[PPT] - A well-balanced reconstruction of wet/dry fronts for the shallow PowerPoint Presentation

SLIDE 1

A well-balanced reconstruction of wet/dry fronts for the shallow water equations

Guoxian Chen

Wuhan University, P.R.China Co-authors: Bollermann, Kurganov, Noelle

May 24, 2014

SLIDE 2

Outline

1

Governing Equations

2

A Central-Upwind Scheme for the Shallow Water Equations

3

A New Reconstruction at the Almost Dry Cells

4

Positivity Preserving and Well-Balancing

5

Numerical Experiments

6

Conclusion and Future Worker

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 2 / 38

SLIDE 3

1

Governing Equations

2

A Central-Upwind Scheme for the Shallow Water Equations

3

A New Reconstruction at the Almost Dry Cells

4

Positivity Preserving and Well-Balancing

5

Numerical Experiments

6

Conclusion and Future Worker

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 3 / 38

SLIDE 4

Saint-Venant system of shallow water equations

Notations: h(x, t) is the fluid depth, u(x, t) is the velocity, g is the gravitational constant, B(x) represents the bottom topography.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 4 / 38

SLIDE 5

Saint-Venant system of shallow water equations

Notations: h(x, t) is the fluid depth, u(x, t) is the velocity, g is the gravitational constant, B(x) represents the bottom topography. In one dimension, the Saint-Venant system reads:    ht + (hu)x = 0, (hu)t +

hu2 + 1

2gh2

x = −ghBx,

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 4 / 38

SLIDE 6

Saint-Venant system of shallow water equations

Notations: h(x, t) is the fluid depth, u(x, t) is the velocity, g is the gravitational constant, B(x) represents the bottom topography. In one dimension, the Saint-Venant system reads:    ht + (hu)x = 0, (hu)t +

hu2 + 1

2gh2

x = −ghBx,

subject to the initial conditions h(x, 0) = h0(x), u(x, 0) = u0(x),

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 4 / 38

SLIDE 7

Numerical difficulties

Figure: Numerical storm over lake Rursee, produced by a naive finite volume scheme

Quasi steady solutions, Coarse grid, Numerical storm etc.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 5 / 38

SLIDE 8

Numerical difficulties

Bj w xj− 1

2

xj+ 1

2

B ˜ B

Quasi steady solutions, Coarse grid, Numerical storm etc. Solution: Well-balanced scheme u = 0, w := h + B = Const.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 6 / 38

SLIDE 9

Numerical difficulties

Bj w xj− 1

2

xj+ 1

2

B ˜ B

Quasi steady solutions, Coarse grid, Numerical storm etc. Solution: Well-balanced scheme u = 0, w := h + B = Const. dry areas (island, shore). Lose strictly hyperbolic in the dry areas (h = 0): u ± √gh; The calculation will simply due to h < 0 break down. Solution: a positive preserving scheme h ≥ 0

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 6 / 38

SLIDE 10

Numerical difficulties

Bj w xj− 1

2

xj+ 1

2

B ˜ B

Quasi steady solutions, Coarse grid, Numerical storm etc. Solution: Well-balanced scheme u = 0, w := h + B = Const. dry areas (island, shore). Lose strictly hyperbolic in the dry areas (h = 0): u ± √gh; The calculation will simply due to h < 0 break down. Solution: a positive preserving scheme h ≥ 0 “dry lake at rest” steady state: hu = 0, h = 0.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 6 / 38

SLIDE 11

1

Governing Equations

2

A Central-Upwind Scheme for the Shallow Water Equations

3

A New Reconstruction at the Almost Dry Cells

4

Positivity Preserving and Well-Balancing

5

Numerical Experiments

6

Conclusion and Future Worker

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 7 / 38

SLIDE 12

Overview of KP scheme: Kurganov and Petrova, 2007

Bj w xj− 1

2

xj+ 1

2

B ˜ B

Bottom is Continuous, piecewise linear approximated by B

B(x) = Bj− 1

2 +

Bj+ 1

2 − Bj− 1 2

·

x − xj− 1

2

∆x , xj− 1

2 ≤ x ≤ xj+ 1 2 .

where Bj+ 1

2 :=

B(xj+ 1

2 + 0) + B(xj+ 1 2 − 0)

2 , then Bj := B(xj) = 1 ∆x

Ij
B(x) dx =

Bj+ 1

2 + Bj− 1 2

2 .

