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

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C. Berthon ¹ , S. Clain ² , F. Foucher ^1,3 , R. Loubère ⁴ , V. Michel-Dansac ⁵

1

Laboratoire de Mathématiques Jean Leray, Université de Nantes

2

Centre of Mathematics, Minho University

3

École Centrale de Nantes

4

CNRS et Institut de Mathématiques de Bordeaux

5

Institut de Mathématiques de Toulouse et INSA Toulouse

Thursday, March 1st, 2018

Séminaire EDPAN, Clermont-Ferrand

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

Several kinds of destructive geophysical ows

Dam failure (Malpasset, France, 1959) Tsunami (T ohoku, Japan, 2011)

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The shallow-water equations and their source terms



 

 

∂ t h + ∂ x (hu) = 0

∂ _t (hu) + ∂ _x

hu ² + 1 2 gh ²

= − gh∂ _x Z − kq | q |

h ⁷ ³ (with q = hu) We can rewrite the equations as ∂ _t W + ∂ _x F (W ) = S(W ) , with W =

h q

.

x h(x, t)

water surface

channel bottom u(x, t)

Z(x)

Z(x) is the known topography k is the Manning coecient

g is the gravitational constant

we label the water

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Steady state solutions

Denition: Steady state solutions

W is a steady state solution i ∂ _t W = 0 , i.e. ∂ _x F(W ) = S(W ) . Taking ∂ t W = 0 in the shallow-water equations leads to



 

 

∂ x q = 0

∂ x

q ² h + 1

2 gh ²

= − gh∂ x Z − kq | q | h ⁷ ³ . The steady state solutions are therefore given by



  q = cst = q 0

∂ q ₀ ²

+ 1 gh ²

= − gh∂ Z − kq ₀ | q ₀ | .

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A real-life simulation:

the 2011 T ohoku tsunami.

The water is close to a steady state at rest far from the tsunami.

This steady state is not preserved by a

non-well-balanced

scheme!

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A real-life simulation:

the 2011 T ohoku tsunami.

The water is close to a steady state at rest far from the tsunami.

This steady state is not preserved by a

non-well-balanced

scheme!

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Objectives

Our goal is to derive a numerical method for the shallow-water model with topography and Manning friction that exactly preserves its stationary solutions on every mesh.

To that end, we seek a numerical scheme that:

1 is well-balanced for the shallow-water equations with

topography and friction, i.e. it exactly preserves and captures the steady states without having to solve the governing nonlinear dierential equation;

2 preserves the non-negativity of the water height;

3 ensures a discrete entropy inequality;

4 can be easily extended for other source terms of the

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A high-order well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction to Godunov-type schemes

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

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Setting: nite volume schemes

Objective: Approximate the solution W (x, t) of the system

∂ _t W + ∂ _x F (W ) = S(W ) , with suitable initial and boundary conditions.

We partition the space domain in cells, of volume ∆x and of evenly spaced centers x _i , and we dene:

x _i−

1

2

and x _i+

1

2

, the boundaries of the cell i ;

W _i ⁿ , an approximation of W (x, t) , constant in the cell i and at time t ⁿ , which is dened as W _i ⁿ = 1

∆x

Z _∆x/2

∆x/2

W (x, t ⁿ )dx .

W (x, t) W

_iⁿ

x x

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Godunov-type scheme (approximate Riemann solver)

As a consequence, at time t ⁿ , we have a succession of Riemann problems (Cauchy problems with discontinuous initial data) at the interfaces between cells:



 

 

∂ _t W + ∂ _x F (W ) = S(W ) W (x, t ⁿ ) =

( W _i ⁿ if x < x _i+

1 2

W _i+1 ⁿ if x > x _i+

1 2

x _i x _i+

1

x _i+1

2

W _i ⁿ W _i+1 ⁿ

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Godunov-type scheme (approximate Riemann solver)

We choose to use an approximate Riemann solver, as follows.

