Matrix Sign Function and Roe Scheme Presented by Ababacar DIAGNE - - PowerPoint PPT Presentation

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

Matrix Sign Function and Roe Scheme

Presented by Ababacar DIAGNE Guided by

• Pr. Enrique Fernandez Nieto

Universit´ e Gaston Berger de Saint-Louis UFR de Sciences Appliqu´ ees et Technologie Laboratoire d’Analyse Num´ erique et d’Informatique

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Outline

1

First Order Roe’s scheme

2

Matrix Sign Function

3

Saint Venant Equations

4

Saint Venant model coupled with sediment transport equation

5

A Two-Layer model coupled with sediment transport equation

6

Numerical Result

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

1

First Order Roe’s scheme

2

Matrix Sign Function

3

Saint Venant Equations

4

Saint Venant model coupled with sediment transport equation

5

A Two-Layer model coupled with sediment transport equation

6

Numerical Result

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

The Roe Scheme

We consider the general form of hyperbolic equation written under the form ∂U ∂t + ∂F(U) ∂x = B(U)∂U ∂x + S(U)∂σ ∂x , (1.1) where the unknown U(x, t) takes values an open convex set D

• f RN, F is a regular function from D to RN and σ(x) a known

function from RN to RN. Considering the trivial equation, ∂tσ = 0, system (1.1) can be presented in the form ∂W ∂t + A(W)∂W ∂x = 0, (1.2)

Roe Scheme

when considering the auxiliar variable W = (U, σ). The matrix A belong to MN+1(R) and can be written as A(W) = A(W) −S(W)

• where A(W) = J(W) − B(W) and J is the jacobian matrix of

the flux function F.

Family of Paths

A family of paths in Ω ⊂ RN is a locally lipschitz map Φ : [0, 1] × Ω × Ω − → Ω such that: Φ(0; WL, WR) = WL, Φ(1; WL, WR) = WR for any WL, WR in For every bounded set O of Ω there exist k such that

• ∂Φ

∂s (s; WL, WR)

• ≤ k |WL − WR|

for any WL, WR in Ω and s ∈ [0, 1]. For every bounded set O of Ω there exists K such that

• ∂Φ

∂s (s; W 1

L , W 1 R) − ∂Φ

∂s (s; W 2

L , W 2 R)

• ≤ K
• 2
• i=1
• W (i)

L

− W (i)

R

• for any W 1

L , W 1 R, W 2 L , W 2 R in O and s ∈ [0, 1].

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

Roe matrix

Given a family of paths Ψ, a matrix function AΨ : Ω × Ω − → MN(R) is called Roe linerizarion if it satisfies: for any WL, WR ∈ Ω, AΨ(WL, WR) has N real distinct eigenvalues; for all W ∈ Ω, AΨ(W, W) = AΨ(W); for any WL, WR ∈ Ω: AΨ(WL, WR).(WL − WR) = 1 A

• Ψ(s; WL, WR)

∂Ψ ∂s (s; WL, WR) ds

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

Numerical flux

As usual, we denote by W n

i the approximation of the cells

averages for the exact solution provided by the numerical scheme: W n

i

≈ 1 ∆x xi+1/2

xi−1/2

W(x, tn) dx. We wil use the following notation, in order to calculate these approximation: we denote the Roe matrix associated to the states W n

i and W n i+1

Ai+1/2 = AΨ(Wi, Wi+1)

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

Numerical flux

As usual, we denote by W n

i the approximation of the cells

averages for the exact solution provided by the numerical scheme: W n

i

≈ 1 ∆x xi+1/2

xi−1/2

W(x, tn) dx. We wil use the following notation, in order to calculate these approximation: we denote the Roe matrix associated to the states W n

i and W n i+1

Ai+1/2 = AΨ(Wi, Wi+1)

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Roe matrix

We also define the matrix A±

i+1/2 = Ki+1/2

   (λi+1/2

1

)± ... (λi+1/2

N

)±    K −1

i+1/2 (1.4)

(1.5) and

• Ai+1/2
• = A+

i+1/2 − A− i+1/2.

Roe Scheme

We show that the approximation at time tn+1 can be obtained by the formula under the hypothesis: xi−1/2 + λi−1/2

1

∆t ≤ xi ≤ xi+1/2 + λi+1/2

N

∆t. W n+1

i

= W n

i − ∆t

∆x

• A+

i−1/2.(W n i − W n i−1) − A+ i+1/2.(W n i+1 − W n i )

• ,(1.6)

which is the general expression of a Roe scheme for (1.2).