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 8 / 38

SLIDE 13

Central-upwind semi-discretization

Cell averages of U := (w, hu)T is defined Uj(t) ≈ 1 ∆x

Ij

U(x, t) dx.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 9 / 38

SLIDE 14

Central-upwind semi-discretization

Cell averages of U := (w, hu)T is defined Uj(t) ≈ 1 ∆x

Ij

U(x, t) dx. Time-dependent ODEs (SSP-RK-3) d dt Uj(t) = − Hj+ 1

2 (t) − Hj− 1 2 (t)

∆x + Sj(t),

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 9 / 38

SLIDE 15

Central-upwind semi-discretization

Cell averages of U := (w, hu)T is defined Uj(t) ≈ 1 ∆x

Ij

U(x, t) dx. Time-dependent ODEs (SSP-RK-3) d dt Uj(t) = − Hj+ 1

2 (t) − Hj− 1 2 (t)

∆x + Sj(t), the cell averages of the source term: Sj(t) ≈ 1 ∆x

Ij

S(U(x, t), B(x)) dx, S := (0, −ghBx)T. i.e S

(2) j

(t) := −ghj Bj+ 1

2 − Bj− 1 2

∆x .

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 9 / 38

SLIDE 16

central-upwind numerical fluxes

The central-upwind numerical fluxes: Hj+ 1

2 (t)

= a+

j+ 1

2 F(U−

j+ 1

2 , Bj+ 1 2 ) − a−

j+ 1

2 F(U+

j+ 1

2 , Bj+ 1 2 )

a+

j+ 1

2 − a−

j+ 1

2

+ a+

j+ 1

2 a−

j+ 1

2

a+

j+ 1

2 − a−

j+ 1

2

U+

j+ 1

2 − U−

j+ 1

2

,

where we use the following flux notation: F(U, B) :=

hu, (hu)2

w − B + g 2 (w − B)2 T .

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 10 / 38

SLIDE 17

central-upwind numerical fluxes

The central-upwind numerical fluxes: Hj+ 1

2 (t)

= a+

j+ 1

2 F(U−

j+ 1

2 , Bj+ 1 2 ) − a−

j+ 1

2 F(U+

j+ 1

2 , Bj+ 1 2 )

a+

j+ 1

2 − a−

j+ 1

2

+ a+

j+ 1

2 a−

j+ 1

2

a+

j+ 1

2 − a−

j+ 1

2

U+

j+ 1

2 − U−

j+ 1

2

,

where we use the following flux notation: F(U, B) :=

hu, (hu)2

w − B + g 2 (w − B)2 T . the local speeds a±

j+ 1

2 in numerical flux are obtained using the eigenvalues of

the Jacobian ∂F

∂U as follows:

a+

j+ 1

2 = max

u+

j+ 1

2 +

gh+

j+ 1

2 , u−

j+ 1

2 +

gh−

j+ 1

2 , 0

,

a−

j+ 1

2 = min

u+

j+ 1

2 −

gh+

j+ 1

2 , u−

j+ 1

2 −

gh−

j+ 1

2 , 0

.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 10 / 38

SLIDE 18

Initial date reconstruction(Surface Gradient Method)

piecewise linear reconstruction q stands for w and u respectively.

q(x) := qj + (qx)j(x − xj),

xj− 1

2 < x < xj+ 1 2 , Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38

SLIDE 19

Initial date reconstruction(Surface Gradient Method)

piecewise linear reconstruction q stands for w and u respectively.

q(x) := qj + (qx)j(x − xj),

xj− 1

2 < x < xj+ 1 2 ,

The generalized minmod limiter: (qx)j = minmod

θqj − qj−1

∆x , qj+1 − qj−1 2∆x , θqj+1 − qj ∆x

,

θ ∈ [1, 2], with minmod(z1, z2, ...) :=    minj{zj}, if zj > 0 ∀j, maxj{zj}, if zj < 0 ∀j, 0,

therwise,

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38

SLIDE 20

Initial date reconstruction(Surface Gradient Method)

piecewise linear reconstruction q stands for w and u respectively.

q(x) := qj + (qx)j(x − xj),

xj− 1

2 < x < xj+ 1 2 ,

The generalized minmod limiter: (qx)j = minmod

θqj − qj−1

∆x , qj+1 − qj−1 2∆x , θqj+1 − qj ∆x

,

θ ∈ [1, 2], with minmod(z1, z2, ...) :=    minj{zj}, if zj > 0 ∀j, maxj{zj}, if zj < 0 ∀j, 0,

therwise,

average velocity is defined as with ǫ = 10−9 uj :=

(hu)j/hj,

if hj ≥ ǫ, 0,

therwise.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38

SLIDE 21

Initial date reconstruction(Surface Gradient Method)

piecewise linear reconstruction q stands for w and u respectively.

q(x) := qj + (qx)j(x − xj),

xj− 1

2 < x < xj+ 1 2 ,

The generalized minmod limiter: (qx)j = minmod

θqj − qj−1

∆x , qj+1 − qj−1 2∆x , θqj+1 − qj ∆x

,

θ ∈ [1, 2], with minmod(z1, z2, ...) :=    minj{zj}, if zj > 0 ∀j, maxj{zj}, if zj < 0 ∀j, 0,

therwise,

average velocity is defined as with ǫ = 10−9 uj :=

(hu)j/hj,

if hj ≥ ǫ, 0,

therwise.

water height is recontructed by h±

j+ 1

2 = w ±

j+ 1

2 − Bj+ 1 2 . Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38

SLIDE 22

Positive correction of water height(Key of KP scheme)

Bj− 1

2

Bj+ 1

2

xj x∗

w

w+

j− 1

2

w−

j+ 1

2

wj w+

j+ 1

2

wj+1 Bj− 1

2

Bj+ 1

2

xj x∗

w

w−

j+ 1

2

wj w+

j+ 1

2

wj+1

If w −

j+ 1

2 < Bj+ 1 2 ,

then set (wx)j := Bj+ 1

2 − w j

∆x/2 , = ⇒ w −

j+ 1

2 = Bj+ 1 2 ,

w +

j− 1

2 = 2w j − Bj+ 1 2 ;

