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

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(1)

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

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

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

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

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

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

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

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

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

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

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

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

Li+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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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

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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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

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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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

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 3x Y = 0,

t Y = 0,

∂ t Z = 0.

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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

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 3x 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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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

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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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

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

(20)

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

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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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

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

(22)

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

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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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

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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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

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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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

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 R1 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 R1 h L

+ g

2 ( h R ) 2( h L ) 2

= S ∆ x.

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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

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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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

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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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

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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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

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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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

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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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

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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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

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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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

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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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

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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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

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

Ri−1 2

λ

Li+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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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

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

(37)

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

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

(38)

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

(39)

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)

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

(41)

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

(42)

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 i0 = ⇒ ∀ 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.

(43)

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)

(44)

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)

(45)

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)

(46)

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

(47)

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

(48)

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.

(49)

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 i1 0 1

θ

ni

ϕ

ni

WB HO

(50)

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

(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)

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;

(53)

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] .

(54)

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,PWB3

h,PWB5

1e3 1e4

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

4 6

1

kqk,PWB3

kqk,PWB5

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

4 6

1

h,PWB3

h,PWB5 1e-6 1e-8

1e-10 4

6 1

kqk,PWB3

kqk,PWB5

(55)

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.

(56)

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

(57)

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

(58)

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

(59)

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

(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)

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.

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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

(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)

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.

(66)

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

Verication of the well-balance: topography

(67)

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

(68)

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

(69)

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

(70)

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

(71)

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 )

(72)

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

(73)

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

(74)

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

(75)

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

Double dry dam-break on a sinusoidal bottom

Figure

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Références

Sujets connexes :
Outline : IVP: