Introduction WAF schemes Numerical tests Conclusions Two-waves PVM-WAF method for non-conservative systems andez Nieto 2 , Gladys Narbona Reina 2 and ıaz 1 , E.D Fern´ Manuel J. Castro D´ on 1 Marc de la Asunci´ 1 Departamento de An´ alisis Matem´ atico University of M´ alaga (Spain) 2 Departmento de Matem´ atica Aplicada University of Sevilla (Spain) Hyp 2012, Padova 25-29 June 2012
Introduction WAF schemes Numerical tests Conclusions Outline Introduction 1 Path-conservative Roe-based schemes PVM methods WAF schemes 2 PVM2U-WAF method Numerical tests 3 Conclusions 4
Introduction WAF schemes Numerical tests Conclusions Model problem Let us consider the system w t + F ( w ) x + B ( w ) · w x = G ( w ) H x , (1) where w ( x , t ) takes values on an open convex set O ⊂ R N , F is a regular function from O to R N , B is a regular matrix function from O to M N × N ( R ) , G is a function from O to R N , and H is a function from R to R . By adding to (1) the equation H t = 0, the system (1) can be rewritten under the form W t + A ( W ) · W x = 0 , (2) where W is the augmented vector � w � ∈ Ω = O × R ⊂ R N + 1 W = H and
Introduction WAF schemes Numerical tests Conclusions Model problem Let us consider the system w t + F ( w ) x + B ( w ) · w x = G ( w ) H x , (1) where w ( x , t ) takes values on an open convex set O ⊂ R N , F is a regular function from O to R N , B is a regular matrix function from O to M N × N ( R ) , G is a function from O to R N , and H is a function from R to R . By adding to (1) the equation H t = 0, the system (1) can be rewritten under the form W t + A ( W ) · W x = 0 , (2) where A ( W ) is the matrix whose block structure is given by: � A ( w ) − G ( w ) � A ( W ) = , 0 0 where being J ( w ) = ∂ F A ( w ) = J ( w ) + B ( w ) , ∂ w ( w ) .
Introduction WAF schemes Numerical tests Conclusions Difficulties Main difficulties Non conservative products A ( W ) · W x . Solutions may develop discontinuities and the concept of weak solution in the sense of distributions cannot be used. The theory introduced by DLM 1995 is used here to define the weak solutions of the system. This theory allows one to give a sense to the non conservative terms of the system as Borel measures provided a prescribed family of paths in the space of states. Derivation of numerical schemes for non-conservative systems: path-conservative numerical schemes (Par´ es 2006). The eigenstructure of systems like bilayer Shallow-Water system or two-phase flow model of Pitman Le are not explicitly known: PVM and/or WAF schemes.
Introduction WAF schemes Numerical tests Conclusions PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: i − ∆ t w n + 1 = w n D + i − 1 / 2 + D − � � , (3) i i + 1 / 2 ∆ x where ∆ x and ∆ t are, for simplicity, assumed to be constant; w n i is the approximation provided by the numerical scheme of the cell average of the exact solution at the i -th cell, I i = [ x i − 1 / 2 , x i + 1 / 2 ] at the n -th time level t n = n ∆ t . H i is the cell average of the function H ( x ) .
Introduction WAF schemes Numerical tests Conclusions PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: i − ∆ t w n + 1 = w n D + i − 1 / 2 + D − � � , (3) i i + 1 / 2 ∆ x 1 D ± � i + 1 / 2 = F ( w i + 1 ) − F ( w i ) + B i + 1 / 2 · ( w i + 1 − w i ) − G i + 1 / 2 ( H i + 1 − H i ) 2 � ± Q i + 1 / 2 ( w i + 1 − w i − A − 1 i + 1 / 2 G i + 1 / 2 ( H i + 1 − H i )) , (4) where A i + 1 / 2 = J i + 1 / 2 + B i + 1 / 2 . Here, J i + 1 / 2 is a Roe matrix of the Jacobian of the flux F in the usual sense: J i + 1 / 2 · ( w i + 1 − w i ) = F ( w i + 1 ) − F ( w i ); � 1 B (Φ w ( s ; W i , W i + 1 )) ∂ Φ w B i + 1 / 2 · ( w i + 1 − w i ) = ∂ s ( s ; W i , W i + 1 ) ds ; 0 � 1 G (Φ w ( s ; W i , W i + 1 )) ∂ Φ H G i + 1 / 2 ( H i + 1 − H i ) = ∂ s ( s ; W i , W i + 1 ) ds ; 0 Q i + 1 / 2 is a numerical viscosity matrix.