Roe Scheme

From (1.6) and through somes algebraic calculations, the scheme writes as follows: Un+1

i

=Un

i + ∆t

∆x

• Fi−1/2 − Fi+1/2
• + ∆t

2∆x

• Bi−1/2(Un

i − Un i−1) + Bi+1/2(Un i+1 − Un i )

• + ∆t

∆x

• P+

i−1/2Si−1/2(σi − σi−1) − P− i+1/2Si+1/2(σi+1 − σi)

• .

where the associated numerical flux is written Fi+1/2 = 1 2

• F(Un

i+1) + F(Un i )

• − 1

2

• Ai+1/2
• Un

i+1 − Un i

• (1.8)

and P±

i+1/2 = 1

2

• I ±
• Ai+1/2
• Ai+1/2
• .

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

1

First Order Roe’s scheme

2

Matrix Sign Function

3

Saint Venant Equations

4

Saint Venant model coupled with sediment transport equation

5

A Two-Layer model coupled with sediment transport equation

6

Numerical Result

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Matrix Sign Function

We consider a matrix A ∈ M(Rn×n) that admits n eigenvalues. A matrix sign function sign(A) is defined by |A| = A × sign(A), sign(A) = K    sign(λ1) ... sign(λn)    K −1, (2.1) where K is a non singular matrix and sign(λi) =    1 if λi > 0 if λi = 0 −1 if λi < 0.

Matrix Sign Function

Our sharp idea is to made this polynomial iteration in a domain containing all the eigenvalues of the matrix A. Since his spectral radius is bounded by the induced norm of the matrix i.e. ρ(A) ≤ ||A||,˙ We define the quantity S = ||A|| and consider the iterative procedure x0 = x ∈ [−S, S] xk+1 = P(xk) (2.2)

Iterative Polynomial

y=P(x) y=x X y

−S S

Figure: Iterative Polynomial

P(x) = x − a(x − S)(x − S). (2.3)

Matrix Sign Function

The iteration procedure A0 = A Ak+1 = P(Ak) (2.4) converges to a matrix A∗ who can be decomposed in the form A∗ = K    S × sign(λ1) ... S × sign(λn)    K −1, (2.5) (2.6) sign(A) = A∗ S (2.7)

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

1

First Order Roe’s scheme

2

Matrix Sign Function

3

Saint Venant Equations

4

Saint Venant model coupled with sediment transport equation

5

A Two-Layer model coupled with sediment transport equation

6

Numerical Result

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Saint Venant Equation

The 1D shallow water equations are : ∂h ∂t + ∂hu ∂x = 0 ∂hu ∂t + ∂hu2 ∂x + 1 2g ∂h2 ∂x = −gh∂b ∂x . (3.1) The equation (3.1), can be written in a more compact way as: ∂U ∂t + ∂F(U) ∂x = S(U)∂b ∂x (3.2)

When applying the detailled Roe scheme (1.7) to (3.2), we write Un+1

i

=Un

i +∆x

∆t

• Fi−1/2 − Fi+1/2
• +∆x

∆t

• P+

i−1/2Si−1/2(bi − bi−1) + P− i+1/2Si+1/2(bi+1 − bi)

• ,

where P±

i+1/2 = 1

2

• I ± sign(Ai+1/2)
• and

Si+2 =    g hi+1 + hi 2    .

Numerical Flux

The flux associated to the scheme can also be written using the sign matrix function of the Roe matrix when using the fact that |A| = A × sign(A). So we write Fi+1/2 = 1 2

• F(Un

i+1) + F(Un i )

• − 1

2sign(Ai+1/2)Ai+1/2

• Un

i+1 − Un i

• .

Taking into account the Roe requirement we get Fi+1/2 = 1 2

• F(Un

i+1) + F(Un i )

• − 1

2sign(Ai+1/2)

• F(Un

i+1) − F(Un i )

• .

The Scheme

Defining the notation F s

i+1/2 =1

2

• F(Un

i+1) + F(Un i )

• −1

2sign(Ai+1/2)

• F(Un

i+1) − F(Un i ) − Si+1/2(bi+1 − bi)

• .

(3.4) Finally, we rewrite the relation (3.3) as Un+1

i

=Un

i + ∆t

∆x

• F s

i−1/2 − F s i+1/2

• + ∆t

2∆x

• Si−1/2(bi − bi−1) + Si+1/2(bi+1 − bi)
• (3.5)

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

1

First Order Roe’s scheme

2

Matrix Sign Function

3

Saint Venant Equations

4

Saint Venant model coupled with sediment transport equation

5

A Two-Layer model coupled with sediment transport equation

6

Numerical Result

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

We add to the previous Saint Venant Equation a morphodynamical model governing the bed evolution defined by (1 − p)∂b ∂t + ∂Qs ∂x = 0. (4.1) For this solid transport, we consider the Grass formula [1] given by Qs = Ag q h

• q

h

• m−1

, 1 ≤ m ≤ 4.