W _i ⁿ W _i+1 ⁿ W ⁿ

i+

¹₂

λ ^L

i+

¹₂

λ ^R _i+

1 2

x _i+

¹

2

x

t

W _i+ ⁿ

1

2

is an approximation of the interaction between W _i ⁿ and W _i+1 ⁿ (i.e. of the solution to the Riemann problem), possibly made of several constant states separated by discontinuities.

λ ^L _i+

1 2

and λ ^R _i+

1 2

are approximations of the largest wave speeds of

the system.

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Godunov-type scheme (approximate Riemann solver)

x t

t

ⁿ⁺¹

t

ⁿ

x

_i

x

_i−1

2

x

_i+1

2

W

_iⁿ

W

_i−ⁿ 1

2

W

_i+ⁿ 1

2

λ

^R

i−¹₂

λ

^L_i+1

2

z }| {

W

^∆

(x, t

ⁿ⁺¹

)

W

_i−1ⁿ

W

_i+1ⁿ

We dene the time update as follows:

W _i ⁿ⁺¹ := 1

∆x Z _x

i+ 12

x

_i−1 2

W ^∆ (x, t ⁿ⁺¹ )dx.

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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

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

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The HLL approximate Riemann solver

To approximate solutions of

∂ _t W + ∂ _x F(W ) = 0 , the HLL approximate Riemann solver (Harten, Lax, van Leer (1983)) may be chosen; it is denoted by W ^∆ and displayed on the right.

W

_HLL

W

L

W

R

λ

_L

x t

λ

_R

−∆x/2 0 ∆x/2

The consistency condition (as per Harten and Lax) holds if:

1 ∆x

Z ∆x/2

−∆x/2

W ^∆ (∆t, x; W L , W R )dx = 1

∆x

Z ∆x/2

−∆x/2

W R (∆t, x; W L , W R )dx,

which gives W _HLL = λ _R W _R − λ _L W _L

λ − λ − F (W _R ) − F (W _L )

λ − λ =

h HLL

q

.

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Modication of the HLL approximate Riemann solver

The shallow-water equations with the topography and friction source terms read as follows:



 

 

∂ _t h + ∂ _x q = 0,

∂ _t q + ∂ _x q ²

h + 1 2 gh ²

+ gh∂ _x Z + kq | q |

h ⁷ ³ = 0.

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Modication of the HLL approximate Riemann solver

With Y (t, x) := x, we can add the equations ∂ t Z = 0 and

∂ _t Y = 0 , which correspond to the xed geometry of the problem:



 



 



∂ _t h + ∂ _x q = 0,

∂ _t q + ∂ _x q ²

h + 1 2 gh ²

+ gh∂ _x Z + kq | q |

h ⁷ ³ ∂ _x Y = 0,

∂ _t Y = 0,

∂ t Z = 0.

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Modication of the HLL approximate Riemann solver

With Y (t, x) := x, we can add the equations ∂ t Z = 0 and

∂ _t Y = 0 , which correspond to the xed geometry of the problem:



 



 



∂ _t h + ∂ _x q = 0,

∂ _t q + ∂ _x q ²

h + 1 2 gh ²

+ gh∂ _x Z + kq | q |

h ⁷ ³ ∂ _x Y = 0,

∂ _t Y = 0,

∂ t Z = 0.

The equations ∂ _t Y = 0 and ∂ _t Z = 0 induce stationary waves associated to the source term (of which q is a Riemann invariant).

To approximate solutions of

∂ _t W + ∂ _x F(W ) = S(W ) , we thus use the approximate Riemann solver displayed on the right

(assuming λ < 0 < λ ). ^W

^L

^W

^R

λ

L

0 λ

R

W

_L^∗

W

_R^∗

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Modication of the HLL approximate Riemann solver

We have 4 unknowns to determine: W _L ^∗ = h ^∗ _L

q _L ^∗

and W _R ^∗ = h ^∗ _R

q _R ^∗

.