Introduction WAF schemes Numerical tests Conclusions PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: i − ∆ t w n + 1 = w n D + i − 1 / 2 + D − � � , (3) i i + 1 / 2 ∆ x 1 D ± � i + 1 / 2 = F ( w i + 1 ) − F ( w i ) + B i + 1 / 2 ( w i + 1 − w i ) − G i + 1 / 2 ( H i + 1 − H i ) 2 � ± Q i + 1 / 2 ( w i + 1 − w i − A − 1 i + 1 / 2 G i + 1 / 2 ( H i + 1 − H i )) , (4) Conservative systems If the system is conservative and F i + 1 / 2 = F ( w i ) + F ( w i + 1 ) − 1 2 Q i + 1 / 2 ( w i + 1 − w i ) 2 is a conservative flux, where Q i + 1 / 2 is defined in terms of J i + 1 / 2 , then D − D + i + 1 / 2 = F i + 1 / 2 − F ( w i ) i + 1 / 2 = F ( w i + 1 ) − F i + 1 / 2 .
Introduction WAF schemes Numerical tests Conclusions PC-Roe-based schemes I Let us consider path-conservative numerical schemes that can be written as follows: i − ∆ t w n + 1 = w n D + i − 1 / 2 + D − � � , (3) i i + 1 / 2 ∆ x 1 D ± � i + 1 / 2 = F ( w i + 1 ) − F ( w i ) + B i + 1 / 2 ( w i + 1 − w i ) − G i + 1 / 2 ( H i + 1 − H i ) 2 � ± Q i + 1 / 2 ( w i + 1 − w i − A − 1 i + 1 / 2 G i + 1 / 2 ( H i + 1 − H i )) , (4) Different numerical schemes can be obtained for different definitions of Q i + 1 / 2
Introduction WAF schemes Numerical tests Conclusions PC-Roe-based schemes II Roe scheme corresponds to the choice Q i + 1 / 2 = | A i + 1 / 2 | , Lax-Friedrichs scheme: Q i + 1 / 2 = ∆ x ∆ t Id , being Id the identity matrix. Lax-Wendroff scheme: Q i + 1 / 2 = ∆ t ∆ xA 2 i + 1 / 2 , FORCE and GFORCE schemes: Q i + 1 / 2 = ( 1 − ω )∆ x ∆ t Id + ω ∆ t ∆ xA 2 i + 1 / 2 , 1 with ω = 0 . 5 and ω = 1 + α , respectively, being α the CFL parameter.
Introduction WAF schemes Numerical tests Conclusions PVM methods We propose a class of finite volume methods defined by Q i + 1 / 2 = P l ( A i + 1 / 2 ) , being P l ( x ) a polinomial of degree l , l � α i + 1 / 2 x j , P l ( x ) = j j = 0 and A i + 1 / 2 a Roe matrix. That is, Q i + 1 / 2 can be seen as a Polynomial Viscosity Matrix (PVM). See also: P. Degond, P.F. Peyrard, G. Russo, Ph. Villedieu. Polynomial upwind schemes for hyperbolic systems. C. R. Acad. Sci. Paris 1 328, 479-483, 1999.
