Sensitivity Analysis for Shallow Water Equations - Advanced Study ...

Sensitivity Analysis for Shallow Water Equations 

Vani Cheruvu 

Advanced Study Program Research Review 

National Center for Atmospheric Research 

June 14, 2007 

Vani Cheruvu (ASP Research Review) Adjoint Sensitivity Analysis 06/14/2007 1 / 30

Organization of the talk 

Motivation 

Adjoint Sensitivity Analysis for SWE 

◮ Direct Sensitivity Method 

◮ Adjoint Sensitivity Method 

◮ SWE and its Adjoint 

◮ Spectral Element Method 

Spectral finite volume method for SWE 

Wavelet based limiter for discontinuous Galerkin Method 

◮ Gibbs phenomenon 

◮ Limiters 

Summary 




Chennai’s coastline before and after tsunami 


Chennai’s coastline before and after tsunami 


Chennai’s Marina beach after tsunami 


Chennai’s Marina beach after tsunami 


Why an Adjoint Equation? 

Diffusion equation defined in the domain x ∈ (0, L) is 

Objective function : 

where r = δ(x − x T ). 

Adjoint Equation 

Objective function: 

where f = Mδ(x − x s ). 

D ∂2 C 

+ f = 0, C(0) = C(L) = 0 

∂x 2 

J = 

J = 

∫ L 

0 

r(x)C(x) dx 

D ∂2 C ∗ 

∂x 2 + r = 0 

∫ L 

0 

f (x)C ∗ (x) dx 


Derivation 

∫ L 

0 

[ ] [ 

C ∗ D ∂2 C 

∂x 2 + f dx = D C ∗ ∂C ] L [ ] ∂C 

∗ L 

− D 

∂x 

0 

∂x C 0 

∫ L 

+ 

[CD ∂2 C ∗ ] 

∂x 2 + C ∗ f dx = 0 

0 

J = 

∫ L 

0 

[CD ∂2 C ∗ 

∂x 2 

] 

+ C ∗ f dx + 

∫ L 

0 

rCdx 

J = 

∫ L 

0 

C 

[D ∂2 C ∗ 

∂x 2 + r ] 

dx + 

∫ L 

0 

fC ∗ dx 


Shallow Water Equations 

Objective Function 

∂η 

∂t + ∂(Hu′ + Ūη) 

∂x 

∂u ′ 

∂t + ∂(Ūu′ + gη) 

∂x 

= 0 

= 0 

J = 

∫ T ∫ L 

0 

0 

r(η, u ′ ; x, t)dxdt 

t= time; x = distance along the channel; η = depth of flow; 

u ′ = discharge per unit width; r = user-defined measuring function 

r = 1 2 [η(x 0, t) − ¯η(x 0 )] 2 δ(x − x 0 ) 


Direct Sensitivity Method 

The basic problem is solved 

A reference value of the objective function J 0 is computed 

A flow variable is perturbed at the boundary at the selected 

perturbation time 

The associated objective function value is computed again 

This procedure is repeated for sufficiently many selections of time 

This yields the temporal evolution of the sensitivities 


Adjoint Sensitivity Method 

Derivation of 1D Adjoint Equations 

Adjoint variables : φ(x, t) and ψ(x, t) 

Multiply the continuity equation by φ(x, t) and the momentum 

equation by ψ(x, t) 

