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

1

Well-balanced schemes

2

Application: Geostrophic Flow

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 2 / 38

Shallow Water Equations

Shallow Water Equations with topography and coriolisforce:   h hu hv  

t

+   hu hu2 + 1

2gh2

huv  

x

+   hv huv hv2 + 1

2gh2

 

y

=   −ghbx − fhv −ghby + fhu   h(x, y, t): waterheight hu(x, y, t): x-momentum hv(x, y, t): y-momentum b(x): topography g: gravitation constant 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 (Uij)t + 1 ∆x (Fi+ 1

2 − Fi− 1 2 ) + 1

∆y (Gj+ 1

2 − Gj− 1 2 ) = Sij

numerical flux (LF, LLF, HLL, kinetic, . . . ) Fi+ 1

2 = F(Ui,r, Ui+1,l)

polynomial reconstruction for η = h + b, hu, and hv reconstruction of surface displacement hrec = ηrec − b(x, y). Sij 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 b(x, y) = sin(2πx) + cos(2πy), η(x, y) = 10 + esin(2πx) cos(2πy), hu(x, y) = sin(cos(2πx)) sin(2πy), hv(x, y) = cos(2πx) cos(sin(2πy)). number h hu hv

• f points

CFL L1 error

• rder

L1 error

• rder

L1 error

• rder

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

initial data: η(x, y) = 1.01 , x ∈ [0.05, 0.15] 1 , otherwise hu(x, y) = 0, hv(x, y) = 0 b(x, y) = 0.8 exp(−5(x − 0.9)2 − 50(y − 0.5)2)

1 2 0.5 1 0.2 0.4 0.6 0.8 x y b

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

1

Well-balanced schemes

2

Application: Geostrophic Flow

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 N L1-error

• rder

L1-error

• rder

L1-error

• rder

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 L1-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

Geostrophically balanced flow: potential vorticity q := ∇×u+f

h

Initial data: width of jet: 2a = 1 potential vorticity: q(x, y, 0) =

• ¯

q + Q sign(ˆ y)(a −

• |ˆ

y| − a

• ), |ˆ

y| < 2a, ¯ q, otherwise, . ˆ y = y − 0.1 sin(2x) + 0.1 sin(3x)

−3 −2 −1 1 2 3 −2 2 2nd order staggered grid, day 0 x 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

−3 −2 −1 1 2 3 −2 2 2nd order staggered grid, day 4 x y

FV4

−2 2 −3 −2 −1 1 2 3 finite volume 4th order, day 4

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

−3 −2 −1 1 2 3 −2 2 2nd order staggered grid, day 8 x y

FV4

−2 2 −3 −2 −1 1 2 3 finite volume 4th order, day 8

potential vorticity after 8 days, 512 x 512 points/cells.

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 17 / 38

Gulfstream Bj¨

• rn 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¨

• rn Gjevik

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 20 / 38

Jet in Shelf Area

Setup: Gjevik, Moe, Ommundsen Domain: 300 × 600 km Gridwidth: ∆x = 1 km Coriolis constant: f = 1.2 × 10−4 Jetshape: vjet := γ(t)v0 exp(−2(x − LB B )2) Growthfactor: γ(t) := (1 − exp(−σt)) where σ = 2.3148 × 10−5 Boundary conditions: south inflow, north absorbing, east and west reflective.

100 200 300 0.1 0.2 0.3 0.4

x [km] y [m/s] jet with a Gaussian profile

100 200 300 200 −200 −600 −1000 −1400 −1800

x [km] shelf profile

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 = vjet, 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 = vjet, 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 = vjet, u = 0

creates unphysical discontinuity in tangential velocity reduced convergence rate for FV4

Sundstr¨

• m’s boundary condition: (NBC) v = vjet, ∂xu = 0

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 22 / 38

Inflow boundary conditions:

Two different boundary condition Naive inflow boundary condition: v = vjet, u = 0

creates unphysical discontinuity in tangential velocity reduced convergence rate for FV4

Sundstr¨

• m’s boundary condition: (NBC) v = vjet, ∂xu = 0

translate this to FV–scheme via characteristic theory hjet = (−

• gh+h+)/(vjet −
• gh+ + v+)

ujet = u+hjet vjet = vjethjet

third order convergence rate for FV4

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 22 / 38

Convergence of inflow boundary conditions

FD2 FV4 FV4 (NBC) N L1-error

• rder

L1-error

• rder

L1-error

• rder

50 6.10e+07 3.19e+06 3.20e+06 100 2.90e+07 1.07 2.11e+05 3.92 2.20e+05 3.86 200 1.38e+07 1.07 1.51e+04 3.80 1.66e+04 3.72 400 6.31e+06 1.13 5.10e+03 1.57 1.32e+03 3.65 800 2.75e+06 1.20 2.67e+03 0.93 1.02e+02 3.69 L1-errors of h and numerical order of convergence, T = 3000 sec

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 23 / 38

Jet in Shelf Area, Day 5

FD2

150 170 190 210 230 50 100 150 200 2nd order SG, day 5 x [km] y [km]

