Preliminaries PVM and RVM methods Approximate OS solvers Conclusions Approximate Osher-Solomon schemes for hyperbolic systems ıaz 1 , Jos´ e M. Gallardo 1 , Antonio Marquina 2 Manuel J. Castro D´ 1 Departament of Mathematical Analysis, Statistics, and Applied Mathematics University of M´ alaga (Spain) 2 Departament of Applied Mathematics University of Valencia (Spain) Granada, abril 2017 M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM and RVM methods Approximate OS solvers Conclusions Outline Preliminaries 1 PVM and RVM methods 2 PVM methods PVM Jacobian free methods RVM methods 3 Approximate OS solvers 4 Conclusions M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM and RVM methods Approximate OS solvers Conclusions Preliminaries Consider a hyperbolic system of conservation laws ∂ t U + ∂ x F ( U ) = 0 where U ( x, t ) takes values on an open convex set O ⊂ R N and F : O → R N is a smooth flux function. We are interested in the numerical solution of the Cauchy problem for the system by means of a finite volume method of the form � � i − ∆ t U n +1 = U n F n i +1 / 2 − F n i i − 1 / 2 ∆ x where U n i denotes the approximation to the average of the exact solution at the cell I i = [ x i − 1 / 2 , x i +1 / 2 ] at time t n = n ∆ t . We consider numerical fluxes that are defined as (we drop the dependence on time) F i + / 2 = F ( U i ) + F ( U i +1 ) − 1 2 Q i +1 / 2 ( U i +1 − U i ) 2 where Q i +1 / 2 is a numerical viscosity matrix . Different numerical methods can be designed depending on the choice of the viscosity matrix. M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM and RVM methods Approximate OS solvers Conclusions Preliminaries Examples: Roe: Q i +1 / 2 = | A i +1 / 2 | where A i +1 / 2 is a Roe matrix for the system. Lax-Friedrichs: Q i +1 / 2 = ∆ x ∆ t I being I the identity matrix. Lax-Wendroff: Q i +1 / 2 = ∆ t ∆ x A 2 i +1 / 2 FORCE and GFORCE: Q i +1 / 2 = (1 − ω ) ∆ x ∆ t Id + ω ∆ t ∆ x A 2 i +1 / 2 with ω = 1 1 2 and ω = 1+ CFL , respectively. M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM and RVM methods Approximate OS solvers Conclusions Preliminaries We propose a class of finite volume methods defined by Q i +1 / 2 = f ( A i +1 / 2 ) where f : R → R is a function and A i +1 / 2 is a Roe matrix or the Jacobian of the flux evaluated at some average state. Some properties of f : f ( x ) � 0 and smooth. f ( A i +1 / 2 ) should be easy to evaluate: no spectral decomposition of A i +1 / 2 needed. L ∞ linear stability: CFL ∆ x ∀ x ∈ [ λ i +1 / 2 , λ i +1 / 2 ∆ t � f ( x ) � | x | , ] , 1 N where λ i +1 / 2 are the eigenvalues of A i +1 / 2 . l f ( x ) should be as close as possible to | x | . M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods PVM methods One possible choice is to set f ( x ) = P d ( x ) , being P d ( x ) a polynomial of degree d . In this case Q i +1 / 2 = P d ( A i +1 / 2 ) that is, Q i +1 / 2 is a Polynomial Viscosity Matrix (PVM). [Castro-Fern´ andez Nieto, SIAM J. Sci. Comput. 34, 2012] PVM methods has been applied to multilayer shallow water equations or the two-phase flow model of Pitman-Le, for which the eigenstructures are not explicitely known . PVM methods can be extended to nonconservative hyperbolic systems , following the theory of path-conservative schemes ([Pares, SIAM J. Numer. Anal. 44, 2006]). Some well-known solvers as Lax-Friedrichs, Rusanov, FORCE/GFORCE, HLL, Roe, Lax-Wendroff, etc., can be recovered as PVM methods. In particular, this allows to build direct extensions of the mentioned solvers to the nonconservative case. M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods HLL Some examples: Degond et al. Lax-Friedrichs, modified Lax-Friedrichs and Rusanov (local LxF): S 0 ∈ { S LF , S mod P 0 ( x ) = S 0 , LF , S Rus } ∆ t and S Rus = max j | λ i +1 / 2 with S LF = ∆ x ∆ t , S mod = CFL ∆ x | . j LF HLL: P 1 ( x ) = α 0 + α 1 x λ 2 . . . . . .