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
Paris
October 15, 2014
Introduction Scale distribution for tsunamis The Tohoku tsunami
Outline
1 Wave equations, multigrid and adaptive meshes
2 Scale distribution for tsunamis
3 The Tohoku tsunami of 11th March 2011
Introduction Scale distribution for tsunamis The Tohoku tsunami
Outline
1 Wave equations, multigrid and adaptive meshes
2 Scale distribution for tsunamis
3 The Tohoku tsunami of 11th March 2011
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
Wave equations: linearised approximations
ε = η/h0 << 1
Fully linearised case, µ = h20/λ
2 << 1: the ∂’Alembert waveequation (1742)
∂2φ
∂x2=∂2φ
∂t2
∂2η
∂x2=∂2η
∂t2
with unit velocity√
gh0.Short linear waves, µ = h2
0/λ2 >> 1: Airy waves (1845)
∇2φ = 0
Free-surface boundary condition
∂2φ
∂t2+∂φ
∂y= 0
This gives the dispersion relation
ω2 = gk tanh(kh0)
Introduction Scale distribution for tsunamis The Tohoku tsunami
Wave equations: linearised approximations
ε = η/h0 << 1Fully linearised case, µ = h2
0/λ2 << 1: the ∂’Alembert wave
equation (1742)∂2φ
∂x2=∂2φ
∂t2
∂2η
∂x2=∂2η
∂t2
with unit velocity√
gh0.
Short linear waves, µ = h20/λ
2 >> 1: Airy waves (1845)
∇2φ = 0
Free-surface boundary condition
∂2φ
∂t2+∂φ
∂y= 0
This gives the dispersion relation
ω2 = gk tanh(kh0)
Introduction Scale distribution for tsunamis The Tohoku tsunami
Wave equations: linearised approximations
ε = η/h0 << 1Fully linearised case, µ = h2
0/λ2 << 1: the ∂’Alembert wave
equation (1742)∂2φ
∂x2=∂2φ
∂t2
∂2η
∂x2=∂2η
∂t2
with unit velocity√
gh0.Short linear waves, µ = h2
0/λ2 >> 1: Airy waves (1845)
∇2φ = 0
Free-surface boundary condition
∂2φ
∂t2+∂φ
∂y= 0
This gives the dispersion relation
ω2 = gk tanh(kh0)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 +
1
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 +
1
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 +
1
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
The Saint-Venant equations
Conservative form
∂t
∫Ω
qdΩ =
∫∂Ω
f(q) · nd∂Ω−∫
Ω
hg∇z
q =
hhuhv
, 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)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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...)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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...)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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...)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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...)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
Unstructured statically refined mesh
Adaptive in spaceMultigrid difficult
Introduction Scale distribution for tsunamis The Tohoku tsunami
Regular Cartesian grid
Not adaptiveMultigrid easy
Introduction Scale distribution for tsunamis The Tohoku tsunami
Dynamic refinement using quadtrees
Adaptive in space and timeMultigrid easy (require storage on non-leaf levels)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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”)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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”)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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”)
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
Does this work?
10000
100000
1e+06
1e+07
1e+08
1e+09
1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 2012
# d
egre
es o
f fr
eedom
Year
ECMWFMetOffice
resolution doubles every 8 yearsresolution doubles every 10 years
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 ) ;
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 . ;
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 . ;
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 ) ;
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 [ ] ;
Introduction Scale distribution for tsunamis The Tohoku tsunami
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 ) ;
Introduction Scale distribution for tsunamis The Tohoku tsunami
The Serre–Green–Naghdi residual
−αd
3∂x(h3∂xDx
)+ h
[αd
(∂xη∂xzb +
h
2∂2x zb
)+ 1
]Dx+
αdh
[(h
2∂2xyzb + ∂xη∂yzb
)Dy +
h
2∂yzb∂xDy −
h2
3∂2xyDy − h∂yDy
(∂xh +
1
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 . ) ) ) ;
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
Instantaneous wave field
Introduction Scale distribution for tsunamis The Tohoku tsunami
Maximum wave height
Introduction Scale distribution for tsunamis The Tohoku tsunami
Comparison with experimental data
Introduction Scale distribution for tsunamis The Tohoku tsunami
Outline
1 Wave equations, multigrid and adaptive meshes
2 Scale distribution for tsunamis
3 The Tohoku tsunami of 11th March 2011
Introduction Scale distribution for tsunamis The Tohoku tsunami
2004 Indian ocean tsunami
Staggered fault displacement model (5 segments)
Introduction Scale distribution for tsunamis The Tohoku tsunami
2004 Indian ocean tsunami
1 km ≤ Spatial resolution ≤ 150 km
Introduction Scale distribution for tsunamis The Tohoku tsunami
Adaptivity
Truncation error of the wave height < 5 cm
Introduction Scale distribution for tsunamis The Tohoku tsunami
Average number of elements as a function of maximumresolution
Introduction Scale distribution for tsunamis The Tohoku tsunami
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.
