High-order well-balanced finite-volume schemes for eddy computations - PowerPoint PPT Presentation

High-order well-balanced finite-volume schemes for eddy computations in barostrophic jets. Algorithms and numerical comparisons Normann Pankratz IGPM — RWTH Aachen Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 1 / 38

Outline Well-balanced schemes 1 Application: Geostrophic Flow 2 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 2 / 38

Shallow Water Equations Shallow Water Equations with topography and coriolisforce:        0  h hu hv hu 2 + 1 2 gh 2 hu + + huv = − ghb x − fhv         hv 2 + 1 2 gh 2 hv huv − ghb y + fhu t x y h ( x , y , t ): waterheight b ( x ): topography hu ( x , y , t ): x -momentum g : gravitation constant hv ( x , y , t ): y -momentum f : coriolisforce constant Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 3 / 38

Well-Balanced Schemes First and second order accuracy: Bermudez, Vazquez 1994 Greenberg, LeRoux 1995 Gosse et al. 1998 LeVeque et al. 2000 Gallouet, Seguin 2000 Kurganov, Levy 2003 Klein et al. 2003 Theorem: Audusse, Bristeau, Bouchut, Klein, Perthame 2004 A suitable hydrostatic spatial reconstruction gives positivity of water height discrete entropy inequality (first order scheme) discrete hydrostatic balance ...and many others Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 4 / 38

Well-Balanced Schemes High-order accuracy: Vukovic, Sopta 2002 Xing, Shu 2004 Castro, Garllardo, Pares 2006 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 5 / 38

Well-Balanced Schemes High-order accuracy: Vukovic, Sopta 2002 Xing, Shu 2004 Castro, Garllardo, Pares 2006 Theorem: [NPPN] 2005 Extrapolation gives arbitrary order of accuracy discrete hydrostatic balance Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 5 / 38

Well-Balanced Schemes High-order accuracy: Vukovic, Sopta 2002 Xing, Shu 2004 Castro, Garllardo, Pares 2006 Theorem: [NPPN] 2005 Extrapolation gives arbitrary order of accuracy discrete hydrostatic balance [NPPN] S. Noelle, N. Pankratz, G. Puppo, J. Natvig. Well-balanced finite-volume schemes of arbitary order of accuracy for shallow water flows, J. Comput. Phys. 213 (2006), pp. 474–499. Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 5 / 38

High-Order Upwind Finite-Volumes Semidiscrete upwind finite-volume scheme + source terms ( U ij ) t + 1 2 ) + 1 ∆ x ( F i + 1 2 − F i − 1 ∆ y ( G j + 1 2 − G j − 1 2 ) = S ij numerical flux (LF, LLF, HLL, kinetic, . . . ) F i + 1 2 = F ( U i , r , U i +1 , l ) polynomial reconstruction for η = h + b , hu , and hv reconstruction of surface displacement h r ec = η r ec − b ( x , y ). S ij discretized with extrapolated well-balanced sourceterm Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 6 / 38

Convergence Smooth data, Xing, Shu (2005) : CFL= 0 . 5, g = 9 . 812 = 10 + e sin(2 π x ) cos(2 π y ) , b ( x , y ) = sin(2 π x ) + cos(2 π y ) , η ( x , y ) hu ( x , y ) = sin(cos(2 π x )) sin(2 π y ) , hv ( x , y ) = cos(2 π x ) cos(sin(2 π y )) . number h hu hv CFL L 1 error L 1 error L 1 error of points order order order 25 0.5 8.77E-03 3.42E-02 6.71E-02 50 0.5 1.10E-03 3.00 2.73E-03 3.65 9.40E-03 2.84 100 0.5 9.84E-05 3.48 1.56E-04 4.13 7.85E-04 3.58 200 0.5 4.91E-06 4.32 6.58E-06 4.57 3.93E-05 4.32 400 0.5 1.82E-07 4.76 2.41E-07 4.77 1.46E-06 4.75 800 0.5 6.06E-09 4.91 7.94E-09 4.92 4.90E-08 4.90 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 7 / 38

A small Perturbation of a Lake at Rest � 1 . 01 , x ∈ [0 . 05 , 0 . 15] initial data: η ( x , y ) = 1 , otherwise hu ( x , y ) = 0 , hv ( x , y ) = 0 0 . 8 exp( − 5( x − 0 . 9) 2 − 50( y − 0 . 5) 2 ) b ( x , y ) = 0.8 0.6 0.4 b 0.2 0 1 2 0.5 1 0 0 y x bottom topography Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 8 / 38

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 9 / 38

Outline Well-balanced schemes 1 Application: Geostrophic Flow 2 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 10 / 38

Ocean flows Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 11 / 38

Ocean flows Main Question: Is there need for modern high resolution schemes? Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 11 / 38