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

The coupled model is ∂h ∂t + ∂q ∂x = 0 ∂q ∂t + ∂ ∂x q2 h

• + 1

2g ∂h2 ∂x + gh∂b ∂x = 0 ∂b ∂t + Ag 1 − p ∂ ∂x q3 h3

• = 0

(4.2) To close this set of equations, suitable initial and boundary conditions will be considered.

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Governing Equation

We define the following notation W = (h, q, b)t, F(W) =           q q2 h + 1 2gh2 Ag 1 − p q3 h3           and B(W) =       −gh       . As previously, we rewrite the system (4.2) in the form ∂W ∂t + ∂F(W) ∂x = B(W)∂W ∂x (4.3)

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

And finally ∂W ∂t + C(W)∂W ∂x = 0 (4.4) where C(W) = A(W) − B(W) and the Jacobian is A(W) =           1 gh − q2 h2 2q h − 3Ag 1 − p q3 h4 3Ag 1 − p q2 h3           . (4.5)

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Roe Scheme for Sediment Transport Flow

The new expression of the flux associated to the Roe scheme can be written as Fi+1/2 =1 2

• Fi+1 + Fi
• −1

2sign(Ci+1/2)

• Fi+1 − Fi − BΦ

i+1/2

• W n

i+1 − W n i

• .

As usual, we write the numerical scheme under the following form: W n+1

i

=W n

i + ∆t

∆x

• Fi−1/2 − Fi+1/2
• + ∆t

2∆x

• Bi−1/2
• W n

i − W n i−1

• + Bi+1/2
• W n

i+1 − W n i

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

1

First Order Roe’s scheme

2

Matrix Sign Function

3

Saint Venant Equations

4

Saint Venant model coupled with sediment transport equation

5

A Two-Layer model coupled with sediment transport equation

6

Numerical Result

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Notations

h1 h2 b u1 u2 Topography

First Layer Second Layer

ρ1 ρ2

Figure: Two-Layer Shallow water flows with variable topography

Governing Equation

The set of equation is ∂h1 ∂t + ∂q1 ∂x = 0 ∂q1 ∂t + ∂ ∂x q2

1

h1

• + 1

2g ∂h2

1

∂x + gh1 ∂b ∂x + gh1 ∂h2 ∂x = 0 ∂h2 ∂t + ∂q2 ∂x = 0 ∂q2 ∂t + ∂ ∂x q2

2

h2

• + 1

2g ∂h2

2

∂x + gh2 ∂b ∂x + ρ1 ρ2 gh2 ∂h1 ∂x = 0 ∂b ∂t + Ag 1 − p ∂ ∂x q3

2

h3

2

• = 0

(5.1)

From (5.1), we define W = (h1 q1 h2 q2 b)t We could write then ∂W ∂t + ∂F(W) ∂x = B(W)∂W ∂x (5.2) When considering the Jacobian matrix A from (5.2) we get ∂W ∂t + M(W)∂W ∂x = 0 (5.3) where M(W) = A(W) − B(W).

The Scheme

For all calculations done the numerical flux with the sign matrix writes as follows Fi+1/2 =1 2

• Fi+1 + Fi
• −1

2sign(Mi+1/2)

• Fi+1 − Fi − BΦ

i+1/2

• W n

i+1 − W n i

• .

(5.4) Thereafter, we write the numerical Roe scheme for solving (5.1) W n+1

i

= W n

i + ∆t

∆x

• Fi−1/2 − Fi+1/2
• + ∆t

2∆x

• Bi−1/2
• W n

i − W n i−1

• + Bi+1/2
• W n

i+1 − W n i

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

1

First Order Roe’s scheme

2

Matrix Sign Function

3

Saint Venant Equations

4

Saint Venant model coupled with sediment transport equation

5

A Two-Layer model coupled with sediment transport equation

6

Numerical Result

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Test 1. Comparison with Exact Solution

In order to test the performance of the rewriting Roe scheme proposed here, we compare the numerical result with an asymptotic and analytical solution obtained by Hudson and Sweby in [4], for Grass model when Ag is smaller than 10−2. The test is performed in a computational domain whose length is L = 1000 m, dsicretized with 500 nodes uniformally. The CFL condition is set to 0.8. The sediment porosity p and the constant Ag of Grass model [1] are set to 0.4 and 0.001

• respectively. The initial conditions are

b(x, 0) =      0.1 + sin2 π(x − 300) 200

• if

300 ≤ x ≤ 500, 0.1

• therwise

h(x, 0) = 10 − b(x, 0), q(x, 0) = 10. (6.1)

Test 1. Comparison with Exact Solution

100 200 300 400 500 600 700 800 900 1000 0.05 0.1 0.15 0.2 0.25 0.3 Abscisses x[m] Sediment Layer thickness [m]