W L W R

λ L 0 λ _R W _L ^∗ W _R ^∗

q is a 0 -Riemann invariant we take q _L ^∗ = q ^∗ _R = q ^∗ (relation 1) The Harten-Lax consistency gives us the following two

relations:

λ

R

h

^∗_R

− λ

L

h

^∗_L

= (λ

R

− λ

L

)h

HLL

(relation 2), q

^∗

= q

HLL

+ S∆x

λ

R

− λ

L

(relation 3), where S ' 1

∆x 1

∆t Z

∆x/2

−∆x/2

Z

∆t

0

S(W

R

(x, t)) dt dx.

next step: obtain a fourth relation

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Modication of the HLL approximate Riemann solver

We have 4 unknowns to determine: W _L ^∗ = h ^∗ _L

q ^∗ _L

and W _R ^∗ = h ^∗ _R

q _R ^∗

.

q is a 0-Riemann invariant we take q _L ^∗ = q _R ^∗ = q ^∗ (relation 1)

The Harten-Lax consistency gives us the following two relations:

λ

R

h

^∗_R

− λ

L

h

^∗_L

= (λ

R

− λ

L

)h

HLL

(relation 2), q

^∗

= q

HLL

+ S∆x

λ

R

− λ

L

(relation 3), where S ' 1

∆x 1

∆t Z

∆x/2

−∆x/2

Z

∆t

0

S(W

R

(x, t)) dt dx.

next step : obtain a fourth relation

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Modication of the HLL approximate Riemann solver

We have 4 unknowns to determine: W _L ^∗ = h ^∗ _L

q ^∗ _L

and W _R ^∗ = h ^∗ _R

q _R ^∗

.

q is a 0-Riemann invariant we take q _L ^∗ = q _R ^∗ = q ^∗ (relation 1) The Harten-Lax consistency gives us the following two relations:

λ

R

h

^∗_R

− λ

L

h

^∗_L

= (λ

R

− λ

L

)h

HLL

(relation 2), q

^∗

= q

HLL

+ S∆x

λ

R

− λ

L

(relation 3), where S ' 1

∆x 1

∆t Z

∆x/2

−∆x/2

Z

∆t

0

S(W

R

(x, t)) dt dx.

next step : obtain a fourth relation

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Modication of the HLL approximate Riemann solver

We have 4 unknowns to determine: W _L ^∗ = h ^∗ _L

q ^∗ _L

and W _R ^∗ = h ^∗ _R

q _R ^∗

.

q is a 0-Riemann invariant we take q _L ^∗ = q _R ^∗ = q ^∗ (relation 1) The Harten-Lax consistency gives us the following two relations:

λ

R

h

^∗_R

− λ

L

h

^∗_L

= (λ

R

− λ

L

)h

HLL

(relation 2),

q

^∗

= q

HLL

+ S∆x λ

R

− λ

L

(relation 3), where S ' 1

∆x 1

∆t Z

∆x/2

−∆x/2

Z

∆t

0

S(W

R

(x, t)) dt dx.

next step : obtain a fourth relation

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Modication of the HLL approximate Riemann solver

We have 4 unknowns to determine: W _L ^∗ = h ^∗ _L

q ^∗ _L

and W _R ^∗ = h ^∗ _R

q _R ^∗

.

q is a 0-Riemann invariant we take q _L ^∗ = q _R ^∗ = q ^∗ (relation 1) The Harten-Lax consistency gives us the following two relations:

λ

R

h

^∗_R

− λ

L

h

^∗_L

= (λ

R

− λ

L

)h

HLL

(relation 2), q

^∗

= q

HLL

+ S∆x

λ

R

− λ

L

(relation 3), where S ' 1

∆x 1

∆t Z

∆x/2

−∆x/2

Z

∆t

0

S(W

R

(x, t)) dt dx.

next step : obtain a fourth relation

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Modication of the HLL approximate Riemann solver

We have 4 unknowns to determine: W _L ^∗ = h ^∗ _L

q ^∗ _L

and W _R ^∗ = h ^∗ _R

q _R ^∗

.

q is a 0-Riemann invariant we take q _L ^∗ = q _R ^∗ = q ^∗ (relation 1) The Harten-Lax consistency gives us the following two relations:

λ

R

h

^∗_R

− λ

L

h

^∗_L

= (λ

R

− λ

L

)h

HLL

(relation 2), q

^∗

= q

HLL

+ S∆x

λ

R

− λ

L

(relation 3), where S ' 1

∆x 1

∆t Z

∆x/2

−∆x/2

Z

∆t

0

S(W

R

(x, t)) dt dx.

next step : obtain a fourth relation

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Obtaining an additional relation

Assume that W _L and W _R dene a steady state, i.e. that they satisfy the following discrete version of the steady relation

∂ _x F ( W ) = S ( W ) (where [ X ] = X _R − X _L ):

1 ∆ x

q ² ₀ 1

h

+ g 2

h ²

= S.