Introduction Scale distribution for tsunamis The Tohoku tsunami
Evolution of the scaling exponent with time
Mandelbrot, How long is the coast of Britain?, Science, 1967
Introduction Scale distribution for tsunamis The Tohoku tsunami
Evolution of the scaling exponent with time
Mandelbrot, How long is the coast of Britain?, Science, 1967
Introduction Scale distribution for tsunamis The Tohoku tsunami
Evolution of the scaling exponent with time
Mandelbrot, How long is the coast of Britain?, Science, 1967
Introduction Scale distribution for tsunamis The Tohoku tsunami
Evolution of the scaling exponent with time
Mandelbrot, How long is the coast of Britain?, Science, 1967
Introduction Scale distribution for tsunamis The Tohoku tsunami
Outline
1 Wave equations, multigrid and adaptive meshes
2 Scale distribution for tsunamis
3 The Tohoku tsunami of 11th March 2011
Introduction Scale distribution for tsunamis The Tohoku tsunami
Tohoku tsunami: bathymetry, DART and GLOSS stations
Introduction Scale distribution for tsunamis The Tohoku tsunami
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)
Introduction Scale distribution for tsunamis The Tohoku tsunami
Evolution of wave elevation (Saint-Venant)
dark blue: -1 metres, dark red: +2 metres
10 hours
Introduction Scale distribution for tsunamis The Tohoku tsunami
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
Introduction Scale distribution for tsunamis The Tohoku tsunami
Detail for the Miyagi prefecture area
220× 180 km, 1 hour after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
Detail for the Miyagi prefecture area
220× 180 km, 2 hours after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
Detail for the Miyagi prefecture area
220× 180 km, 4 hours after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
Evolution of the number of grid points
10000
100000
1e+06
1e+07
1e+08
1e+09
1e+10
0 2 4 6 8 10
Num
ber
of grid p
oin
ts
Time (hours)
adaptive2
24
230
Single-CPU runtime ≈ 3 hours, 5× 105 updates/sec
Introduction Scale distribution for tsunamis The Tohoku tsunami
Maximum elevation reached over 10 hours
Introduction Scale distribution for tsunamis The Tohoku tsunami
Maximum elevation reached over 10 hours
Miyako Ofunato
Miyagi Fukushima
Introduction Scale distribution for tsunamis The Tohoku tsunami
Long distance propagation: DART buoys
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10
Wa
ve
he
igh
t (m
)
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
Wa
ve
he
igh
t (m
)
BuoyModel
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10
Wa
ve
he
igh
t (m
)
BuoyModel
Introduction Scale distribution for tsunamis The Tohoku tsunami
Inshore propagation: GLOSS tide gauges
Ofunato, 99 km
-15
-10
-5
0
5
10
15
20
25
0 2 4 6 8 10
Wave h
eig
ht (m
)
StationModel
Hanasaki, 588 km
-2-1.5
-1-0.5
0 0.5
1 1.5
2 2.5
3
0 2 4 6 8 10
Wave h
eig
ht (m
)
StationModel
Wake island, 3145 km
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10
Wave h
eig
ht (m
)
StationModel
Introduction Scale distribution for tsunamis The Tohoku tsunami
Flooding: comparison with Synthetic Aperture Radar
Introduction Scale distribution for tsunamis The Tohoku tsunami
Flooding: comparison with field surveys
Tohoku Earthquake Tsunami Joint Survey Group: > 5000 GPS records
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
30 minutes after the eventColorscale ±2 metres
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
60 minutes after the eventColorscale ±2 metres
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
90 minutes after the eventColorscale ±2 metres
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
120 minutes after the eventColorscale ±2 metres
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
150 minutes after the eventColorscale ±2 metres
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
30 minutes after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
60 minutes after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
90 minutes after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
120 minutes after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
Are dispersive terms important?
Saint-Venant Serre–Green–Naghdi
150 minutes after the event
Introduction Scale distribution for tsunamis The Tohoku tsunami
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