Central-Upwind Schemes for Shallow Water Models Alexander Kurganov - PowerPoint PPT Presentation

Central-Upwind Schemes for Shallow Water Models Alexander Kurganov Southern University of Science and Technology, China and Tulane University, USA www.math.tulane.edu/ ∼ kurganov Supported by NSFC and NSF

Finite-Volume Methods 1-D System: U t + F ( U ) x = 0 � 1 U j ( t ) ≈ U ( x, t ) d x : cell averages over C j := ( x j − 1 2 , x j + 1 2 ) ∆ x C j This solution is approximated by a piecewise linear (conservative, second-order accurate, non-oscillatory) reconstruction: � U ( x ) = U j + ( U x ) j ( x − x j ) for x ∈ C j 2

x x x x j−1 j j+1 j+2 For example,    θ U j − U j − 1 , U j +1 − U j − 1 , θ U j +1 − U j  ( U x ) j = minmod θ ∈ [1 , 2] ∆ x 2∆ x ∆ x where the minmod function is defined as:  min j { z j } , if z j > 0 ∀ j   minmod( z 1 , z 2 , ... ) := max j { z j } , if z j < 0 ∀ j   0 , otherwise 3

Godunov-type upwind schemes are designed by integrating U t + f ( U ) x = 0 2 ] × [ t n , t n +1 ] over the space-time control volumes [ x j − 1 2 , x j + 1 t n+1 t n x x x x x j−1 j−1/2 j j+1/2 j+1 4

t n+1 t n x x x x x j−1 j−1/2 j j+1/2 j+1 t n +1 � � � 1 U n +1 = U n j − f ( U ( x j + 1 2 , t )) − f ( U ( x j − 1 2 , t )) d t j ∆ x t n In order to evaluate the flux integrals on the RHS, one has to (approximately) solve the generalized Riemann problem. This may be hard or even impossible... 5

t t n+1 t n x x x x x x j−1 j−1/2 j j+1/2 j+1 n u(x,t ) x x x x x x j−1 j−1/2 j j+1/2 j+1 6

✄☎ ✄ ☎ ✆ ✆ ✝ ✝ Nessyahu-Tadmor Scheme The Nessyahu-Tadmor [Nessyahu, Tadmor; 1990] scheme is a central Godunov-type scheme. It is designed by integrating U t + f ( U ) x = 0 over the different set of staggered space-time control volumes [ x j , x j +1 ] × [ t n , t n +1 ] containing the Riemann fans �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ t n+1 ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� t n x x x j j+1/2 j+1 7

☎ ✆ ✄☎ ✄ ✝ ✝ ✆ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� t n+1 �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁�✁� t n x x x j j+1/2 j+1 x j +1 t n +1 � � � � 1 1 U n +1 U n ( x ) d x − � 2 = f ( U ( x j +1 , t )) − f ( U ( x j , t )) d t j + 1 ∆ x ∆ x x j t n Due to the finite speed of propagation, this can be reduced to: 2 = U n j + U n � � � � n + 1 n + 1 + ∆ x − ∆ t U n +1 j +1 ( U x ) n j − ( U x ) n 2 2 f ( U j +1 ) − f ( U ) j + 1 j +1 j 2 8 ∆ x

Values of U at t = t n + 1 2 are approximated using the Taylor expansion: n + 1 U n ( x j ) + ∆ t 2 U t ( x j , t n ) ≈ � 2 U j U n ( x ) = U n U n ( x j ) = U n • � j + ( U x ) n ⇒ � j ( x − x j ) = j U t ( x j , t n ) = − f ( U n • j ) x The space derivatives f x are computed using the (minmod) limiter:   θ f ( U n j ) − f ( U n , f ( U n j +1 ) − f ( U n j − 1 ) j − 1 ) f ( U n j ) x = minmod , ∆ x 2∆ x  θ f ( U n j +1 ) − f ( U n j )  ∆ x 9

Higher-Order and Multidimensional Staggered Central Schemes [Arminjon, Viallon, Madrane; 1997] [Jiang, Tadmor; 1998] [Liu, Tadmor; 1998] [Bianco, Puppo, Russo; 1999] [Levy, Puppo, Russo; 1999, 2000, 2002] [Lie, Noelle; 2000] 10

Central-Upwind Schemes Goal: to reduce numerical dissipation of central schemes Example — Numerical Dissipation of the Staggered LxF Scheme u n j +1 + u n � � − ∆ t j u n +1 f ( u n j +1 ) − f ( u n = j ) j + 1 2 ∆ x 2 u n j +1 − 2 u n + u n � � j + 1 j + ∆ t u n +1 − u n f ( u n j +1 ) − f ( u n 2 j ) = j + 1 j + 1 ∆ x 2 2 2 u n +1 − u n u n j +1 − 2 u n + u n f ( u n j +1 ) − f ( u n j ) = (∆ x ) 2 j + 1 j + 1 j + 1 j 2 2 2 + · (∆ x/ 2) 2 ∆ t ∆ x 8∆ t • As ∆ t decreases, the numerical dissipation increases • As ∆ t ∼ (∆ x ) 2 , the LxF scheme is inconsistent • As ∆ t → 0, the numerical dissipation blows up 11

Central-Upwind Schemes Godunov-type central schemes with a built-in upwind nature [Kurganov, Tadmor; 2000] [Kurganov, Petrova; 2001] [Kurganov, Noelle, Petrova; 2001] [Kurganov, Tadmor; 2002] [Kurganov, Petrova; 2005] [Kurganov, Lin; 2007] [Kurganov, Prugger, Wu; 2017] 12

x x x x j−1 j j+1 j+2 U n ( x ) = U n j + ( U x ) n � j ( x − x j ) for x ∈ C j j + ∆ x U ( x, t n ) = U n U − 2 ( U x ) n � := lim j j + 1 x → x j +1 − 2 2 j +1 − ∆ x U ( x, t n ) = U n U + 2 ( U x ) n � := lim j +1 j + 1 x → x j +1 + 2 2

x x x x j−1 j j+1 j+2 The discontinuities appearing at the reconstruction step at the interface points { x j + 1 2 } propagate at finite speeds estimated by: � � ∂ F � � ∂ F � � a + ∂ U ( U − ∂ U ( U + := max λ N ) , λ N ) , 0 j + 1 j + 1 j + 1 2 2 2 � � ∂ F � � ∂ F � � a − ∂ U ( U − ∂ U ( U + := min λ 1 ) , λ 1 ) , 0 j + 1 j + 1 j + 1 2 2 2 λ 1 < λ 2 < . . . < λ N : N eigenvalues of the Jacobian ∂ F ∂ U