For the steady state to be preserved, it is sucient to have h ^∗ _L = h _L , h ^∗ _R = h _R

and q ^∗ = q ₀ . ^W

^L

^W

^R

λ

L

0 λ

R

W

L

W

_R

Assuming a steady state, we show that q ^∗ = q ₀ , as follows:

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Obtaining an additional relation

In order to determine an additional relation, we consider the discrete steady relation, satised when W _L and W _R dene a steady state:

q ₀ ² 1

h _R − ¹ h _L

+ g

2 ( h _R ) ² − ⁽ h _L ) ²

= S ∆ x.

To ensure that h ^∗ _L = h _L and h ^∗ _R = h _R , we impose that h ^∗ _L and h ^∗ _R satisfy the above relation, as follows:

q ₀ ² 1

h ^∗ _R − ¹ h ^∗ _L

+ g

2 ( h ^∗ _R ) ² − ⁽ h ^∗ _L ) ²

= S ∆ x.

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Determination of h ^∗ _L and h ^∗ _R

The intermediate water heights satisfy the following relation:

− q ² ₀

h ^∗ _R − h ^∗ _L h ^∗ _L h ^∗ _R

+ g

2 ( h ^∗ _L + h ^∗ _R )( h ^∗ _R − h ^∗ _L ) = S ∆ x.

Recall that q ^∗ is known and is equal to q ₀ for a steady state.

Instead of the above relation, we choose the following linearization:

− ⁽ q ^∗ ) ²

h _L h _R ⁽ h ^∗ _R − h ^∗ _L ) + g

2 ( h L + h R )( h ^∗ _R − h ^∗ _L ) = S ∆ x, which can be rewritten as follows:

− ⁽ q ^∗ ) ² h _L h _R ⁺

g

2 ( h _L + h _R )

( h ^∗ _R − h ^∗ _L ) = S ∆ x.

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Determination of h ^∗ _L and h ^∗ _R

With the consistency relation between h ^∗ _L and h ^∗ _R , the intermediate water heights satisfy the following linear system:

( α ( h ^∗ _R − h ^∗ _L ) = S ∆ x,

λ R h ^∗ _R − λ L h ^∗ _L = ( λ R − λ L ) h HLL . Using both relations linking h ^∗ _L and h ^∗ _R , we obtain



 

 

 

 

h ^∗ _L = h _HLL − λ _R S ∆ x α ( λ _R − λ _L ) , h ^∗ _R = h HLL − λ _L S ∆ x

α ( λ _R − λ _L ) , where α =

− ⁽ q ^∗ ) ² h h ⁺

g

2 ( h _L + h _R )

with q ^∗ = q _HLL + S ∆ x

λ − λ .

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Correction to ensure non-negative h ^∗ _L and h ^∗ _R

However, these expressions of h ^∗ _L and h ^∗ _R do not guarantee that the intermediate heights are non-negative: instead, we use the following cuto (see Audusse, Chalons, Ung (2015)):



 

 

 

 

h ^∗ _L = min

h _HLL − λ _R S ∆ x α ( λ _R − λ _L )

+

,

1 − λ _R λ _L

h _HLL

, h ^∗ _R = min

h _HLL − λ _L S ∆ x α ( λ _R − λ _L )

+

,

1 − λ _L λ _R

h _HLL

.

