incompressible, 3D

Inviscid, irrotational fluid: potential flow solution

u = ∇φ

Incompressibility∇2φ = 0

Momentum equation

∂tφ+1

2∇ · φ2 + gη = 0

∂yφ = ∂tη + ∂xφ∂xη

Wave equations: linearised approximations

ε = η/h0 << 1

Fully linearised case, µ = h20/λ

2 << 1: the ∂’Alembert waveequation (1742)

∂2φ

∂x2=∂2φ

∂2η

∂x2=∂2η

with unit velocity√

gh0.Short linear waves, µ = h2

0/λ2 >> 1: Airy waves (1845)

∇2φ = 0

∂2φ

∂t2+∂φ

∂y= 0

This gives the dispersion relation

ω2 = gk tanh(kh0)

ε = η/h0 << 1Fully linearised case, µ = h2

0/λ2 << 1: the ∂’Alembert wave

equation (1742)∂2φ

∂x2=∂2φ

∂2η

∂x2=∂2η

Short linear waves, µ = h20/λ

2 >> 1: Airy waves (1845)

∇2φ = 0

∂2φ

∂t2+∂φ

∂y= 0

ω2 = gk tanh(kh0)

ε = η/h0 << 1Fully linearised case, µ = h2

0/λ2 << 1: the ∂’Alembert wave

equation (1742)∂2φ

∂x2=∂2φ

∂2η

∂x2=∂2η

gh0.Short linear waves, µ = h2

0/λ2 >> 1: Airy waves (1845)

∇2φ = 0

∂2φ

∂t2+∂φ

∂y= 0

ω2 = gk tanh(kh0)

Wave equations: non-linear waves

Long waves: η arbitrary, µ = h20/λ

2 << 1. The Saint-Venant (1871)or shallow-water equations:

∂tu + u∂xu = −g∂xη

∂tη + ∂x [(h0 + η)u] = 0

Balance between dispersion and non-linearity, the Korteweg–de Vriesequation (1895)

∂tη +3

2η∂xη +

6∂3x3η = 0

gives solitary waves

Dispersive corrections to shallow-water: The (weakly non-linear)Boussinesq equations (1871)

∂tu + u∂xu = −g∂xh +h2

2∂3x2tu

∂tη + ∂x(hu) =h3

6∂3x3 u

recovers KdV

∂tη + ∂x [(h0 + η)u] = 0

∂tη +3

2η∂xη +

6∂3x3η = 0

2∂3x2tu

∂tη + ∂x(hu) =h3

6∂3x3 u

recovers KdV

∂tη + ∂x [(h0 + η)u] = 0

∂tη +3

2η∂xη +

6∂3x3η = 0

2∂3x2tu

∂tη + ∂x(hu) =h3

6∂3x3 u

recovers KdV

The Saint-Venant equations

Conservative form

qdΩ =

∫∂Ω

f(q) · nd∂Ω−∫

hg∇z

, f(q) =

hu hvhu2 + 1

2gh2 huv

huv hv 2 + 12gh2

System of conservation laws (with source terms)

Analogous to the 2D compressible Euler equations (with γ = 2)

Hyperbolic → characteristic solutions

Godunov-type (colocated) 2nd-order finite-volume, shock-capturing

Wetting/drying, positivity, lake-at-rest balance: scheme of Audusseet al (2004)

Wave equations: higher-order expansions

Start from the exact Zakharov–Craig–Sulem (1968, 1993) inviscidEuler formulation

∂tη +∇ · (hV ) = 0

∂t∇ψ +∇η +ε

2∇|∇ψ|2 − εµ∇ (−∇ · (hV ) +∇(εη) · ∇ψ)2

2(1 + ε2µ|∇η|2)= 0

Asymptotic expansion of ψ = ψ0 + µψ1 + µ2ψ2 . . .

At first-order ∇ψ = V + O(µ) → Saint-Venant

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = 0

Next order O(µ2): The Serre (1953, 1D), Green and Nagdhi (1976,2D) equations. Similar formulations rediscovered in the 1990s(Nwogu, 1993, Wei and Kirby, 1995 etc...)