and If w +

j− 1

2 < Bj− 1 2 ,

then set (wx)j := w j − Bj− 1

2

∆x/2 , = ⇒ w −

j+ 1

2 = 2w j − Bj− 1 2 ,

w +

j− 1

2 = Bj− 1 2 . Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 12 / 38

SLIDE 23

1

Governing Equations

2

A Central-Upwind Scheme for the Shallow Water Equations

3

A New Reconstruction at the Almost Dry Cells

4

Positivity Preserving and Well-Balancing

5

Numerical Experiments

6

Conclusion and Future Worker

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 13 / 38

SLIDE 24

Problem

Bj− 1

2

Bj+ 1

2

xj x∗

w

w+

j− 1

2

w−

j+ 1

2

wj w+

j+ 1

2

wj+1 Bj− 1

2

Bj+ 1

2

xj x∗

w

w−

j+ 1

2

wj w+

j+ 1

2

wj+1

ne may clearly see in the same figure, the obtained reconstruction is not

well-balanced since the reconstructed values w −

j+ 1

2 and w +

j+ 1

2 are not the same. Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 14 / 38

SLIDE 25

Correction: Cell average hj to water surface wj

xj− 1

2

xj+ 1

2

B wj ∆x · hj x∗

w

xj− 1

2

xj+ 1

2

B wj ∆x · hj

Fully flooded cell: hj ≥ ∆x

2 |(Bx)j|,

wj(x) = w j,

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 15 / 38

SLIDE 26

Correction: Cell average hj to water surface wj

xj− 1

2

xj+ 1

2

B wj ∆x · hj x∗

w

xj− 1

2

xj+ 1

2

B wj ∆x · hj

Fully flooded cell: hj ≥ ∆x

2 |(Bx)j|,

wj(x) = w j, Partially flooded cell: wj(x) =

Bj(x),

if x < x⋆

w,

wj,

therwise,

where x⋆

w is the boundary point separating the dry and wet parts in the cell j.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 15 / 38

SLIDE 27

Flat surface calculation

It can be determined by the mass conservation, ∆x · hj = xj+ 1

2

xj− 1

2

(wj(x) − Bj(x))dx = xj+ 1

2

x⋆

w

(wj − Bj(x))dx = ∆x⋆

w

2 (wj − Bj+ 1

2 ) = ∆x⋆

w

2 (B(x⋆

w) − Bj+ 1

2 ) = −(∆x⋆

w)2

2 (Bx)j, where ∆x⋆

w = xj+ 1

2 − x⋆

w, thus

∆x⋆

w =

2∆xhj

−(Bx)j =

2hj

Bj− 1

2 − Bj+ 1 2

∆x, resulting in the free surface wj formula for the wet/dry cells, wj = Bj+ 1

2 +

2hj(Bj− 1

2 − Bj+ 1 2 ) Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 16 / 38

SLIDE 28

Piecewise Linear

Bj− 1

2

Bj+ 1

2

wj w+

j−1/2

w±

j+ 1

2

xj− 1

2

xj+ 1

2

x∗

w

xj Bj− 1

2

Bj+ 1

2

wj w±

j+ 1

2

xj− 1

2

xj+ 1

2

x∗

w

xj x∗

j

set w −

j+ 1

2 = w +

j+ 1

2 and determine the reconstruction of w in cell j Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 17 / 38

SLIDE 29

Piecewise Linear

Bj− 1

2

Bj+ 1

2

wj w+

j−1/2

w±

j+ 1

2

xj− 1

2

xj+ 1

2

x∗

w

xj Bj− 1

2

Bj+ 1

2

wj w±

j+ 1

2

xj− 1

2

xj+ 1

2

x∗

w

xj x∗

j

set w −

j+ 1

2 = w +

j+ 1

2 and determine the reconstruction of w in cell j

Case 1: the amount of water in cell j is sufficiently large hj = 1 2(h−

j+ 1

2 + h+

j− 1

2 ),

btain w +

j− 1

2 = h+

j− 1

2 + Bj− 1 2 . Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 17 / 38

SLIDE 30

Piecewise Linear

Bj− 1

2

Bj+ 1

2

wj w+

j−1/2

w±

j+ 1

2

xj− 1

2

xj+ 1

2

x∗

w

xj Bj− 1

2

Bj+ 1

2

wj w±

j+ 1

2

xj− 1

2

xj+ 1

2

x∗

w

xj x∗

j

set w −

j+ 1

2 = w +

j+ 1

2 and determine the reconstruction of w in cell j

Case 1: the amount of water in cell j is sufficiently large hj = 1 2(h−

j+ 1

2 + h+

j− 1

2 ),

btain w +

j− 1

2 = h+

j− 1

2 + Bj− 1 2 .

