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

(1)

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

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

1

Laboratoire de Mathématiques Jean Leray, Université de Nantes

2

Centre of Mathematics, Minho University

3

École Centrale de Nantes

4

INSA Toulouse

Monday, September 25th, 2017

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

Several kinds of destructive geophysical flows

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

Flood (La Faute sur Mer, France, 2010) Mudslide (Madeira, Portugal, 2010)

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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 coefficient

g is the gravitational constant

we label the water

discharge q := hu

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Steady state solutions

Definition: Steady state solutions

W is a steady state solution iff ∂ _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

∂ _x q ₀ ²

h + 1 2 gh ²

= − gh∂ _x Z − kq ₀ | q ₀ | h ⁷ ³ .

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Topography steady state not captured in 1D

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

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Topography steady state not captured in 1D

The classical HLL numerical scheme converges towards a numerical steady state which does not correspond to the physical one.

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Topography steady state not captured in 1D

The classical HLL numerical scheme converges towards a numerical steady state which does not correspond to the physical one.

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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.

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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 differential equation;

2 preserves the non-negativity of the water height;

3 can be easily extended for other source terms of the shallow-water equations (e.g. breadth).

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1 Brief introduction to Godunov-type schemes

2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension

4 Numerical simulations

5 Conclusion and perspectives

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A well-balanced scheme for the shallow-water equations with topography and Manning friction Brief introduction to Godunov-type schemes

1 Brief introduction to Godunov-type schemes

2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension

4 Numerical simulations

5 Conclusion and perspectives

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Setting

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 [a, b] in cells , of volume ∆x and of evenly spaced centers x _i , and we define:

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 defined as W _i ⁿ = 1

∆x

Z _∆x/2

∆x/2

W (x, t ⁿ )dx.

x

_i

W (x, t)

x

_i−1

2

x

_i+1

2

W

_iⁿ

x x

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Using an 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

W _i+1 ⁿ if x > x _i+

21 2

x _i x _i+

1

x _i+1

2

W _i ⁿ W _i+1 ⁿ

For S(W ) 6 = 0, the exact solution to these Riemann problems is unknown or costly to compute we require an approximation.

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Using an approximate Riemann solver

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

W

_iⁿ

W

_i+1ⁿ

W

_i+ⁿ1

2

λ

^L_i+1

2

λ

^R_i+1

2

x

_i+¹

2

W ⁿ

i+

¹₂

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+

¹₂

and λ ^R

i+

¹₂

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 define 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.

Since W ⁿ

i −

¹₂

and W ⁿ

i+

¹₂

are made of constant states, the above integral is easy to compute.

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A well-balanced scheme for the shallow-water equations with topography and Manning friction Derivation of a generic first-order well-balanced scheme

1 Brief introduction to Godunov-type schemes

2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension

4 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 _R − λ _L − F (W _R ) − F (W _L ) λ _R − λ _L =

h HLL

q _HLL

. Note that, if h

L

> 0 and h

R

> 0, then h

HLL

> 0 for | λ

L

| and | λ

R

| large enough.

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Modification 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 + k q | q | h ⁷ ³ = 0.

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Modification 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 fixed geometry of the problem:

 

 

 

 



∂ _t h + ∂ _x q = 0,

∂ _t q + ∂ _x q ²

h + 1 2 gh ²

+ gh∂ _x Z + k q | q |

h ⁷ ³ ∂ _x Y = 0,

∂ _t Z = 0,

∂ t Y = 0.

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Modification 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 fixed geometry of the problem:

 

 

 

 



∂ _t h + ∂ _x q = 0,

∂ _t q + ∂ _x q ²

h + 1 2 gh ²

+ gh∂ _x Z + k q | q |

h ⁷ ³ ∂ _x Y = 0,

∂ _t Z = 0,

∂ t Y = 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 _L < 0 < λ _R ).

W

L

W

R

λ

L

0 λ

_R

W

_L^∗

W

_R^∗

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Modification 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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Modification 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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Modification 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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Modification 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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Modification 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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Modification 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 define 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 sufficient to have h ^∗ _L = h _L , h ^∗ _R = h _R and q ^∗ = q ₀ .

0 W

L

W

R

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

q ^∗ = q _HLL + S ∆ x

λ R − λ L = q ₀ − ¹ λ R − λ L

q ² ₀

1 h

+ g 2

h ²

− S ∆ x

= q ₀ .