∂tη +∇ · (hV ) = 0

∂t∇ψ +∇η +ε

2∇|∇ψ|2 − εµ∇ (−∇ · (hV ) +∇(εη) · ∇ψ)2

2(1 + ε2µ|∇η|2)= 0

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = 0

∂tη +∇ · (hV ) = 0

∂t∇ψ +∇η +ε

2∇|∇ψ|2 − εµ∇ (−∇ · (hV ) +∇(εη) · ∇ψ)2

2(1 + ε2µ|∇η|2)= 0

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = 0

∂tη +∇ · (hV ) = 0

∂t∇ψ +∇η +ε

2∇|∇ψ|2 − εµ∇ (−∇ · (hV ) +∇(εη) · ∇ψ)2

2(1 + ε2µ|∇η|2)= 0

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = 0

The Serre–Green–Naghdi equations

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = D

(I + µT )(D) = µQ(V )

withT (D) = ∇(h3∇ · D) + . . .

and Q(V ) a (complicated) function of the first- and second-derivatives ofV .

No source term in the mass equation

Requires the inversion of a (spatially-coupled), time-dependent,2nd-order linear system for (vector) D

Can be recast into two scalar tridiagonal systems (only on regulargrids) but this is complicated

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = D

(I + µT )(D) = µQ(V )

withT (D) = ∇(h3∇ · D) + . . .

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = D

(I + µT )(D) = µQ(V )

withT (D) = ∇(h3∇ · D) + . . .

∂tη +∇ · (hV ) = 0

∂tV + εV · ∇V +∇η = D

(I + µT )(D) = µQ(V )

withT (D) = ∇(h3∇ · D) + . . .

Geometric multigrid

Fedorenko (1961), Brandt (1977)

Convergence acceleration technique for iterative solvers

e.g. Gauss–Seidel converges in O(λ/∆) iterations ⇒ wavelengthdecomposition of the problem on different grids

1 Given an initial guess u?

2 Compute residual on fine grid: R∆ = β − L(u?)3 Restrict residual to coarser grid: R∆ → R2∆

4 Solve on coarse grid: L(δu2∆) = R2∆

5 Prolongate the correction onto fine grid: δu2∆ → δu?∆

6 Smooth the correction (using e.g. Gauss–Seidel iterations)7 Correct the initial guess: u = u? + δu∆

Full multigrid has optimal computational cost O(N)

Similar to Fourier (frequency domain) and closely-related to waveletdecomposition of the signal

Unstructured statically refined mesh

Adaptive in spaceMultigrid difficult

Regular Cartesian grid

Not adaptiveMultigrid easy

Dynamic refinement using quadtrees

Adaptive in space and timeMultigrid easy (require storage on non-leaf levels)

A natural multi-scale/frequency representation

⇒ Efficient multigrid solvers⇒ A large collection of other efficient “divide-and-conquer” algorithms:spatial indexing, compression etc...⇒ Formally linked to wavelets/multifractals (“multiresolution analysis”)

Why adaptivity? Scaling of solution cost

Number of degrees of freedom scales like

C ∆−4

(4 = 3 spatial dimensions + time)Moore’s law

Computing power doubles every two years

combined with the above scaling gives

Spatial resolution of e.g. climate models doublesevery eight years

Does this work?

100000

1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 2012

ECMWFMetOffice

resolution doubles every 8 yearsresolution doubles every 10 years

Basilisk: a new quadtree-adaptive framework

Free Software (GPL): basilisk.fr

Principal objectives: Precision – Simplicity – Performance

“Generalised Cartesian grids”: Cartesian schemes are turned“seamlessly” into quadtree-adaptive schemes

Basic abstraction: operations only on local stencils

a[−1,1] a[0,1] a[1,1]

a[0,0]a[−1,0]

a[−1,−1] a[0,−1] a[1,−1]

a[1,0]

Code example: b = ∇2a

f o r e a c h ( )b [ 0 , 0 ] = ( a [ 0 , 1 ] + a [ 1 , 0 ] + a [0 ,−1] + a [−1 ,0] − 4 .∗ a [ 0 , 0 ] )

/ sq ( D e l t a ) ;

Restriction

vo id r e s t r i c t i o n ( s c a l a r v )

v [ ] = ( f i n e ( v , 0 , 0 ) + f i n e ( v , 1 , 0 ) + f i n e ( v , 0 , 1 ) + f i n e ( v , 1 , 1 ) ) / 4 . ;

Prolongation

vo id p r o l o n g a t i o n ( s c a l a r v )