Case 2: h+

j− 1

2 < 0 The breaking point between the “wet” and “dry” pieces

will be denoted by x⋆

j and it will be determined from the conservation

requirement, which in this case reads ∆x · hj = ∆x⋆

j

2 h−

j+ 1

2 ,

where

⋆

⋆
Guoxian Chen (WHU)

well-balanced reconstruction of wet/dry fronts May 24, 2014 17 / 38

SLIDE 31

Summation

Combining the above two cases, we obtain the reconstructed value h+

j− 1

2 = max

0, 2hj − h−

j+ 1

2

We also generalize the definition of ∆x⋆

j and set

∆x⋆

j := ∆x · min

2 hj

h−

j+ 1

2

, 1

,

which will be used in the proofs of the positivity and well-balancing of the resulting central-upwind scheme

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 18 / 38

SLIDE 32

1

Governing Equations

2

A Central-Upwind Scheme for the Shallow Water Equations

3

A New Reconstruction at the Almost Dry Cells

4

Positivity Preserving and Well-Balancing

5

Numerical Experiments

6

Conclusion and Future Worker

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 19 / 38

SLIDE 33

Lemma: Assume that the system of ODEs is solved by the forward Euler method and that for all j, h

n j ≥ 0. Then, for all j, h n+1 j

≥ 0, provided ∆t ≤ min

j

∆x⋆

j

2aj

,

aj := max{a+

j+ 1

2 , −a−

j+ 1

2 }. Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 20 / 38

SLIDE 34

Lemma: Assume that the system of ODEs is solved by the forward Euler method and that for all j, h

n j ≥ 0. Then, for all j, h n+1 j

≥ 0, provided ∆t ≤ min

j

∆x⋆

j

2aj

,

aj := max{a+

j+ 1

2 , −a−

j+ 1

2 }.

Proof: h

n+1 j

=

∆x⋆

j

2∆x − λa+

j+ 1

2

u−

j+ 1

2 − a−

j+ 1

2

a+

j+ 1

2 − a−

j+ 1

2

h−

j+ 1

2

− λa−

j+ 1

2

a+

j+ 1

2 − u+

j+ 1

2

a+

j+ 1

2 − a−

j+ 1

2

h+

j+ 1

2 + λa+

j− 1

2

u−

j− 1

2 − a−

j− 1

2

a+

j− 1

2 − a−

j− 1

2

h−

j− 1

2 , Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 20 / 38

SLIDE 35

Draining time(Bollermann, Noelle, Lukacova, 2011 )

Wet region Dry region tn+1= tn+∆t t

κ∆t

tn xl xr

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 21 / 38

SLIDE 36

Draining time estimation

We need to ensure that the first component of discrete system satisfies h

n+1 j

= h

n j − ∆t

H(1)

j+ 1

2 − H(1)

j− 1

2

∆x ≥ 0.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 22 / 38

SLIDE 37

Draining time estimation

We need to ensure that the first component of discrete system satisfies h

n+1 j

= h

n j − ∆t

H(1)

j+ 1

2 − H(1)

j− 1

2

∆x ≥ 0. Draining time step, ∆tj,drain = ∆xh

n j

max(0, H(1)

j+ 1

2 ) + max(0, −H(1)

j− 1

2 )

, which describes the time when the water contained in cell j in the beginning

f the time step has left via the outflow fluxes.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 22 / 38

SLIDE 38

Draining time estimation

We need to ensure that the first component of discrete system satisfies h

n+1 j

= h

n j − ∆t

H(1)

j+ 1

2 − H(1)

j− 1

2

∆x ≥ 0. Draining time step, ∆tj,drain = ∆xh

n j

max(0, H(1)

j+ 1

2 ) + max(0, −H(1)

j− 1

2 )

, which describes the time when the water contained in cell j in the beginning

f the time step has left via the outflow fluxes.