Note that this cuto does not interfere with:

the consistency condition λ _R h ^∗ _R − λ _L h ^∗ _L = ( λ _R − λ _L ) h _HLL ;

the well-balance property, since it is not activated when W _L and

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Summary

The two-state approximate Riemann solver with intermediate states W _L ^∗ =

h ^∗ _L q ^∗

and W _R ^∗ = h ^∗ _R

q ^∗

given by



 



 



q ^∗ = q _HLL + S ∆ x λ _R − λ _L , h ^∗ _L = min

h _HLL − λ _R S ∆ x α ( λ _R − λ _L )

+

,

1 − λ _R λ _L

h _HLL

, h ^∗ _R = min

h _HLL − λ _L S ∆ x α ( λ _R − λ _L )

+

,

1 − λ _L λ _R

h _HLL

, is consistent, non-negativity-preserving, entropy preserving and well-balanced.

next step: determination of S according to the source term

denition (topography or friction).

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The topography source term

We now consider S ( W ) = S ^t ( W ) = − gh∂ _x Z : the smooth steady states are governed by

∂ _x q ₀ ²

h

+ g 2 ∂ _x h ²

= − gh∂ _x Z, q ₀ ²

2 ∂ x

1 h ²

+ g∂ x ( h + Z ) = 0 ,



 

 

 

 

−−−−−−−→

discretization



 



 

 q ₀ ²

1 h

+ g 2

h ²

= S ^t ∆ x, q ₀ ²

2 1

h ²

+ g [ h + Z ] = 0 . We can exhibit an expression of q ₀ ² and thus obtain

S ^t = − g ² h L h R

h _L + h _R [ Z ]

∆ x ⁺ g 2∆ x

[ h ] ³ h _L + h _R .

However, when Z _L = Z _R , we have S ^t 6 ⁼ O ^(∆ x ) , i.e. a loss of

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The topography source term

Instead, we set, for some constant C > 0 ,



 

 

 

 

S ^t = − g ² h _L h _R h _L + h _R

[ Z ]

∆ x ⁺ g 2∆ x

[ h ] ³ _c h _L + h _R , [ h ] c =

( h _R − h _L if | h _R − h _L | ≤ C ∆ x, sgn( h _R − h _L ) C ∆ x otherwise .

Theorem: Well-balance for the topography source term

If W L and W R dene a smooth steady state, i.e. if they satisfy q ₀ ²

2 1

h ²

+ g [ h + Z ] = 0 ,

then we have W _L ^∗ = W _L _and W _R ^∗ = W _R and the approximate

Riemann solver is well-balanced. By construction, the Godunov-type

scheme using this approximate Riemann solver is consistent,

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The friction source term

We consider, in this case, S ( W ) = S ^f ( W ) = − kq | q | h ^−η , where we have set η = ⁷ 3 .

The average of S ^f we choose is S ^f = − k q ¯ | q ^¯ | h ^−η , with

q ¯ the harmonic mean of q _L _and q _R _{(note that} q ¯ = q ₀ _{at the} equilibrium);

h ^−η a well-chosen discretization of h ^−η , depending on h L and h _R , and ensuring the well-balance property.

We determine h ^−η using the same technique (with µ 0 = sgn( q 0 ) ):

∂

x

q

²₀

h

+ g

2 ∂

x

h

²

= − kq

0

| q

0

| h

^−η

,

2

∂

x

h

^η−1

∂

x

h

^η+2



 

 −−−−−−−→

discretization



 

 q

²₀

1 h

+ g 2

h

²

= − kµ

0

q

₀²

h

^−η

∆x,

2

h

^η−1

h

^η+2

2

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The friction source term

The expression for q ₀ ² we obtained is now used to get:

h ^−η = [ h ² ] 2

η + 2 [ h ^η+2 ] − µ ₀

k ∆ x 1

h

+ [ h ² ] 2

[ h ^η−1 ] η − ¹

η + 2 [ h ^η+2 ]

, which gives S ^f = − k q ¯ | q ^¯ | h ^−η ( h ^−η is consistent with h ^−η if a cuto is applied to the second term of h ^−η ).