/∗ b i l i n e a r i n t e r p o l a t i o n from p a r e n t ∗/v [ ] = ( 9 .∗ c o a r s e ( v , 0 , 0 ) +

3 .∗ ( c o a r s e ( v , c h i l d . x , 0 ) + c o a r s e ( v , 0 , c h i l d . y ) ) +c o a r s e ( v , c h i l d . x , c h i l d . y ) ) / 1 6 . ;

Boundary conditions

Guarantees stencil consistency independently of neighborhood resolution

active points

restriction

prolongation

vo id boundary ( s c a l a r v , i n t l e v e l )

f o r ( i n t l = l e v e l − 1 ; l <= 0 ; l−−)f o r e a c h l e v e l ( l )

r e s t r i c t i o n ( v ) ;f o r ( i n t l = 0 ; l <= l e v e l ; l ++)

f o r e a c h h a l o l e v e l ( l )p r o l o n g a t i o n ( v ) ;

Generic multigrid implementation in Basilisk

vo id m g c y c l e ( s c a l a r a , s c a l a r r e s , s c a l a r dp ,vo id (∗ r e l a x ) ( s c a l a r dp , s c a l a r r e s , i n t depth ) ,i n t n r e l a x , i n t m i n l e v e l )

/∗ r e s t r i c t r e s i d u a l ∗/f o r ( i n t l = depth ( ) − 1 ; l <= m i n l e v e l ; l−−)

f o r e a c h l e v e l ( l )r e s t r i c t i o n ( p o i n t , r e s ) ;

/∗ m u l t i g r i d t r a v e r s a l ∗/f o r ( i n t l = m i n l e v e l ; l <= depth ( ) ; l ++)

i f ( l == m i n l e v e l )/∗ i n i t i a l g u e s s on c o a r s e s t l e v e l ∗/f o r e a c h l e v e l ( l )

dp [ ] = 0 . ;e l s e

/∗ p r o l o n g a t i o n from c o a r s e r l e v e l ∗/f o r e a c h l e v e l ( l )

p r o l o n g a t i o n ( dp ) ;boundary ( dp , l ) ;/∗ r e l a x a t i o n ∗/f o r ( i n t i = 0 ; i < n r e l a x ; i ++)

r e l a x ( dp , r e s , l ) ;boundary ( dp , l ) ;

/∗ c o r r e c t i o n ∗/f o r e a c h ( )

a [ ] += dp [ ] ;

Application to Poisson equation ∇2a = b

The relaxation operator is simply

vo id r e l a x ( s c a l a r a , s c a l a r b , i n t l )

f o r e a c h l e v e l ( l )a [ ] = ( a [ 1 , 0 ] + a [−1 ,0] + a [ 0 , 1 ] + a [0 ,−1] − sq ( D e l t a )∗b [ ] ) / 4 . ;

The corresponding residual is

vo id r e s i d u a l ( s c a l a r a , s c a l a r b , s c a l a r r e s )

f o r e a c h ( )r e s [ ] = b [ ] − ( a [ 1 , 0 ] + a [−1 ,0] + a [ 0 , 1 ] + a [0 ,−1]

− 4 .∗ a [ ] ) / sq ( D e l t a ) ;

The Serre–Green–Naghdi residual

−αd

3∂x(h3∂xDx

(∂xη∂xzb +

2∂2x zb

2∂2xyzb + ∂xη∂yzb

2∂yzb∂xDy −

3∂2xyDy − h∂yDy

(∂xh +

2∂xzb

)]= bx

vo id r e s i d u a l ( v e c t o r D, v e c t o r b , v e c t o r r e s )

f o r e a c h ( )f o r e a c h d i m e n s i o n ( )

double hc = h [ ] , dxh = dx ( h ) , dxzb = dx ( zb ) , d x e t a = dxzb + dxh ;double h l 3 = ( hc + h [ −1 , 0 ] ) / 2 . ; h l 3 = cube ( h l 3 ) ;double hr3 = ( hc + h [ 1 , 0 ] ) / 2 . ; hr3 = cube ( hr3 ) ;

r e s . x [ ] = b . x [ ] −(−a l p h a d / 3 .∗ ( hr3∗D. x [ 1 , 0 ] + h l 3∗D. x [−1 ,0] −