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Obtaining an additional relation

In order to determine an additional relation, we consider the discrete steady relation, satisfied when W _L and W _R define 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 )

| {z } α

( 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 _L h _R ⁺

g

2 ( h _L + h _R )

with q ^∗ = q _HLL + S ∆ x λ _R − λ _L .

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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 cutoff (see Audusse, Chalons, Ung (2014)):

 

 



 

 

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 cutoff 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 W _R define a steady state.

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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 and well-balanced.

next step : determination of S according to the source term definition (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 consistency with S ^t (see for instance Berthon, Chalons (2016)).

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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 define 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, fully well-balanced and positivity-preserving.

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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

,

q

₀²

∂

x

h

^η⁻¹

η − 1 − g ∂

x

h

^η+2

η + 2 = kq

0

| q

0

| ,

 

 

 



−−−−−−−→

discretization

 

 

 

 q

²₀

1 h

+ g 2

h

²

= − kµ

0

q

₀²

h

^−η

∆x, q

²₀

h

^η−1

η − 1 − g

h

^η+2

η + 2 = kµ

0

q

₀²

∆x.

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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 cutoff is applied to the second term of h ^−η ).

Theorem: Well-balance for the friction source term If W _L and W _R define 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, fully well-balanced and positivity-preserving.

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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 α ( λ _R − λ _L )

+

,

1 − λ _L λ _R

h HLL

.

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(38)

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−^R1

2

W

_i+^L1

2

λ

^R

i−¹2

λ

^L_i+1 2

z W

^∆

(x, t }|

ⁿ⁺¹

) {

We recall W _i ⁿ⁺¹ = 1

∆ x Z x

_{i+ 1}

2

x

_i₋1 2

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

∆ x h

λ ^L _i+

1 2

W _i+ ^L

1

2

− W _i ⁿ

− λ ^R _i−

1 2

W _i− ^R

1

2

− W _i ⁿ i , which can be rewritten, after straightforward computations,

W

_iⁿ⁺¹

= W

_iⁿ

− ∆t

∆x F

_i+ⁿ 1

2

− F

_iⁿ₋1 2

+ ∆t









0 (S

^t

)

ⁿ_i−1

2

+(S

^t

)

ⁿ_i+1 2

2 

+





0 (S

^f

)

ⁿ_i

−¹₂

+(S

^f

)

ⁿ_i+1 2

2 





.

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(39)

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;

is not able to correctly approximate wet/dry interfaces due to the stiffness of the friction: the friction term should be treated implicitly.

next step : introduction of this semi-implicit scheme

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(40)

Semi-implicit finite volume scheme

We use a splitting method with an explicit treatment of the flux 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

i

2

= 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⁺ _i

¹²

q _i ⁿ⁺¹

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(41)

Semi-implicit finite volume scheme

Solving the IVP yields:

q _i ⁿ⁺¹ = ( h ⁿ⁺¹ _i ) ^η q ⁿ⁺

1

i

2

( h ⁿ⁺¹ _i ) ^η + k ∆ t q ⁿ⁺ _i

¹²

.

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

i

2

q _i ⁿ .

semi-implicit treatment of the friction source term scheme able to model wet/dry transitions

scheme still well-balanced and non-negativity-preserving

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(42)

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

1 Brief introduction to Godunov-type schemes

2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension

4 Numerical simulations

5 Conclusion and perspectives

(43)

Second-order extension

x _i−1 x _i x _i+1

V _i+1 ⁿ

V _i−1 ⁿ

V _i ⁿ

x x _i+

¹

x _i−

¹ 2 2

29 / 39

(44)

Second-order extension

x _i−1 x _i x _i+1

V _i+1 ⁿ

V _i−1 ⁿ

V _i ⁿ V c _i ⁿ (x)

V _i,− ⁿ

V _i,+ ⁿ

x x _i+

¹

x _i−

¹ 2 2

29 / 39

(45)

Second-order extension

For the second-order MUSCL procedure, we introduce the vector V = ^t ( h, q, h + Z )

of reconstructed variables. Then, with σ ⁿ _i a limited slope, a linear reconstruction of the constant state V _i ⁿ in each cell i is given by:

V _i,± ⁿ = V c _i ⁿ

x _i ± ^∆ x 2

= V _i ⁿ ± ^∆ x 2 σ _i ⁿ . Two remarks follow from this definition:

1 If q = 0 and h + Z is constant in the cells i − 1, i _and i _{+ 1,} they remain constant after the reconstruction: the lake at rest steady state is naturally preserved.