We now replace the evolution step with h

n+1 j

= h

n j −

∆tj+ 1

2 H(1)

j+ 1

2 − ∆tj− 1 2 H(1)

j− 1

2

∆x , where we set ∆tj+ 1

2 = min(∆t, ∆ti,drain),

i = j + 1 2 − sgn

H(1)

j+ 1

2

2

.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 22 / 38

SLIDE 39

The draining time step technique is only used close to the dry boundary. Away from these regions, we have ∆tj+ 1

2 = ∆t and therefore the finite volume updates

are not changed.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 23 / 38

SLIDE 40

Momentum update with draining time

Split the flux in its advective and gravity driven parts: Fa(V) :=

hu, (hu)2

h T and Fg(V) :=

0, g

2 h2T ,

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 24 / 38

SLIDE 41

Momentum update with draining time

Split the flux in its advective and gravity driven parts: Fa(V) :=

hu, (hu)2

h T and Fg(V) :=

0, g

2 h2T , The corresponding advective and gravity driven parts of the central-upwind fluxes then read Hg

j+ 1

2 (t) =

a+

j+ 1

2 Fg(V−

j+ 1

2 ) − a−

j+ 1

2 Fg(V+

j+ 1

2 )

a+

j+ 1

2 − a−

j+ 1

2

+ a+

j+ 1

2 a−

j+ 1

2

a+

j+ 1

2 − a−

j+ 1

2

V+

j+ 1

2 − V−

j+ 1

2

,

and Ha

j+ 1

2 (t) =

a+

j+ 1

2 Fa(V−

j+ 1

2 ) − a−

j+ 1

2 Fa(V+

j+ 1

2 )

a+

j+ 1

2 − a−

j+ 1

2

,

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 24 / 38

SLIDE 42

Momentum update with draining time

Split the flux in its advective and gravity driven parts: Fa(V) :=

hu, (hu)2

h T and Fg(V) :=

0, g

2 h2T , The corresponding advective and gravity driven parts of the central-upwind fluxes then read Hg

j+ 1

2 (t) =

a+

j+ 1

2 Fg(V−

j+ 1

2 ) − a−

j+ 1

2 Fg(V+

j+ 1

2 )

a+

j+ 1

2 − a−

j+ 1

2

+ a+

j+ 1

2 a−

j+ 1

2

a+

j+ 1

2 − a−

j+ 1

2

V+

j+ 1

2 − V−

j+ 1

2

,

and Ha

j+ 1

2 (t) =

a+

j+ 1

2 Fa(V−

j+ 1

2 ) − a−

j+ 1

2 Fa(V+

j+ 1

2 )

a+

j+ 1

2 − a−

j+ 1

2

, modified finite volume update: V

n+1 j

= V

n j −

∆tj+ 1

2 Ha

j+ 1

2 − ∆tj− 1 2 Ha

j− 1

2

∆x

− ∆t

Hg

j+ 1

2 − Hg

j− 1

2

∆x + S

n j

.

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 24 / 38

SLIDE 43

Theorem(well-balance)

Theorem: The steady state which is a combination of the “lake at rest” and “dry lake” states are preserved, that is, the scheme is well-balanced V(tn+1) = V(tn).

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 25 / 38

SLIDE 44

Theorem(well-balance)

Theorem: The steady state which is a combination of the “lake at rest” and “dry lake” states are preserved, that is, the scheme is well-balanced V(tn+1) = V(tn). Proof hn+1

j

=hn

j + ∆t

∆x (Fh

j+ 1

2 − Fh

j− 1

2 ) = hn

j

(hu)n+1

j

=(hu)n

j + ∆t

∆x 1 2g(h−

j+ 1

2 )2 − 1

2g(h+

j− 1

2 )2

+ ∆t

∆x h−

j+ 1

2 + h+

j− 1

2

2 (b−

j+ 1

2 − b+

j− 1

2 )

=(hu)n

j + ∆t

∆x h−

j+ 1

2 + h+

j− 1

2

2 ((b−

j+ 1

2 + h−

j+ 1

2 ) − (b+

j− 1

2 + h+

j− 1

2 ))