FV4

150 170 190 210 230 50 100 150 200 4th order FV, day 5 x [km] y [km]

FV4 (NBC)

150 170 190 210 230 50 100 150 200 4th order FV (NBC), day 5 x [km] y [km]

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 24 / 38

Jet in Shelf Area, Day 10

FD2

150 170 190 210 230 50 100 150 200 2nd order SG, day 10 x [km] y [km]

FV4

150 170 190 210 230 50 100 150 200 4th order FV, day 10 x [km] y [km]

FV4 (NBC)

150 170 190 210 230 50 100 150 200 4th order FV (NBC), day 10 x [km] y [km]

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 25 / 38

Jet in Shelf Area, Day 15

FD2

150 170 190 210 230 50 100 150 200 2nd order SG, day 15 x [km] y [km]

FV4

150 170 190 210 230 50 100 150 200 4th order FV, day 15 x [km] y [km]

FV4 (NBC)

150 170 190 210 230 50 100 150 200 4th order FV (NBC), day 15 x [km] y [km]

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 26 / 38

Discussion of Jet in Shelf Area

Jet in shelf Area: new inflow boundary condition improves eddy formation

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 27 / 38

Discussion of Jet in Shelf Area

Jet in shelf Area: new inflow boundary condition improves eddy formation need further improvement for absorbing outflow boundary

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 27 / 38

Discussion of Jet in Shelf Area

Jet in shelf Area: new inflow boundary condition improves eddy formation need further improvement for absorbing outflow boundary FD2 and FV4 qualitatively the same

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 27 / 38

Discussion of Jet in Shelf Area

Jet in shelf Area: new inflow boundary condition improves eddy formation need further improvement for absorbing outflow boundary FD2 and FV4 qualitatively the same convergence studies for the full jet problem still out of reach due to problemsize

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 27 / 38

Conclusion

Finite-Differences: + easy to implement + computationally cheap − less accurate than FV − dispersive oscillations - needs artificial diffusion − breaks down at shocks

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 28 / 38

Conclusion

Finite-Differences: + easy to implement + computationally cheap − less accurate than FV − dispersive oscillations - needs artificial diffusion − breaks down at shocks Finite-Volume: + highly accurate + can track waves over long distances and times + no dispersive oscillations + very stable at shocks − computationally expensive (costs > 50 × FD2)

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 28 / 38

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 29 / 38

Finite-Differences for Shallow Water Equation

Conservation of mass: ηt + hux + hvy = 0, Time and Space discretization leads to: ηn+ 1

2 = ηn− 1 2 − ∆t[µyδxhun + µxδyhvn].

δx, δy finite-difference µx, µy average

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 30 / 38

Update Surface Displacement

ηn+ 1

2 = ηn− 1 2 − ∆t[µyδxhun + µxδyhvn].

B-Grid y x hui−1

2,j−1 2

hui−1

2,j+1 2

hui+1

2,j+1 2

hui+1

2,j−1 2

t j ηn−1

2

ηn+1

2

i i + 1

2

i − 1

2

n n + 1

2

n − 1

2

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 31 / 38

Update Momentum

(hu)n+1 = (hu)n − ∆t

• δx

(µx(hu)n)2 µyhn+ 1

2

+ δy µy(hu)n µy(hv)n µxhn+ 1

2

+ gµxµyhn+ 1

2 δxµyηn+ 1 2 − f (hv)n Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 32 / 38

Update Momentum

(hu)n+1 = (hu)n − ∆t

• δx

(µx(hu)n)2 µyhn+ 1

2

+ δy µy(hu)n µy(hv)n µxhn+ 1

2

+ gµxµyhn+ 1

2 δxµyηn+ 1 2 − f (hv)n

FD1

x i + 1 i t hui+1

2,j+1 2

n + 1

2

n + 1 n

Symmetry in time lost ⇒ only first order accurate

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 32 / 38

Update Momentum

(hu)n+1 = (hu)n − ∆t

• δx

(µx(hu)n)2 µyhn+ 1

2

+ δy µy(hu)n µy(hv)n µxhn+ 1

2

+ gµxµyhn+ 1

2 δxµyηn+ 1 2 − f (hv)n

FD1

x i + 1 i t hui+1

2,j+1 2

n + 1

2

n + 1 n

Symmetry in time lost ⇒ only first order accurate FD2

x i + 1 i t hui+1

2,j+1 2

n + 1

2

n + 1 n

Symm in time and accuracy recovered

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 32 / 38

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 33 / 38

contour lines of surface, T= 0.12 sec

x y

0.5 1 1.5 2 0.5 1

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 34 / 38

x y

contour lines of surface, T= 0.24 sec 0.5 1 1.5 2 0.5 1

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 35 / 38

x y

contour lines of surface, T= 0.36 sec 0.5 1 1.5 2 0.5 1

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 36 / 38

x y

contour lines of surface, T= 0.48 sec 0.5 1 1.5 2 0.5 1

Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 37 / 38

x y

contour lines of surface, T= 0.6 sec 0.5 1 1.5 2 0.5 1

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