2 We have V _i ⁿ = V _i, ⁿ ₋ + V _i,+ ⁿ

2 .

30 / 39

(46)

Second-order extension

x _i−1 x _i x _i+1

V _i+1 ⁿ

V _i−1 ⁿ

V _i ⁿ V c _i ⁿ (x)

V _i,− ⁿ

V _i,+ ⁿ

x x _i+

¹

x _i−

¹ 2 2

31 / 39

(47)

Second-order extension

x _i−1 x _i x _i+1

V _i+1 ⁿ

V _i−1 ⁿ

V _i ⁿ V c _i ⁿ (x)

V _i,− ⁿ

V _i,+ ⁿ

x x _i+

¹

x _i−

¹ 2 2

V _i+1,− ⁿ V _i−1,− ⁿ

31 / 39

(48)

Second-order extension

x _i−1 x _i x _i+1

V _i,− ⁿ

V _i,+ ⁿ

x V _i−1,+ ⁿ

V _i+1,− ⁿ

x _i−

¹

2

x _i+

¹

2

31 / 39

(49)

Second-order extension

x

i−1

x

i

x

i+1

W

_i,−ⁿ

W

_i,+ⁿ

x W

_i−1,+ⁿ

W

_i+1,−ⁿ

x

_i−1

2

x

_i+1

2

For simplicity, we rewrite the first-order scheme:

W _i ⁿ⁺¹ = H ⁽ W _i ⁿ ₋ ₁ , W _i ⁿ , W _i+1 ⁿ ) . The MUSCL update, in the subcells ( x _i ₋

¹

2

, x i ) and ( x i , x _i+

¹

2

) , reads:

W _i, ⁿ⁺¹ ₋ = H ⁽ W _i ⁿ ₋ _1,+ , W _i, ⁿ ₋ , W _i,+ ⁿ ) and W _i,+ ⁿ⁺¹ = H ⁽ W _i, ⁿ ₋ , W _i,+ ⁿ , W _i+1, ⁿ ₋ ) . We then take W _i ⁿ⁺¹ = ( W _i,− ⁿ⁺¹ + W _i,+ ⁿ⁺¹ ) / 2 . This update is a

convex combination: we exhibit the same robustness results as the first-order scheme as soon as the CFL constraint is halved.

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(50)

Second-order extension: well-balance recovery

reconstruction procedure scheme no longer preserves steady states with q ₀ 6 ^{= 0} Well-balance recovery

We suggest a convex combination between the second-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 second-order scheme is used.

next step : derive a suitable expression for θ _i ⁿ

33 / 39

(51)

Second-order extension: well-balance recovery

Steady state detector steady state solution:

 



q _L = q _R = q 0 , E ^:= q ₀ ²

h _R − q ₀ ² 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 ⁿ _i ₋ ₁ [ E ^] ⁿ _i ₋

1

2





2 +



 q ⁿ _i+1 − q _i ⁿ [ E ^] ⁿ _i+

1

2





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

m∆x M ∆x

θ

_iⁿ

ϕ

ⁿ_i

34 / 39

(52)

A well-balanced scheme for the shallow-water equations with topography and Manning friction Numerical simulations

1 Brief introduction to Godunov-type schemes

2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension

4 Numerical simulations

5 Conclusion and perspectives

(53)

Verification of the well-balance: topography

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

35 / 39

(54)

Verification of the well-balance: topography

The non-well-balanced HLL scheme converges towards a numerical steady state which does not correspond to the physical one.

35 / 39

(55)

Verification of the well-balance: topography

The non-well-balanced HLL scheme converges towards a numerical steady state which does not correspond to the physical one. The

well-balanced scheme converges towards the physical steady state.

^{35 / 39}

(56)

Verification of the well-balance: topography

The non-well-balanced HLL scheme converges towards a numerical steady state which does not correspond to the physical one. The

well-balanced scheme converges towards the physical steady state.

^{35 / 39}

(57)

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] .

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(58)

Order of accuracy assessment

The errors are collected in the graphs below.

1e3 1e4

1e-2

1e-3

1e-4 1 2

1 L ^∞ errors on h , order 1 L ^∞ errors on h , order 2

1e3 1e4

1e-1 1e-2 1e-3 1e-4

1 2 1

L ^∞ errors on k q k ^{, order 1} L ^∞ errors on k q k , order 2 We note that the first-order scheme is first-order accurate, while the second-order scheme is second-order accurate.