=(hu)n

j = 0

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 25 / 38

SLIDE 45

1

Governing Equations

2

A Central-Upwind Scheme for the Shallow Water Equations

3

A New Reconstruction at the Almost Dry Cells

4

Positivity Preserving and Well-Balancing

5

Numerical Experiments

6

Conclusion and Future Worker

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 26 / 38

SLIDE 46

Example 1. Numerical accuracy order(continuous example)

With computational domain [0, 1], the problem is subject to the gravitational constant g = 9.812, the bottom topography B(x) = sin2(πx), the initial data h(x, 0) = 5 + ecos(2πx), hu(x, 0) = sin(cos(2πx)), and the periodic boundary conditions. # points h error EOC hu error EOC 25 5.30e-2 2.33e-1 50 1.51e-2 1.81 1.38e-1 0.76 100 4.86e-3 1.63 4.43e-2 1.64 200 1.40e-3 1.80 1.14e-2 1.95 400 3.59e-4 1.96 2.84e-3 2.01 800 8.93e-5 2.01 7.05e-4 2.01

Table: The reference solution is computed on a grid with 12800 cells. Time t=0.1.

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SLIDE 47

Example 2. Still lake

The “lake at rest” and “dry lake” are combined in the domain [0, 1] with the bottom topography given by B(x) = 1 4 − 1 4 cos((2x − 1)π), and the following initial data: h(x, 0) = max (0, 0.4 − B(x)) , u(x, 0) ≡ 0. scheme L∞ error of h L∞ error of hu KP 7.88e-5 9.08e-5 BCKN 3.33e-16 5.43e-16

Table: Numerical solutions with 200 points at the final time T = 19.87

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SLIDE 48

x w=h+B

0.1 0.15 0.2 0.25 0.3 0.3998 0.4 0.4002 KP BCKN Bed

x hu

0.2 0.4 0.6 0.8 1

0.0001
5E-05

5E-05 0.0001 KP BCKN

Figure: Lake at rest. Left: free surface h + B; Right: Discharge hu.

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SLIDE 49

Example 3. oscillating lake

We now consider a sinusoidal perturbation of the steady state by taking h(x, 0) = max

0, 0.4 + sin (4x − 2 − max(0, −0.4 + B(x)))

25 − B(x)

.

we set the final time to be T = 19.87

x w=h+B 0.2 0.4 0.6 0.8 1 0.35 0.4 0.45 0.5

KP BCKN

Ref. sol.

Bed

x hu 0.2 0.4 0.6 0.8 1 0.002 0.004 0.006 0.008 0.01 0.012 0.014

KP BCKN

Ref. Sol.

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

x w=h+B 0.74 0.76 0.78 0.8 0.82 0.84 0.35 0.355 0.36 0.365 0.37 0.375

200 pts 400 pts 800 pts

Ref. Sol.

bed x + x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

x hu 0.74 0.76 0.78 0.8 0.82 0.84 0.002 0.004 0.006 0.008 0.01 0.012 0.014

200 pts 400 pts 800 pts

Ref. sol.

x +

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SLIDE 50

Numerical order

# points h error EOC hu error EOC 25 9.48e-3 1.47e-2 50 2.81e-3 1.75 7.26e-3 1.02 100 1.65e-3 0.77 2.46e-3 1.56 200 7.88e-4 1.06 1.59e-3 0.63 400 3.33e-4 1.24 6.19e-4 1.36 800 1.26e-4 1.40 2.27e-4 1.45 KP scheme 25 7.55e-3 1.31e-2 50 2.27e-3 1.74 6.04e-3 1.11 100 1.45e-3 0.65 2.35e-3 1.36 200 6.77e-4 1.09 1.31e-3 0.84 400 2.71e-4 1.32 5.04e-4 1.38 800 1.04e-4 1.38 1.87e-4 1.43 BCKN scheme

Table: Oscillating lake: Experimental order of convergence measured in the L1-norm.

