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 First Order Roe’s scheme 1 Matrix Sign Function 2 Saint Venant Equations 3 Saint Venant model coupled with sediment transport 4 equation A Two-Layer model coupled with sediment transport 5 equation Numerical Result 6

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 First Order Roe’s scheme 1 Matrix Sign Function 2 Saint Venant Equations 3 Saint Venant model coupled with sediment transport 4 equation A Two-Layer model coupled with sediment transport 5 equation Numerical Result 6 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 ) = B ( U ) ∂ U ∂ x + S ( U ) ∂σ ∂ x , (1.1) ∂ x where the unknown U ( x , t ) takes values an open convex set D of R N , F is a regular function from D to R N and σ ( x ) a known function from R N to R N . Considering the trivial equation, ∂ t σ = 0, system (1.1) can be presented in the form ∂ W ∂ t + A ( W ) ∂ W = 0 , (1.2) ∂ x

Roe Scheme when considering the auxiliar variable W = ( U , σ ) . The matrix A belong to M N + 1 ( R ) and can be written as � A ( W ) � − S ( W ) A ( W ) = 0 0 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 ⊂ R N is a locally lipschitz map Φ : [ 0 , 1 ] × Ω × Ω − → Ω such that: Φ( 0 ; W L , W R ) = W L , Φ( 1 ; W L , W R ) = W R for any W L , W R in For every bounded set O of Ω there exist k such that � ∂ Φ � � � ∂ s ( s ; W L , W R ) � ≤ k | W L − W R | � � � for any W L , W R in Ω and s ∈ [ 0 , 1 ] . For every bounded set O of Ω there exists K such that 2 � � ∂ Φ R ) − ∂ Φ � � � � � W ( i ) − W ( i ) ∂ s ( s ; W 1 L , W 1 ∂ s ( s ; W 2 L , W 2 � � � R ) � ≤ K � � � � L R � � i = 1 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 A Ψ : Ω × Ω − → M N ( R ) is called Roe linerizarion if it satisfies: for any W L , W R ∈ Ω , A Ψ ( W L , W R ) has N real distinct eigenvalues; for all W ∈ Ω , A Ψ ( W , W ) = A Ψ ( W ); for any W L , W R ∈ Ω : � 1 � ∂ Ψ � A Ψ ( W L , W R ) . ( W L − W R ) = A Ψ( s ; W L , W R ) ∂ s ( s ; W L , W R ) ds 0 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: � x i + 1 / 2 1 W n W ( x , t n ) dx . ≈ i ∆ x x i − 1 / 2 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 A i + 1 / 2 = A Ψ ( W i , W i + 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: � x i + 1 / 2 1 W n W ( x , t n ) dx . ≈ i ∆ x x i − 1 / 2 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 A i + 1 / 2 = A Ψ ( W i , W i + 1 ) Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Roe matrix We also define the matrix ( λ i + 1 / 2 ) ±   0 1 ... A ±  K − 1 i + 1 / 2 = K i + 1 / 2 i + 1 / 2 (1.4)    ( λ i + 1 / 2 ) ± 0 N (1.5) and � = A + � � i + 1 / 2 − A − � A i + 1 / 2 i + 1 / 2 .

Roe Scheme We show that the approximation at time t n + 1 can be obtained by the formula under the hypothesis: x i − 1 / 2 + λ i − 1 / 2 ∆ t ≤ x i ≤ x i + 1 / 2 + λ i + 1 / 2 ∆ t . 1 N i − ∆ t � � W n + 1 = W n A + i − 1 / 2 . ( W n i − W n i − 1 ) − A + i + 1 / 2 . ( W n i + 1 − W n i ) , (1.6) i ∆ x 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: i + ∆ t � � U n + 1 = U n F i − 1 / 2 − F i + 1 / 2 i ∆ x + ∆ t � � B i − 1 / 2 ( U n i − U n i − 1 ) + B i + 1 / 2 ( U n i + 1 − U n i ) 2 ∆ x + ∆ t � � P + i − 1 / 2 S i − 1 / 2 ( σ i − σ i − 1 ) − P − i + 1 / 2 S i + 1 / 2 ( σ i + 1 − σ i ) . ∆ x where the associated numerical flux is written F i + 1 / 2 = 1 − 1 � � � � F ( U n i + 1 ) + F ( U n U n i + 1 − U n � � i ) � A i + 1 / 2 (1.8) � i 2 2 and i + 1 / 2 = 1 � � P ± � A i + 1 / 2 � � I ± � A i + 1 / 2 . 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 First Order Roe’s scheme 1 Matrix Sign Function 2 Saint Venant Equations 3 Saint Venant model coupled with sediment transport 4 equation A Two-Layer model coupled with sediment transport 5 equation Numerical Result 6 Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

Matrix Sign Function We consider a matrix A ∈ M ( R n × n ) that admits n eigenvalues. A matrix sign function sign ( A ) is defined by | A | = A × sign ( A ) ,   sign ( λ 1 ) 0 ...  K − 1 , sign ( A ) = K (2.1)    0 sign ( λ n ) where K is a non singular matrix and  1 if λ i > 0  sign ( λ i ) = 0 if λ i = 0 − 1 if λ i < 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 x 0 = x ∈ [ − S , S ] (2.2) x k + 1 = P ( x k )

Iterative Polynomial y y=P(x) y=x −S X S Figure: Iterative Polynomial P ( x ) = x − a ( x − S )( x − S ) . (2.3)

Matrix Sign Function The iteration procedure A 0 = A (2.4) A k + 1 = P ( A k ) converges to a matrix A ∗ who can be decomposed in the form   S × sign ( λ 1 ) 0 A ∗ = K ...  K − 1 , (2.5)    0 S × sign ( λ n ) (2.6) sign ( A ) = A ∗ (2.7) S

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 First Order Roe’s scheme 1 Matrix Sign Function 2 Saint Venant Equations 3 Saint Venant model coupled with sediment transport 4 equation A Two-Layer model coupled with sediment transport 5 equation Numerical Result 6 Ababacar DIAGNE, Pr. Enrique F. Nieto New Improvement of Roe’s Scheme

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

When applying the detailled Roe scheme (1.7) to (3.2), we write i +∆ x � � U n + 1 = U n F i − 1 / 2 − F i + 1 / 2 i ∆ t +∆ x � � P + i − 1 / 2 S i − 1 / 2 ( b i − b i − 1 ) + P − i + 1 / 2 S i + 1 / 2 ( b i + 1 − b i ) , ∆ t where 0   i + 1 / 2 = 1 � � P ± I ± sign ( A i + 1 / 2 ) and S i + 2 =  .   2 g h i + 1 + h i  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 F i + 1 / 2 = 1 − 1 � � � � F ( U n i + 1 ) + F ( U n U n i + 1 − U n i ) 2 sign ( A i + 1 / 2 ) A i + 1 / 2 . i 2 Taking into account the Roe requirement we get F i + 1 / 2 = 1 − 1 � � � � F ( U n i + 1 ) + F ( U n F ( U n i + 1 ) − F ( U n i ) 2 sign ( A i + 1 / 2 ) i ) . 2