Ocean flows Main Question: Is there need for modern high resolution schemes? We consider geophysical flows a traditional central difference scheme (FD1),(FD2) high-order accurate upwind finite-volume scheme (FV4) Comparison of stability, accuracy, efficiency with J. Natvig, B. Gjevik, S. Noelle (2006) Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 11 / 38

Review Traditional Central Difference Scheme: von Neumann - Richtmayer (1949) used in Lagrangian gas dynamics staggered finite-differences second order accurate Problem: dispersive oscillations Solution: add artificial viscosity Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 12 / 38

Comparison: Convergence Rate Smooth data ( Xing, Shu ) and coriolis force parameter f = 10 . 0 FD1 FD2 FV4 L 1 -error L 1 -error L 1 -error N order order order 25 4.56E-02 3.27E-02 6.70E-03 50 1.69E-02 1.43 8.45E-03 1.95 8.46E-04 2.96 100 7.20E-03 1.23 2.10E-03 2.01 6.84E-05 3.63 200 3.35E-03 1.10 5.26E-04 2.00 3.06E-06 4.48 400 1.63E-03 1.04 1.32E-04 2.00 1.11E-07 4.79 800 8.02E-04 1.02 3.29E-05 2.00 3.66E-09 4.91 L 1 -errors in η and numerical order of accuracy at T = 0 . 05 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 13 / 38

Dispersive Oscillation FD2 FV4 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 14 / 38

Dispersive Oscillation FD2 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 14 / 38

Dispersive Oscillation FV4 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 14 / 38

Eddies in Doubly Periodic Domain 2 nd order staggered grid, day 0 Geostrophically balanced flow: potential vorticity q := ∇× u + f 3 h Initial data: 2 width of jet: 2 a = 1 potential vorticity: 1 q ( x , y , 0) = 0 x � � � q + Q sign(ˆ ¯ y )( a − � | ˆ y | − a � ) , | ˆ y | < 2 a , −1 . q , otherwise , ¯ −2 ˆ y = y − 0 . 1 sin(2 x ) + 0 . 1 sin(3 x ) −3 2 0 −2 y D. Dritschel, L. Polvani, A. Mohebalhojeh, The Contour-Advective Semi-Lagrangian Algorithm for the Shallow Water Equations, Monthly Weather Review , 127, (1999) pp. 1551 - 1565. Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 15 / 38

Eddies in Doubly Periodic Domain, Day 4 FD2 FV4 2 nd order staggered grid, day 4 finite volume 4 th order, day 4 3 3 2 2 1 1 0 0 x −1 −1 −2 −2 −3 −3 2 0 −2 2 0 −2 y potential vorticity after 4 days, 512 x 512 points/cells. Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 16 / 38

Eddies in Doubly Periodic Domain, Day 8 FD2 FV4 2 nd order staggered grid, day 8 finite volume 4 th order, day 8 3 3 2 2 1 1 0 0 x −1 −1 −2 −2 −3 −3 2 0 −2 2 0 −2 y potential vorticity after 8 days, 512 x 512 points/cells. Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 17 / 38

Gulfstream Bj¨ orn Gjevik Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 18 / 38

North Sea Region Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 19 / 38

Coastal Region Bj¨ orn Gjevik Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 20 / 38

Jet in Shelf Area jet with a Gaussian profile Setup: Gjevik, Moe, Ommundsen 0.4 Domain: 300 × 600 km Gridwidth: ∆ x = 1 km 0.3 Coriolis constant: f = 1 . 2 × 10 − 4 y [m/s] 0.2 Jetshape: v jet := γ ( t ) v 0 exp( − 2( x − L B 0.1 ) 2 ) B 0 0 100 200 300 x [km] Growthfactor: shelf profile γ ( t ) := (1 − exp( − σ t )) 200 −200 where σ = 2 . 3148 × 10 − 5 −600 Boundary conditions: −1000 south inflow, −1400 north absorbing, −1800 0 100 200 300 east and west reflective. x [km] Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 21 / 38

Inflow boundary conditions: Two different boundary condition Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 22 / 38

Inflow boundary conditions: Two different boundary condition Naive inflow boundary condition: v = v jet , u = 0 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 22 / 38

Inflow boundary conditions: Two different boundary condition Naive inflow boundary condition: v = v jet , u = 0 creates unphysical discontinuity in tangential velocity reduced convergence rate for FV4 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 22 / 38

Inflow boundary conditions: Two different boundary condition Naive inflow boundary condition: v = v jet , u = 0 creates unphysical discontinuity in tangential velocity reduced convergence rate for FV4 Sundstr¨ om’s boundary condition: (NBC) v = v jet , ∂ x u = 0 Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 22 / 38