Introduction WAF schemes Numerical tests Conclusions PVM methods We propose a class of finite volume methods defined by Q i + 1 / 2 = P l ( A i + 1 / 2 ) , being P l ( x ) a polinomial of degree l , l � α i + 1 / 2 x j , P l ( x ) = j j = 0 and A i + 1 / 2 a Roe matrix. That is, Q i + 1 / 2 can be seen as a Polynomial Viscosity Matrix (PVM). Q i + 1 / 2 has the same eigenvectors than A i + 1 / 2 and if λ i + 1 / 2 is an eigenvalue of A i + 1 / 2 , then P l ( λ i + 1 / 2 ) is an eigenvalue of Q i + 1 / 2 .
Introduction WAF schemes Numerical tests Conclusions PVM methods We propose a class of finite volume methods defined by Q i + 1 / 2 = P l ( A i + 1 / 2 ) , being P l ( x ) a polinomial of degree l , l � α i + 1 / 2 x j , P l ( x ) = j j = 0 and A i + 1 / 2 a Roe matrix. That is, Q i + 1 / 2 can be seen as a Polynomial Viscosity Matrix (PVM). Some well-known solvers as Lax-Friedrichs, Rusanov, FORCE/GFORCE, HLL, Roe, Lax-Wendroff, ... can be recovered as PVM methods
Introduction WAF schemes Numerical tests Conclusions PVM-1U( S L , S R ) or HLL method P 1 ( x ) = α 0 + α 1 x such as P 1 ( S L ) = | S L | , P 1 ( S R ) = | S R | . Q i + 1 / 2 = α 0 Id + α 1 A i + 1 / 2 PVM − 1U(S L ,S R ) ... λ N S L λ 1 λ 2 λ j S R
Introduction WAF schemes Numerical tests Conclusions PVM-1U( S L , S R ) or HLL method The usual HLL scheme coincides with PVM-1U( S L , S R ) in the case of conservative systems. Let us suppose that the system is conservative. Then, the conservative flux associated to PVM-1U( S L , S R ) is F i + 1 / 2 = D − i + 1 / 2 + F ( w i ) . Taking into account that α 0 = S R | S L | − S L | S R | α 1 = | S R | − | S L | , S R − S L , S R − S L then F ( w i )( S R + | S R | − S L − | S L | ) + F ( w i + 1 )( S R − | S R | − S L + | S L | ) F i + 1 / 2 = 2 S R − 2 S L − ( S R | S L | − S L | S R | )( w i + 1 − w i ) 2 S R − 2 S L S + R F ( w i ) − S − L F ( w i + 1 ) + ( S + R S − L )( w i + 1 − w i ) = S + R − S − L which is a compact definition of the HLL flux, being S + R = max ( S R , 0 ) and S − L = min ( S L , 0 ) .
Introduction WAF schemes Numerical tests Conclusions PVM-2U( S L , S R ) method P 2 ( x ) = α 0 + α 1 x + α 2 x 2 , such as P 2 ( S m ) = | S m | , P 2 ( S M ) = | S M | , P ′ 2 ( S M ) = sgn ( S M ) , where � S L � S R if | S L | ≥ | S R | , if | S L | ≥ | S R | , S M = S m = if | S L | < | S R | . if | S L | < | S R | . S R S L PVM − 2U(S L ,S R ) ... λ N S L λ 1 λ 2 λ j S R
Introduction WAF schemes Numerical tests Conclusions WAF method Let us consider the following Riemann problem: ∂ w ∂ t + ∂ F ( w ) = 0 ; ∂ x � w i x < 0 ; w ( x , 0 ) = w i + 1 x > 0 . We denote by S i for i = 1 , · · · , N some approximation of the characteristic velocities. t S 1 S 2 ! t " 2 ! t / 2 " " 3 1 x 0 #! ! x /2 x /2 � ∆ x / 2 F ( w ( x , ∆ t 1 WAF F i + 1 / 2 = 2 )) dx . ∆ x − ∆ x / 2
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