The sum of these two products is then integrated over space and time 

Flow control is desired, hence the variation with respect to η and u ′ 

Variation of the objective function with respect to η and u ′ 

Add the variations 

Choose initial condition for the adjoint problem 

Choose boundary values for the adjoint variables 


Shallow water equations and its Adjoint 

Adjoint 

∂η 

∂t + Ū ∂η 

∂x + ¯V ∂η 

∂y + H ∂u′ 

∂x + H ∂v ′ 

∂y 

∂u ′ 

∂t + Ū ∂u′ 

∂x + ¯V ∂u′ 

∂y + g ∂η 

∂x 

∂v ′ 

∂t + Ū ∂v ′ 

∂x + ¯V ∂v ′ 

∂y + g ∂η 

∂y 

= 0 

= 0 

= 0 

∂φ 

∂τ − Ū ∂φ 

∂x − ¯V ∂φ 

∂y − g ∂ψ 1 

∂x − g ∂ψ 2 

∂y + ∂r 

∂η 

= 0 

∂ψ 1 

∂τ − Ū ∂ψ 1 

∂x − ¯V ∂ψ 1 

∂y − H ∂φ 

∂x + ∂r 

∂u ′ = 0 

∂ψ 2 

∂τ − Ū ∂ψ 2 

∂x − ¯V ∂ψ 2 

∂y − H ∂φ 

∂y + ∂r 

∂v ′ = 0 


Solution of Shallow Water Equations 

Contour lines of Gaussian Hill; 1000 time steps with ∆t = 0.0005 

400 

350 

300 

250 

200 

150 

100 

50 

50 100 150 200 250 300 350 400 


Solution of Adjoint Shallow Water Equations 

Contour lines of Gaussian Hill; 500 time steps with ∆t = 0.0005 

400 

350 

300 

250 

200 

150 

100 

50 

50 100 150 200 250 300 350 400 


Spectral Element Method 

The domain [a, b] is partitioned into N x non-overlapping intervals 

called elements. 

I j = [x j−1/2 , x j+1/2 ], j = 1, . . . , N x , and ∆x j = (x j+1/2 − x j−1/2 ) 

I 

j!1 

I j 

I j+1 

x x x x 

j!3/2 

j!1/2 

j+1/2 

j+3/2 

Map every Element I j onto the reference element [−1, +1]. 

Introduce a local coordinate ξ ∈ [−1, +1] s.t., 

ξ = 2 (x − x j) 

∆x j 

, x j = (x j−1/2 + x j+1/2 )/2 ⇒ ∂ 

∂x = 2 

∆x j 

∂ 

∂ξ . 


SE-1D: Space Discretization 

Regular Element 

I j 

Reference Element 

! 

x j!1/2 

x j+1/2 

! 1 

GLL Grid 

+1 

How do the neighboring elements talk? 

Nodal - Averaging the values on the element boundaries 

Time Integration Scheme 

Leapfrog Scheme u n+1 = u n−1 − 2∆tF (u n ) 

The solution within each element is interpolated with a high-order 

polynomial using nodal basis set. 


SFV in 1D 

For a given order of accuracy k, each SV is divided into k control 

volumes (CV) 

The ith CV of jth SV is : C j,i = [x j,i−1/2 , x j,i+1/2 ] and with cell width 

∆x j,i = x j,i+1/2 − x j,i−1/2 

x 

CV 

j,i!1/2 

x j,i+1/2 

x j!1/2 

SV 

I j 

x j+1/2 

Discontinuous Galerkin: Flux evaluation - Riemann Solver 

Spectral Finite Volume: Numerical flux at element boundaries and 

analytical flux at the cell boundaries 


SFV-1D: Flux term 

( 

_ 

) ( + ) (_ 

) ( + ) 

L R L R 

x j!1/2 

I j 

x j+1/2 

Flux function F (U) is not uniquely defined at x j±1/2 

F (U) is replaced by a numerical flux function ˆF (U), dependent on 

the left and right limits of the discontinuous function U 

Typical flux formulae: Godunov, Lax-Friedrichs, Roe, HLLC, etc 


2D Grid 

X 

X 

X 

X 

X 

X 

CV 

X 

X 

X 

X 

X 

X 

Given SV is partitioned into 4 × 4 CVs using 5 × 5 GLL nodes 

3 GL nodes are used for the evaluation of boundary integral on each 

side of a CV 

Ū ij are centered at each CV 


Sensitivity Equations 

Sensitivities required for subcritical flow control are given by 

δJ 

δu ′ (0,t) and 

δJ 

δη(L,t) 

Sensitivities required for supercritical flow control are given by 

δJ 

δu ′ (0,t) and 

δJ 

δη(0,t) 