( hr3 + h l 3 )∗D. x [ ] ) / sq ( D e l t a ) +hc ∗( a l p h a d ∗( d x e t a∗dxzb + hc /2.∗ d2x ( zb ) ) + 1 . )∗D. x [ ] +a l p h a d∗hc ∗ ( ( hc /2 .∗ d2xy ( zb ) + d x e t a∗dy ( zb ))∗D. y [ ] +hc /2.∗ dy ( zb )∗ dx (D. y ) − sq ( hc ) / 3 .∗ d2xy (D. y )− hc∗dy (D. y )∗ ( dxh + dxzb / 2 . ) ) ) ;

Example of validation: wave propagation over an ellipsoidalshoal

Wave tank experiments of Berkhoff et al, Coastal Engineering, 1982Tests both non-linearities and dispersive effects

see http://basilisk.fr/src/examples/shoal.c

Instantaneous wave field

Maximum wave height

Comparison with experimental data

Outline

2004 Indian ocean tsunami

Staggered fault displacement model (5 segments)

2004 Indian ocean tsunami

1 km ≤ Spatial resolution ≤ 150 km

Adaptivity

Truncation error of the wave height < 5 cm

Average number of elements as a function of maximumresolution

Connection with fractal dimension

Classical example: the Sierpinski triangle

has a fractal (Minkowski–Bouligand or “box-counting” or “information”)dimension of ≈ 1.6.In other words, the cost of describing such an object using quadtreeswould scale as ∆−1.6 not ∆−2.

Evolution of the scaling exponent with time

Mandelbrot, How long is the coast of Britain?, Science, 1967

Outline

Tohoku tsunami: bathymetry, DART and GLOSS stations

Source model and pressure gauges

Source model from seismic inversion only, Shao, Li and Ji, UCSB190 Okada subfaults

Available a few days after the event (March 14th 2011)

Evolution of wave elevation (Saint-Venant)

dark blue: -1 metres, dark red: +2 metres

10 hours

Evolution of spatial resolution

dark blue: 5 levels, ≈ 2.3, yellow: 12 levels, ≈ 1 arc-minutedark red: 15 levels, ≈ 250 metres

E(h) < ε = 2.5 cm

Detail for the Miyagi prefecture area

220× 180 km, 1 hour after the event

220× 180 km, 2 hours after the event

220× 180 km, 4 hours after the event

Evolution of the number of grid points

100000

0 2 4 6 8 10

of grid p

Time (hours)

adaptive2

Single-CPU runtime ≈ 3 hours, 5× 105 updates/sec

Maximum elevation reached over 10 hours

Miyako Ofunato

Miyagi Fukushima

Long distance propagation: DART buoys

0 2 4 6 8 10

BuoyModel

-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

0 2 4 6 8 10

BuoyModel

0 2 4 6 8 10

BuoyModel

Inshore propagation: GLOSS tide gauges

Ofunato, 99 km

0 2 4 6 8 10

Wave h

StationModel

Hanasaki, 588 km

-2-1.5

-1-0.5

0 2 4 6 8 10

Wave h

StationModel

Wake island, 3145 km

0 2 4 6 8 10

Wave h

StationModel

Flooding: comparison with Synthetic Aperture Radar

Flooding: comparison with field surveys

Tohoku Earthquake Tsunami Joint Survey Group: > 5000 GPS records

Are dispersive terms important?

Saint-Venant Serre–Green–Naghdi

30 minutes after the eventColorscale ±2 metres

30 minutes after the event

Conclusions

The Serre–Green–Naghdi dispersive model can be implemented as amomentum source added to an existing Saint-Venant model

Preserves the well-balancing, positivity of water depth(wetting/drying) of the orginal scheme

Multigrid is simple and efficient for inverting the SGN operator onadaptive quadtree grids

Validation for the Tohoku tsunami using a source model obtainedonly from seismic data (Popinet, 2012, NHESS).

The effective number of degrees of freedom of the physical problemscales like

C ∆d

with d the effective (or “fractal” or “information”) dimension.

This leads to large gains in computational cost – for a given errorthreshold – for a wide range of problems (including wave dynamics).Current developments

4th-order quadtree-adaptive discretisationsMPI parallelism (dynamic load-balancing etc...)GPUs

A quadtree-adaptive multigrid solver for the Serre-Green ...A quadtree-adaptive multigrid solver for...

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