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(59)

2011 T¯ ohoku tsunami

2D Cartesian scheme obtained from using the 1D scheme at each interface.

Tsunami simulation on a Cartesian mesh: 13 million cells, Fortran code parallelized with OpenMP , run on 48 cores.

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(60)

2011 T¯ ohoku tsunami

0 500 1 , 000 1 , 500 2 , 000 2 , 500

− ⁸

− ⁶

− ⁴

− ² 0

Russia (Vladivostok)

Sea of Japan

Japan (Hokkaid¯o island)

Kuril trench

Pacific Ocean

1D slice of the topography (unit: kilometers).

_{37 / 39}

(61)

2011 T¯ ohoku tsunami

physical time of the simulation: 1 hour

first-order scheme CPU time: ∼ ¹ . 1 hour

second-order scheme CPU time: ∼ ² . 7 hours

37 / 39

(62)

2011 T¯ ohoku tsunami

physical time of the simulation: 1 hour

first-order scheme CPU time: ∼ ¹ . 1 hour

second-order scheme CPU time: ∼ ² . 7 hours

37 / 39

(63)

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

Sensor #1

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

Sensor #3

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(64)

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

1 Brief introduction to Godunov-type schemes

2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension

4 Numerical simulations

5 Conclusion and perspectives

(65)

Conclusion

We have presented a well-balanced and non-negativity-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 a 2D well-balanced 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):568–593, 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. 335:115–154, 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.

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Perspectives

Work in progress or completed

application to other source terms:

Coriolis force source term (work in progress) breadth variation source term (work in progress)

high-order extensions (order 6 achieved, application to large-scale phenomena in progress)

Long-term perspectives

stability of the scheme: values of C , λ _L and λ _R to ensure the entropy preservation

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

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(67)

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

Thank you for your attention!

(68)

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

Second-order extension

xi−1 xi xi+1

W_i,−ⁿ W_i,+ⁿ

x W_i−1,+ⁿ

W_i+1,−ⁿ

x_i−1

2 x_i+1

2

W _i,− ⁿ⁺

¹²

= W _i,− ⁿ − ^∆ t

∆x 2

F ⁽ W _i,− ⁿ , W _i,+ ⁿ ) − F ⁽ W _i− ⁿ _1,+ , W _i,− ⁿ )

+ ∆ t 2

s ^t ( W _i ⁿ ₋ _1,+ , W _i, ⁿ ₋ + s ^t ( W _i, ⁿ ₋ , W _i,+ ⁿ ) W ⁿ⁺

1

i,+

2

= W _i,+ ⁿ − ^∆ t

∆x 2

F ⁽ W _i,+ ⁿ , W _i+1, ⁿ ₋ ) − F ⁽ W _i, ⁿ ₋ , W _i,+ ⁿ )

+ ∆ t 2

s ^t ( W _i, ⁿ ₋ , W _i,+ ⁿ + s ^t ( W _i,+ ⁿ , W _i+1, ⁿ ₋ )

W ⁿ⁺

1

i

2

= W ⁿ⁺

1

i, −

2

+ W ⁿ⁺

1

i,+

2

2 W ⁿ⁺

1

i

2

= W _i ⁿ − ^∆ t

∆ x

F ⁽ W _i, ⁿ ₋ , W _i,+ ⁿ ) − F ⁽ W _i ⁿ ₋ _1,+ , W _i, ⁿ ₋ ) + ∆ t

4 s ^t ( W _i ⁿ ₋ _1,+ , W _i, ⁿ ₋ ) + 2 s ^t ( W _i, ⁿ ₋ , W _i,+ ⁿ ) + s ^t ( W _i,+ ⁿ , W _i+1, ⁿ ₋ )

(69)

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

(70)

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 ) , the scheme reads:

W ⁿ⁺

1

i

2

= 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 _i ⁿ⁺

¹²

with a splitting strategy:

( ∂ _t h = 0

∂ _t q = − k q k q k h ⁻ ^η

 

 



 

 

h ⁿ⁺¹ _i = h ⁿ⁺ _i

¹²

q ⁿ⁺¹ _i = ( h ^η ) ⁿ⁺¹ _i q ⁿ⁺ _i

¹²

( h ^η ) ⁿ⁺¹ _i + k ∆ t q ⁿ⁺

1

i

2

(71)

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.

(72)

Verification of the well-balance: topography

transcritical flow 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 well-balanced scheme for the shallow-water equations with topography and Manning friction