λ N S R S L λ 1 λ j such that P 1 ( S L ) = | S L | , P 1 ( S R ) = | S R | , S L and S R being approximations to the minimum and maximum wave speeds. FORCE: P 2 ( x ) = α 0 + α 2 x 2 such that P 2 ( S 0 ) = S 0 and P ′ 2 ( S 0 ) = 1 , with S 0 = S LF . The related solver proposed in [Degond et al., C.R. Acad. Sci. Paris S´ er. I 328, 1999] can be viewed as a PVM method based on a second order polynomial. The incomplete Riemann solver based on Krylov subspace approximations of | x | in [Torrilhon, SIAM J. Sci. Comput. 34, 2012] can also be interpreted as a PVM scheme. M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods PVM-Force type iterative method Consider the polynomials defined as follows P 2 n − 1 ( x ) − x 2 P 0 ( x ) = 1 , P n ( x ) = P n − 1 ( x ) − , n = 1 , 2 , . . . 2 Viscosity matrix: � � 1 Q i +1 / 2 = | λ max | P n | λ max | A i +1 / 2 ≈ | A i +1 / 2 | where λ max is the eigenvalue of A i +1 / 2 with maximum modulus. Observe that the viscosity matrix obtained with n = 1 is Q i +1 / 2 = λ max 1 A 2 Id + i +1 / 2 . 2 2 λ max M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 |x| 0.2 n=1 n=4 n=7 0.1 n=10 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure: PVM-Force type iterative method: P n ( x ) for n = 1 , 4 , 7 and 10 . M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods PVM-Chebyshev Chebyshev polynomials provide optimal uniform approximation to | x | in [ − 1 , 1] : � ∞ ( − 1) k +1 | x | = 2 4 π + (2 k − 1)(2 k + 1) T 2 k ( x ) , x ∈ [ − 1 , 1] , π k =1 where the Chebyshev polynomials of even degree T 2 k ( x ) are recursively defined as T 2 ( x ) = 2 x 2 − 1 , T 0 ( x ) = 1 , T 2 k ( x ) = 2 T 2 ( x ) T 2 k − 2 ( x ) − T 2 k − 4 ( x ) . Viscosity matrix: � � 1 Q i +1 / 2 = P 2 p ( A i +1 / 2 ) = | λ max | τ 2 p | λ max | A i +1 / 2 ≈ | A i +1 / 2 | where λ max is the eigenvalue of A i +1 / 2 with maximum modulus. [Castro, Gallardo, Marquina, J. Sci. Comput. 60, 2014] M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 | x | 0.2 0.2 τ 4 ( x ) | x | τ 6 ( x ) τ 8 ( x ) τ ε τ 8 ( x ) 8 ( x ) 0.0 0.0 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 Figure: Left: Chebyshev approximations τ 2 p ( x ) for p = 2 , 3 , 4 . Right: τ 8 ( x ) and τ ε 8 ( x ) . Notice that τ 2 p ( x ) do not satisfy the stability condition τ 2 p ( x ) � | x | . This drawback can be avoided by using τ ε 2 p ( x ) = τ 2 p ( x ) + ε such that τ ε 2 p ( x ) � | x | . M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods PVM-Sign approximations We could also consider methods based on the polynomial approximation of the sign function to define an approximation of | A i +1 / 2 | . Let us consider the Newton-Schulz iterative procedure to approximate the sign of x , x ∈ [ − 1 , 1] x n = x n − 1 � � 3 − x 2 x 0 = x, , n = 1 , 2 , 3 , . . . n − 1 2 then we could define � � 1 Q i +1 / 2 = A i +1 / 2 P n | λ max | A i +1 / 2 ≈ | A i +1 / 2 | where P n ( x ) is the polynomial defined by � � P n ( x ) = P n − 1 ( x ) 3 − P 2 P 0 ( x ) = x, n − 1 ( x ) , n = 1 , 2 , . . . 2 M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM methods 1 0.2 0.9 0.18 0.8 0.16 0.7 0.14 0.6 0.12 0.5 0.1 0.4 0.08 0.3 0.06 0.2 0.04 !x! !x! n=4 n=4 0.1 0.02 n=7 n=7 n=10 n=10 0 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 Figure: Left: PVM sign approximation for n = 4 , 7 and 10 . Right: PVM sign approximation Zoom at [ − 0 . 2 , 0 . 2] M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
Preliminaries PVM methods PVM and RVM methods PVM Jacobian free methods Approximate OS solvers RVM methods Conclusions PVM-Chebyshev Jacobian free method Chebyshev Jacobian free implementation Note that the viscosity matrix Q i +1 / 2 need not to be computed explicitly, but only the vector Q i +1 / 2 ∆ U , where ∆ U = U i +1 − U i : � � p � α j U [2 j ] Q i +1 / 2 ∆ U = | λ max | α 0 ∆ U + j =1 ( − 1) j +1 with α 0 = 2 π , α j = 4 (2 j − 1)(2 j +1) for j � 1 and π � � U [2 j ] = T 2 j | λ max | − 1 A i +1 / 2 ∆ U. M. J. Castro, J.M. Gallardo, A. Marquina Granada, 20 abril 2017
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