• Rew. Roe

Exact Solution

Figure: Comparison with Exact Solution

Test 2. Control of a channel modeled by Saint-Venant-Exner Equations

Our simulation are based on the nonlinear Saint Venant Equation coupled with sediment transport equation model. We test i n the sequel a boundary control problem. For a given steady state (¯ h, ¯ u, ¯ b) as a set point such that, there exist α : ¯ u

• g¯

h ≤ α < 1. The system (4.2) converges to the desired set point under a stabilizing boundary control law. These latter feedbacks are given by q0(t) = −gαh0(t)h0(t) − ¯ h ¯ u + h0(t)¯ u qL(t) = gαhL(t)hL(t) + bL(t) − ¯ h − ¯ b ¯ u + hL(t)¯ u. (6.2)

Test 2. Control of a channel modeled by Saint-Venant-Exner Equations

[]

• Bathym. b

[]

• Bathym. b

[]

[]

Figure: Profil of the solution at T = 2500.

Test 3. Internal dam break

Abscisse x[m] Elevation [m]

Abscisse x[m] Elevation [m]

Initial Conditions and Solution at T=25 of the internal dam break problem

Test 4. Internal dam break with sediment transport

We consider the following initial conditions b(x, 0) = 0.5

• 1 + 0.75 exp
• −
• x−50

2

10

• h1(x, 0) =

   1 if x ≥ 50, 2 if x < 50, h2(x, 0) =    2 − b(x, 0) if x ≥ 50, 1 − b(x, 0) if x < 50, q1(x, 0) = q2(x, 0) = 0. (6.3)

First Order Roe’s scheme Matrix Sign Function Saint Venant Equations Saint Venant model coupled with sediment transport equation A Two-Layer model coupled with sediment transport equation Numerical Result

Numerical Result

To approximate the solution of (5.1) with these initial data, we introduce as usual as space discretization with ∆x = 0.5 where the CFL condition is set to 0.9. The sediment porosity coefficient is fised to 0.4. The following figures 5 and 6 are depicted at T = 12 when the ratio r takes values in the set {0, 0.25, 0.5, 0.75, 0.9, 0.998} with Ag = 0.1 and Ag = 0.001

Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

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[]

[]

[]

Figure: Profil of the solution with r=0, 0.25and Ag = 0.1.

[]

[]

[]

[]

Figure: Profil of the solution with r=0.5, 0.75 and Ag = 0.1.

[]

[]

[]

[]

Figure: Profil of the solution with r=0.9, 0,998 and Ag = 0.1.

[]

[]

[]

[]

Figure: Profil of the solution with r=0, 0.25 and Ag = 0.001.

[]

[]

[]

[]

Figure: Profil of the solution with r=0.5, 0.75 and Ag = 0.001

[]

[]

[]

[]

Figure: Profil of the solution with r= 0.9, 0,998 and Ag = 0.001

A few References

A.J. GRASS, Sediment transport by waves and currents, SERC London, Cent. Mar. Technol. Report No.: FL29, 1981. CASTRO M.J. D´

IAZ AND E.D. FERN´ ANDEZ-NIETO AND

A.M. FERREIRO, Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods, Computers Fluids, 37, 3, pp. 299 - 316, 2008. CASTRO, MANUEL AND MAC´

IAS, JORGE AND PAR´ ES,

CARLOS, A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system, Mathematical Modelling and Numerical Analysis,35, 2001, 1, pp. 107–127. CASTRO M.J. AND A.M. FERREIRO FERREIRO AND J.A. GARC´

IA-RODR´ IGUEZ AND J.M. GONZ´ ALEZ-VIDA AND J.

MAC´

IAS AND C. PAR´ ES AND M. ELENA

V ´

• C

´ ”, The numerical treatment of wet/dry

Dal Maso, G. Lefloch, P . Murat F ., Definition and weak stability of a non-conservative product, J. Maths. Pures et Appliqu´ es, Vol. 74, Num. 6, pp. 483 − −548, 1995.

• E. MEYER-PETER AND R. MULLER, Formula for bed-load

transport, in Rep. 2nd Meeting Int. Assoc. Hydraul. Struct.

• Res. Stockholm (1948).

Par´ es, Carlos and Castro, Manuel, On the well-balance property of Roe’s method for nonconservative hyperbolic

• systems. Applications to shallow-water systems,

Mathematical Modelling and Numerical Analysis, 38, 2004, 5, pp. 821–852.

• J. Hudson, P

. K. Sweby, Formulations for numerically approximating hyperbolic systems governing sediment transport, Journal of Scientific Computing, 2003, Vol.19. ROE P. L., Approximate Riemannn solvers, parameter vectors and difference schemes, J. Comput. Phys. 43 (1981 ).

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