Theorem: Well-balance for the friction source term If W _L _and W _R dene a smooth steady state, i.e. verify

q ² ₀ h ^η−1

η − ¹ ⁺ g h ^η+2

η + 2 = − kq ₀ | q ₀ | ^∆ x,

then we have W _L ^∗ = W _L _and W _R ^∗ = W _R and the approximate

Riemann solver is well-balanced. By construction, the Godunov-type

scheme using this approximate Riemann solver is consistent,

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Friction and topography source terms

With both source terms, the scheme preserves the following discretization of the steady relation ∂ x F ( W ) = S ( W ) :

q ₀ ² 1

h

+ g 2

h ²

= S ^t ∆ x + S ^f ∆ x.

The intermediate states are therefore given by:



 



 



q ^∗ = q _HLL + ( S ^t + S ^f )∆ x λ _R − λ _L ^; h ^∗ _L = min

h _HLL − λ _R ( S ^t + S ^f )∆ x α ( λ _R − λ _L )

+

,

1 − λ _R λ _L

h _HLL

;

h ^∗ _R = min

h HLL − λ _L ( S ^t + S ^f )∆ x ,

1 − λ _L h HLL

.

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The full Godunov-type scheme

x t

t

ⁿ⁺¹

t

ⁿ

x

i

x

_i−¹

2

x

_i+¹

2

W

_iⁿ

W

_i−^R,∗1

2

W

^L,∗

i+¹2

λ

^R_i−1 2

λ

^L_i+1 2

z }| {

W

^∆

(x, t

ⁿ⁺¹

)

We recall W _i ⁿ⁺¹ = 1

∆ x Z _x

i+ 12

x

_i−1 2

W ^∆ ( x, t ⁿ⁺¹ ) dx : then W _i ⁿ⁺¹ = W _i ⁿ − ^∆ t

∆ x

λ ^L _i+

1 2

W ^L,∗

i+

¹₂

− W _i ⁿ

− λ ^R _i−

1 2

W ^R,∗

i−

¹₂

− W _i ⁿ

, which can be rewritten, after straightforward computations,

n+1 n

∆t

n n

 

0

t n t n

 

0

f n f n

 

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Summary

We have presented a scheme that:

is consistent with the shallow-water equations with friction and topography;

is well-balanced for friction and topography steady states;

preserves the non-negativity of the water height;

ensures a discrete entropy inequality;

is not able to correctly approximate wet/dry interfaces due to the stiness of the friction kq | q | h ⁻ ⁷ ³ : the friction term should be treated implicitly.

next step: introduction of this semi-implicit scheme

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Semi-implicit nite volume scheme

We use a splitting method with an explicit treatment of the ux and the topography and an implicit treatment of the friction.

1 explicitly solve ∂ _t W + ∂ _x F ( W ) = S ^t ( W ) as follows:

W ⁿ⁺

1 2

i = W _i ⁿ − ^∆ t

∆ x F _i+ ⁿ

¹

2

− F _i− ⁿ

¹

2

+ ∆ t 1 ⁰

2 ( S ^t ⁾ ⁿ _i−

¹

2

+ ( S ^t ⁾ ⁿ _i+

¹

2

!

2 implicitly solve ∂ _t W = S ^f ( W ) as follows:



 

 

 

 

h ⁿ⁺¹ _i = h ⁿ⁺

1 2

i

IVP:

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

1

i

2

q _i ⁿ⁺¹

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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

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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

(40)

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

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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

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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.

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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)

(44)

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)

(45)

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)

(46)

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

(47)

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

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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.

(49)

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

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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

(51)

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

(52)

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;

(53)

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] .

(54)

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

(55)

2011 T ohoku tsunami

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

code parallelized with OpenMP, run on 48 cores.

(56)

2011 T ohoku tsunami

− ⁸

− ⁶

− ⁴

− ² 0

Russia (Vladivostok)

Sea of Japan

Japan (Hokkaid¯ o island)

Kuril trench

Pacific Ocean

(57)

2011 T ohoku tsunami

(58)

2011 T ohoku tsunami

physical time of the simulation: ¹ hour

(59)

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

(60)

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

(61)

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.

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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

(63)

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

Thank you for your attention!

(64)

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 )

(65)

Verication of the well-balance: topography

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

boundary.

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Verication of the well-balance: topography

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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

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Verication of the well-balance: topography

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

(69)

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

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