IVP:

( ∂ _t q = − kq | q | ⁽ h ⁿ⁺¹ _i ) ^−η q ( x _i , t ⁿ ) = q ⁿ⁺

1

i

2

q _i ⁿ⁺¹

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Derivation of a 1D rst-order well-balanced scheme

Semi-implicit nite volume scheme

Solving the IVP yields:

q _i ⁿ⁺¹ = ( h ⁿ⁺¹ _i ) ^η q ⁿ⁺

1 2

i

( h ⁿ⁺¹ _i ) ^η + k ∆ t q ⁿ⁺

1

i

2

.

We use the following approximation of ( h ⁿ⁺¹ _i ) ^η , which provides us with an expression of q _i ⁿ⁺¹ that is equal to q 0 at the equilibrium:

( h ^η ) ⁿ⁺¹ _i = 2 µ ⁿ⁺

1 2

i µ ⁿ _i

h ^−η n+1 i−

¹₂

+

h ^−η n+1 i+

¹₂

+ k ∆ t µ ⁿ⁺

1 2

i q _i ⁿ .

semi-implicit treatment of the friction source term

scheme able to model wet/dry transitions

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

1 Introduction to Godunov-type schemes

2 Derivation of a 1D rst-order well-balanced scheme 3 Two-dimensional and high-order extensions

4 2D and high-order numerical simulations

5 Conclusion and perspectives

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

Two-dimensional extension

2D shallow-water model: ∂ t W + ∇ · F ( W ) = S ^t ( W ) + S ^f ( W )



 



 



∂ _t h + ∇ · q = 0

∂ _t q + ∇ ·

q ⊗ q h ⁺

1 2 gh ² I 2

= − gh∇Z − kq k q k h ^η

to the right: simulation

of the 2011 Japan

tsunami

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

Two-dimensional extension

space discretization: Cartesian mesh x

_i

c

i

e

ij

c

j

n

ij

With F _ij ^{n =} F ⁽ W _i ⁿ , W _j ⁿ ; n ij ) and ν i the neighbors of c i , the scheme reads:

W ⁿ⁺

1 2

i = W _i ⁿ − ^∆ t X

j∈ν

i

| e _ij |

| c _i | F ij ⁿ + ∆ t 2

X

j∈ν

i

( S ^t ₎ ⁿ _ij .

W _i ⁿ⁺¹ is obtained from W ⁿ⁺

1

i

2

with a splitting strategy:

( ∂ _t h = 0

∂ _t q = − k q k q k h ^−η



 

 

 

 

h ⁿ⁺¹ _i = h ⁿ⁺

1

i

2

q ⁿ⁺¹ _i = ( h ^η ) ⁿ⁺¹ _i q ⁿ⁺

1

i

2

( h ^η ) ⁿ⁺¹ + k ∆ t q ⁿ⁺

1 2

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

Two-dimensional extension

The 2D scheme is:

non-negativity-preserving for the water height:

∀ i ∈ Z , h ⁿ _i ≥ ^{0 =} ⇒ ∀ i ∈ Z , h ⁿ⁺¹ _i ≥ 0;

able to deal with wet/dry transitions thanks to the semi-implicitation with the splitting method;

well-balanced by direction for the shallow-water equations with friction and/or topography, i.e.:

it preserves all steady states at rest,

it preserves friction and/or topography steady states in the x-direction and the y-direction,

it does not preserve the fully 2D steady states.

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

High-order extension: the basics, in 1D

x i−1 x i x i+1

W _i+1 ⁿ

W _i−1 ⁿ

W _i ⁿ

x x _i+

¹

2

x _i−

¹

2

W _i ⁿ ∈ P ⁰ : constant (order 1 scheme)

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

High-order extension: the basics, in 1D

x _i−1 x _i x _i+1

W _i+1 ⁿ

W _i−1 ⁿ

W _i ⁿ W d _i ⁿ (x)

W _i,− ⁿ

W _i,+ ⁿ

x x _i+

¹

x _i−

¹ 2 2

W ⁿ ∈ P ¹ : linear (order 2 scheme)

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

High-order extension: the basics, in 1D

x _i−1 x _i x _i+1

W _i+1 ⁿ

W _i−1 ⁿ

W _i ⁿ W d _i ⁿ (x)

W _i,− ⁿ

W _i,+ ⁿ

x x _i+

¹

x _i−

¹ 2 2

W d _i ⁿ ∈ P d : polynomial (order d + 1 scheme)

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

High-order extension: the polynomial reconstruction

polynomial reconstruction (see Diot, Clain, Loubère (2012)):

W c _i ⁿ ( x ) = W _i ⁿ +

d

X

|k|=1

α ^k _i h

( x − x _i ) ^k − M _i ^k i

We have M _i ^k = 1

| c _i | Z

c

i

( x − x _i ) ^k dx such that the conservation property is veried: 1

| c i | Z

c

i

W c _i ⁿ ( x ) dx = W _i ⁿ _.

