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

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)

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

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} | .

### 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!

### 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!

### 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

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

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

_{i}

^{n}

### x x

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}

### 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+

^{1}

_{2}

### λ ^{L}

### i+

^{1}

_{2}

### 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.

### Godunov-type scheme (approximate Riemann solver)

### x t

### t

^{n+1}

### t

^{n}

### x

_{i}

### x

_{i−}1

2

### x

_{i+}1

2

### W

_{i}

^{n}

### W

_{i−}

^{n}1

2

### W

_{i+}

^{n}1

2

### λ

^{R}

i−^{1}_{2}

### λ

^{L}

_{i+}1

2

### z }| {

### W

^{∆}

### (x, t

^{n+1}

### )

### W

_{i−1}

^{n}

### W

_{i+1}

^{n}

### 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.

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

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

### .

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.

### 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.

### 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}

^{∗}

### 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

∆t0

### S(W

R### (x, t)) dt dx.

### next step: obtain a fourth relation

### 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

∆t0

### S(W

R### (x, t)) dt dx.

### next step : obtain a fourth relation

### 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

∆t0

### S(W

R### (x, t)) dt dx.

### next step : obtain a fourth relation

### 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

∆t0

### S(W

R### (x, t)) dt dx.

### next step : obtain a fourth relation

### 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

∆t0

### S(W

R### (x, t)) dt dx.

### next step : obtain a fourth relation

### 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

∆t0

### S(W

R### (x, t)) dt dx.

### next step : obtain a fourth relation

### 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:

### 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.

### 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.

### 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 .

### 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

### 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).

### 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

### 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,

### 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

^{2}

_{0}

### h

### + g

### 2 ∂

x### h

^{2}

### = − kq

0### | q

0### | h

^{−η}

### ,

2

### ∂

x### h

^{η−1}

### ∂

x### h

^{η+2}

###

###

### −−−−−−−→

discretization###

###

### q

^{2}

_{0}

### 1

### h

### + g 2

### h

^{2}

### = − kµ

0### q

_{0}

^{2}

### h

^{−η}

### ∆x,

2

### h

^{η−1}

### h

^{η+2}

2

### 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,

### 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

### .

### The full Godunov-type scheme

### x t

### t

^{n+1}

### t

^{n}

### x

i### x

_{i−}

^{1}

2

### x

_{i+}

^{1}

2

### W

_{i}

^{n}

### W

_{i−}

^{R,∗}1

2

### W

^{L,∗}

i+^{1}2

### 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+

^{1}

_{2}

### − W _{i} ^{n}

### − λ ^{R} _{i−}

1
2
### W ^{R,∗}

### i−

^{1}

_{2}

### − 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

###

### 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

### 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}

### 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−

^{1}

_{2}

### +

### h ^{−η} n+1 i+

^{1}

_{2}

### + k ∆ t µ ^{n+}

1 2

### i q _{i} ^{n} .

### semi-implicit treatment of the friction source term

### scheme able to model wet/dry transitions

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

### 1 Introduction to Godunov-type schemes

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

### 4 2D and high-order numerical simulations

### 5 Conclusion and perspectives

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

### Two-dimensional extension

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

###

###

###

###

###

### ∂ _{t} h + ∇ · q = 0

### ∂ _{t} q + ∇ ·

### q ⊗ q h ^{+}

### 1 2 gh ^{2} I 2

### = − gh∇Z − kq k q k h ^{η}

### to the right: simulation

### of the 2011 Japan

### tsunami

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

### Two-dimensional extension

### space discretization: Cartesian mesh x

_{i}

### c

i### e

ij### c

j### n

ij### With F _{ij} ^{n} ^{=} F ^{(} W _{i} ^{n} , W _{j} ^{n} ; n ij ) and ν i the neighbors of c i , the scheme reads:

### W ^{n+}

1 2

### i = W _{i} ^{n} − ^{∆} t X

### j∈ν

i### | e _{ij} |

### | c _{i} | F ij ^{n} + ∆ t 2

### X

### j∈ν

i### ( S ^{t} _{)} ^{n} _{ij} .