To derive expressions for sensitivities in terms of the adjoint variables 

Initial condition for the basic problem 

η(x, 0) = η 0 (x), u ′ (x, 0) = u ′ 0 (x); δη(x, 0) = 0, δu′ (x, 0) = 0 

Initial conditions in the adjoint problem 

φ(x, T ) = 0 = ψ(x, T ) 

Selection of boundary conditions for the adjoint problem is motivated 

by the boundary conditions of the basic problem 

Expressions are then derived for sensitivities to these boundary values 


Nonreflective Outflow Boundary 

Adjoint boundary conditions: 

ψ(0, t) = 0 and 

φ(L, t) + 2 u′ (L, t) 

ψ(L, t) = 0 

η(L, t) 

Sensitivities for the subcritical flow then are: 

δJ 

δu ′ (0, t p ) 

δJ 

δη(L, t p ) 

= φ(0, t p ) 

[ (u ′ ) 

(L, t p ) 2 

= 

− gη(L, t p )] 

ψ(L, t p ) 

η(L, t p ) 


Cubed-Sphere Geometry 

Sadourny (1972) 

HOMME High-Order Method Modeling Environment 

Z 

P 1 

Y 

P5 

(Top) 

X 

P4 

P1 P2 P3 

Figure: Cube inscribed in a 

Sphere 

P6 

(Bottom) 

Figure: Faces of the Cube 


Neighboring elements that share same cube edge 

Face ! A 

Face ! B 

y 

x 

x 

y 

Figure: (a) 

Figure: (b) 

To compute the flux on the edge of the cubed-sphere, both the left and 

right components F − n and F + n are required. 

A l 

" 

F + 

l 

G + 

l 

# 

» 

F 

+ 

= s 

G s 

+ 

– » 

, A −1 F 

+ 

s 

n G s 

+ 

– » 

F 

+ 

= n 

G n 

+ 

– 

⇒ 

A −1 

n A l 

" 

F + 

l 

G + 

l 

# 

. 


Spectral Finite Volume Results 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

!1 !0.8 !0.6 !0.4 !0.2 0 0.2 0.4 0.6 0.8 1 


Gibbs Phenomenon 

(sin(x) + 1 3 sin(3x) + 1 5 sin(5x) + . . . ) 

f (x) = 4 π 

1.5 

S1 

1 

S3 

0.5 

0 

!0.5 

S2 

!1 

!1.5 

!4 !3 !2 !1 0 1 2 3 4 


DG-1D : Numerical Examples 

U t + F x (U) = 0, Ω ≡ [−1, 1], periodic domain. For linear advection (square wave) F (U) = U without limiter (left panel) and 

with limiter (right panel) 

1 

1 

0.5 

0.5 

0 

0 

!1 !0.5 0 0.5 1 

!1 !0.5 0 0.5 1 


DG-1D : Numerical Examples 

U t + F x (U) = 0, Ω ≡ [−1, 1], periodic domain. For linear advection (sine wave) F (U) = U without limiter (left panel) and 

with limiter (right panel) 

1 

1 

0.8 

0.8 

0.6 

0.6 

0.4 

0.4 

0.2 

0.2 

0 

0 

!0.2 

!0.2 

!0.4 

!0.4 

!0.6 

!0.6 

!0.8 

!0.8 

!1 

!1 !0.8 !0.6 !0.4 !0.2 0 0.2 0.4 0.6 0.8 1 

!1 

!1 !0.8 !0.6 !0.4 !0.2 0 0.2 0.4 0.6 0.8 1 

Figure: Without Limiter 

Figure: With Limiter 


Summary 

Adjoint sensitivity method for SWE using high-order spectral element 

method 

◮ Fast and accurate tool for computing the sensitivities 

◮ Efficient in determining how a specific condition is dependent on many 

different parameters which influence it 

◮ Adjoint sensitivity method is more efficient than Direct sensitivity 

method 

Wavelets can be used for limiting the discontinuous Galerkin solutions 

“Numerical Analysis is service to science and engineering” - 

Gene Golub 

Acknowledgements: Maura Hagan, Al Kellie, Joe Tribbia

Sensitivity Analysis for Shallow Water Equations - Advanced Study ...

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