x

i

W

_i+1ⁿ

W

_i−1ⁿ

W

_iⁿ

W d

_iⁿ

(x)

x x

_i+¹

x

_i−¹

ci

∈S² ∈/S²

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

High-order extension: the scheme

High-order space accuracy W _i ⁿ⁺¹ = W _i ⁿ − ^∆ t X

j∈ν

i

| e _ij |

| c _i |

R

X

r=0

ξ _r F ij,r ⁿ +∆ t

Q

X

q=0

η _q

( S ^{t ) n} i,q + ( S ^{f ) n} i,q

F ij,r ⁿ = F ⁽ c W _i ⁿ ( σ _r ) , W c _j ⁿ ( σ _r ); n _ij )

( S ^{t ) n} _i,q ⁼ S ^t ( W c _i ⁿ ( x q )) and ( S ^{f ) n} _i,q ⁼ S ^f ( W c _i ⁿ ( x q )) We have set:

( ξ _r , σ _r ) r , a quadrature rule on the edge e _{ij ;} ( η _q , x _q ) q , a quadrature rule on the cell c _{i .}

The high-order time accuracy is achieved by the use of SSPRK

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

Well-balance recovery (1D): a convex combination

reconstruction procedure the scheme no longer preserves steady states

Well-balance recovery

We suggest a convex combination between the high-order scheme W _HO and the well-balanced scheme W _{W B :}

W _i ⁿ⁺¹ = θ _i ⁿ ( W _HO ) ⁿ⁺¹ _i + (1 − θ ⁿ _i )( W _{W B} ) ⁿ⁺¹ _i , with θ _i ⁿ the parameter of the convex combination, such that:

if θ ⁿ _i = 0, then the well-balanced scheme is used;

if θ ⁿ _i = 1, then the high-order scheme is used.

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

Well-balance recovery (1D): a steady state detector

Steady state detector steady state solution:







q _L = q _R = q ₀ , E ^:= q ² ₀

h _R − q ² ₀ h _L ⁺

g

2 h ² _R − h ² _L

− ⁽ S ^t + S ^f )∆ x = 0

steady state detector: ϕ ⁿ _i =





q _i ⁿ − q _i−1 ⁿ [ E ^{] n} _i−

¹

2



 2

+





q _i+1 ⁿ − q ⁿ _i [ E ^{] n} _i+

¹

2



 2

ϕ ⁿ _i = 0 if there is a steady state between W _i−1 ⁿ , W _i ⁿ and W _i+1 ⁿ

in this case, we take θ ⁿ _i = 0

otherwise, we take 0 < θ ⁿ _i ≤ ¹ 0 1

θ

ⁿ_i

ϕ

ⁿ_i

WB HO

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Two-dimensional and high-order extensions

MOOD method

High-order schemes induce oscillations: we use the MOOD method to get rid of the oscillations and to restore the non-negativity preservation (see Clain, Diot, Loubère (2011)).

MOOD loop

1 compute a candidate solution W ^c with the high-order scheme

2 determine whether W ^c is admissible, i.e.

if h

^c

is non-negative (PAD criterion)

if W

^c

does not present spurious oscillations (DMP and u2 criteria)

3 where necessary, decrease the degree of the reconstruction

4 compute a new candidate solution

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

1 Introduction to Godunov-type schemes

2 Derivation of a 1D rst-order well-balanced scheme 3 Two-dimensional and high-order extensions

4 2D and high-order numerical simulations

5 Conclusion and perspectives

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

Pseudo-1D double dry dam-break on a sinusoidal bottom

The P ^WB 5 scheme is used in the whole domain:

near the boundaries, steady state at rest well-balanced scheme;

away from the boundaries, far from steady state high-order scheme;

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

Order of accuracy assessment

To assess the order of accuracy, we take the following exact steady solution of the 2D shallow-water system, where r = ^t ( x, y ) :

h = 1 ; q = r

k r k ^; Z = 2 k k r k − ¹ 2 g k r k ² .

With k = 10 , this solution is depicted below on the space domain

[ − ⁰ . 3 , 0 . 3] × ^[0 . 4 , 1] .

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

Order of accuracy assessment

L ² errors with respect to the number of cells top graphs:

2D steady solution with topography bottom graphs:

2D steady solution with friction and topography

1e3 1e4

1e-4 1e-6 1e-8 1e-10

4 6

1

h,P^WB3

h,P^WB5

1e3 1e4

1e-4 1e-6 1e-8 1e-10

4 6

1

kqk,P^WB3

kqk,P^WB5

1e-6 1e-8 1e-10 1e-12

4 6

1

h,P^WB3

h,P^WB5 1e-6 1e-8

1e-10 4

6 1

kqk,P^WB3

kqk,P^WB5

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

2011 T ohoku tsunami

Tsunami simulation on a Cartesian mesh: 13 million cells, Fortran

code parallelized with OpenMP, run on 48 cores.

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

2011 T ohoku tsunami

− ⁸

− ⁶

− ⁴

− ² 0

Russia (Vladivostok)

Sea of Japan

Japan (Hokkaid¯ o island)

Kuril trench

Pacific Ocean

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

2011 T ohoku tsunami

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

2011 T ohoku tsunami

physical time of the simulation: ¹ hour

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction 2D and high-order numerical simulations

2011 T ohoku tsunami

Water depth at the sensors:

#1: 5700 m;

#2: 6100 m;

#3: 4400 m.

Graphs of the time variation of the water height (in meters).

data in black, order 1 in blue, order 2 in red

0 1 , 200 2 , 400 3 , 600

− ⁰ . 2 0 0 . 2 0 . 4 0 . 6

0 1 , 200 2 , 400 3 , 600 0

0 . 1 0 . 2

Sensor #2

0 1 , 200 2 , 400 3 , 600 0

0 . 1

0 . 2

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Conclusion and perspectives

1 Introduction to Godunov-type schemes

2 Derivation of a 1D rst-order well-balanced scheme 3 Two-dimensional and high-order extensions

4 2D and high-order numerical simulations

5 Conclusion and perspectives

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Conclusion and perspectives

Conclusion

We have presented a well-balanced, non-negativity-preserving and en-tropy preserving numerical scheme for the shallow-water equations with topography and Manning friction, able to be applied to other source terms or combinations of source terms.

We have also displayed results from the 2D high-order extension of this numerical method, coded in Fortran and parallelized with OpenMP.

This work has been published:

V. M.-D., C. Berthon, S. Clain and F. Foucher.

A well-balanced scheme for the shallow-water equations with topography.

Comput. Math. Appl. 72(3):568593, 2016.

V. M.-D., C. Berthon, S. Clain and F. Foucher.

A well-balanced scheme for the shallow-water equations with topography or Manning friction. J. Comput. Phys. 335:115154, 2017.

C. Berthon, R. Loubère, and V. M.-D.

A second-order well-balanced scheme for the shallow-water equations with topography. Accepted in Springer Proc. Math. Stat., 2017.

C. Berthon and V. M.-D.

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Conclusion and perspectives

Perspectives

Work in progress

high-order simulation of the 2011 T ohoku tsunami application to other source terms:

Coriolis force source term breadth variation source term

Long-term perspectives

ensure the entropy preservation for the high-order scheme (use of an e-MOOD method)

simulation of rogue waves

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Thanks!

Thank you for your attention!

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

The discrete entropy inequality

The following non-conservative entropy inequality is satised by the shallow-water system:

∂ _t η ( W ) + ∂ _x G ( W ) ≤ q

h S ( W ); η ( W ) = q ² 2 h ⁺

gh ²

2 ; G ( W ) = q h

q ² 2 h ⁺ gh ²

. At the discrete level, we show that:

λ R ( η ^∗ _R − η R ) − λ L ( η ^∗ _L − η L )+( G R − G L ) ≤ q _HLL

h _HLL S ∆ x + O ^(∆ x ² ) . main ingredients: h ^∗ _L = h _HLL − S ∆ x λ _R

α ( λ _R − λ _L ) (and similar expressions for h ^∗ _{R and} q ^∗ ₎

( λ _R − λ _L ) η _HLL ≤ λ _R η _R − λ _L η _L − ⁽ G _R − G _L )

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Verication of the well-balance: topography

The initial condition is at rest; water is injected through the left

boundary.

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Verication of the well-balance: topography

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Verication of the well-balance: topography

The non-well-balanced HLL scheme yields a numerical steady state

which does not correspond to the physical one. The well-balanced

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Verication of the well-balance: topography

The non-well-balanced HLL scheme yields a numerical steady state

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Verication of the well-balance: topography

transcritical ow test case (see Goutal, Maurel (1997))

left panel: initial free surface at rest; water is injected from the left boundary right panel: free surface for the steady state solution, after a transient state

Φ = u ²

+ g ( h + Z )

L ¹ L ² L ^∞

errors on q 1.47e-14 1.58e-14 2.04e-14

errors on Φ 1.67e-14 2.13e-14 4.26e-14

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Verication of the well-balance: friction

left panel: water height for the subcritical steady state solution

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Verication of the well-balance: friction

left panel: convergence to the unperturbed steady state

right panel: errors to the steady state (solid: h , dashed: q )

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Perturbed pseudo-1D friction and topography steady state

h k q k

L ¹ L ² L ^∞ L ¹ L ² L ^∞

P 0 1.22e-15 1.71e-15 6.27e-15 2.34e-15 3.02e-15 9.10e-15

P ⁵ 5.01e-05 1.47e-04 1.16e-03 2.32e-04 2.63e-04 1.18e-03

P ^WB 8.50e-14 1.05e-13 3.35e-13 2.82e-13 3.37e-13 6.76e-13

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Riemann problems between two wet areas

left: k = 0 left: k = 10

both Riemann problems have initial data W _L = 6

0

and

W _R = 1

, on [0 , 5] , with 200 points, and nal time 0 . 2 s

A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Appendices

Riemann problems with a wet/dry transition

left: k = 0 left: k = 10

both Riemann problems have initial data W _L = 6

0

and

Dans le document A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction (Page 37-75)