### W _{i} ^{n+1} is obtained from W ^{n+}

1

### i

2### with a splitting strategy:

### ( ∂ _{t} h = 0

### ∂ _{t} q = − k q k q k h ^{−η}

###

###

###

###

###

### h ^{n+1} _{i} = h ^{n+}

1

### i

2### q ^{n+1} _{i} = ( h ^{η} ) ^{n+1} _{i} q ^{n+}

1

### i

2### ( h ^{η} ) ^{n+1} + k ∆ t q ^{n+}

1 2

### 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.

### 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)

### 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)

### 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)

### 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+1}

^{n}

### W

_{i−1}

^{n}

### W

_{i}

^{n}

### W d

_{i}

^{n}

### (x)

### x x

_{i+}

^{1}

### x

_{i−}

^{1}

ci

∈S^{2} ∈/S^{2}

### 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

### 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.

### 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

### MOOD method

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

### MOOD loop

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

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

### if h

^{c}

### is non-negative (PAD criterion)

### if W

^{c}

### does not present spurious oscillations (DMP and u2 criteria)

### 3 where necessary, decrease the degree of the reconstruction

### 4 compute a new candidate solution

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

### 1 Introduction to Godunov-type schemes

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

### 4 2D and high-order numerical simulations

### 5 Conclusion and perspectives

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

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

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

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

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

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

### Order of accuracy assessment

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

### h = 1 ; q = r

### k r k ^{;} Z = 2 k k r k − ^{1} 2 g k r k ^{2} .

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

### [ − ^{0} . 3 , 0 . 3] × ^{[0} . 4 , 1] .

### 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^{WB}3

h,P^{WB}5

1e3 1e4

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

4 6

1

kqk,P^{WB}3

kqk,P^{WB}5

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

4 6

1

h,P^{WB}3

h,P^{WB}5 1e-6
1e-8

1e-10 4

6 1

kqk,P^{WB}3

kqk,P^{WB}5

### 2011 T ohoku tsunami

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

### code parallelized with OpenMP, run on 48 cores.

### 2011 T ohoku tsunami

### − ^{8}

### − ^{6}

### − ^{4}

### − ^{2} 0

### Russia (Vladivostok)

### Sea of Japan

### Japan (Hokkaid¯ o island)

### Kuril trench

### Pacific Ocean

### 2011 T ohoku tsunami

### 2011 T ohoku tsunami

### physical time of the simulation: ^{1} hour

### 2011 T ohoku tsunami

### Water depth at the sensors:

### #1: 5700 m;

### #2: 6100 m;

### #3: 4400 m.

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

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

### 0 1 , 200 2 , 400 3 , 600

### − ^{0} . 2 0 0 . 2 0 . 4 0 . 6

### 0 1 , 200 2 , 400 3 , 600 0

### 0 . 1 0 . 2

### Sensor #2

### 0 1 , 200 2 , 400 3 , 600 0

### 0 . 1

### 0 . 2

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

### 1 Introduction to Godunov-type schemes

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

### 4 2D and high-order numerical simulations

### 5 Conclusion and perspectives

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

### Conclusion

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

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

### This work has been published:

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

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

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

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

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

### C. Berthon, R. Loubère, and V. M.-D.

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

### C. Berthon and V. M.-D.

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

### Perspectives

### Work in progress

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

### Coriolis force source term breadth variation source term

### Long-term perspectives

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

### simulation of rogue waves

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

## Thank you for your attention!

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

### The discrete entropy inequality

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

### ∂ _{t} η ( W ) + ∂ _{x} G ( W ) ≤ q

### h S ( W ); η ( W ) = q ^{2} 2 h ^{+}

### gh ^{2}

### 2 ; G ( W ) = q h

### q ^{2} 2 h ^{+} gh ^{2}

### . At the discrete level, we show that:

### λ R ( η ^{∗} _{R} − η R ) − λ L ( η ^{∗} _{L} − η L )+( G R − G L ) ≤ q _{HLL}

### h _{HLL} S ∆ x + O ^{(∆} x ^{2} ) . main ingredients: h ^{∗} _{L} = h _{HLL} − S ∆ x λ _{R}

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

### ( λ _{R} − λ _{L} ) η _{HLL} ≤ λ _{R} η _{R} − λ _{L} η _{L} − ^{(} G _{R} − G _{L} )

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

### Verication of the well-balance: topography

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

### boundary.

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