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SLIDE 51

Example 4. Wave run-up on a sloping shore

The initial data are H0(x) = max

D + δ sech2(γ(x − xa)), B(x)
,

u0(x) = g D H0(x), and the bottom topography is B(x) =    0, if x < 2xa, x − 2xa 19.85 ,

therwise.

We set D = 1, δ = 0.019, γ =

3δ

4D , xa =

4D

3δ arccosh √ 20

.

The computational domain is [0, 80] and the number of grid cells is 200.

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SLIDE 52

Example 4.

x w=h+B 35 40 45 50 55 60 1 1.05 1.1

KP BCKN

Ref. sol.

bed t=17 t=80 t=23 t=28

x hu 20 25 30 35 40 45 50 55 60

0.06
0.04
0.02

0.02

KP BCKN

Ref. sol.

t=17 t=80 t=23 t=28

Figure: Left: free surface w = h + B; Right: discharge hu.

t log(||w-max(1,B)||∞) 50 100 150 200 250 300

14
12
10
8
6
4
2

KP BCKN 200 pts 100 pts 800 pts 400 pts

t log(||hu||∞) 50 100 150 200 250 300

12
10
8
6
4
2

KP BCKN 200 pts 100 pts 800 pts 400 pts

Figure: deviation from stationary state. Left: free surface log(||w − max(1, B)||∞); Right: discharge ln(||hu||∞). Long time convergence of KP scheme stalls.

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SLIDE 53

Example 5. Dam-break over a plane

x α=0

10
5

5 10

1

1 2 3 4 5 6 7

w u bed

t front position 0.5 1 1.5 2 2 4 6 8 10 12 14

BCKN Exact solution

t front velocity 0.5 1 1.5 2 2 4 6 8

BCKN Exact solution

x α=π/6

10
5

5 10

1

1 2 3 4 5 6

w u bed

t front position 0.5 1 1.5 2 2 4 6 8 10 12 14

BCKN Exact solution

t front velocity 0.5 1 1.5 2 2 4 6 8

BCKN Exact solution

x α=-π/6

10
5

5 10 2 4 6 8

w u bed

t front position 0.5 1 1.5 2 2 4 6 8 10 12 14

BCKN Exact solution

t front velocity 0.5 1 1.5 2 2 4 6 8

BCKN Exact solution

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 34 / 38

SLIDE 54

Example 6. Laboratory dam-break over a triangular hump

dam Reservoir 0.4m 0.75m 15.5m 10m 6m 6.5m water surface

t h

20 40 60 80 0.1 0.2 0.3 0.4 0.5 0.6 0.7

BCKN Lab measured

GP2: h(t)

t h

20 40 60 80 0.1 0.2 0.3 0.4 0.5 0.6 0.7

BCKN Lab measured

GP13: h(t)

t h

20 40 60 80 0.1 0.2 0.3 0.4 0.5 0.6 0.7

BCKN Lab measured

GP20: h(t)

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SLIDE 55

1

Governing Equations

2

A Central-Upwind Scheme for the Shallow Water Equations

3

A New Reconstruction at the Almost Dry Cells

4

Positivity Preserving and Well-Balancing

5

Numerical Experiments

6

Conclusion and Future Worker

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 36 / 38

SLIDE 56

Designed a special reconstruction of the water level at wet/dry fronts: conservative, well-balanced and positivity preserving

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 37 / 38

SLIDE 57

Designed a special reconstruction of the water level at wet/dry fronts: conservative, well-balanced and positivity preserving Draining time method is implied to enlarge the time step

Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 37 / 38

SLIDE 58

Designed a special reconstruction of the water level at wet/dry fronts: conservative, well-balanced and positivity preserving Draining time method is implied to enlarge the time step Two dimensional extension.

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SLIDE 59

Thank you!

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