A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element...

Intro SWE cG(1)cG(1) Tests

A Stable Equal Order Finite Element Discretization ofthe Shallow Water Equations of the Ocean

Erich L Foster

13 January 2014

E. L. Foster (BCAM) cG(1)cG(1) for SWE 1 / 13

Characteristics of the Earth’s Oceans

Simulated Sea Surface Temperature

Absorbs energy from theSun and stores it.

Transports heat from theequator towards the poles.

71% of Eath’s surface iscovered by the oceans.

1000 times the heat capacityof the atmosphere.

Most of the Ocean’s KE iscontained in meso-scaleeddies (<100km).

Challenges

Complex domain, coastlines and undersea mountain ranges.

Small spatial scales, yet long time scales.

Long memory, due to heat capacity and inertia, requiring severalthousand simulated years for “spin up.”

resolution or higher needed to capture the bulk of the energycontained in the meso-scale eddy field.

Large amounts of data, ∼1TB per simulated year for 0.1

gridresolution.

Challenges

gridresolution.

Challenges

gridresolution.

Challenges

gridresolution.

Challenges

gridresolution.

Shallow Water Equations (SWE)

Standard test problem for Ocean Modelling.

Like Navier-Stokes, suffers from spurious computational modes.

ηt + Θ−1H∇ · u = 0

ut + (u · ∇) u +Ro−1u⊥ + Fr−2Θ∇η −Re−1∆u = 0on Ω (1)

u · n = 0 on δΩ (2)

Why finite elements?

Finite Difference grid of GIOMAS Finite Element mesh of SLIM

Some Known Issues with Finite Elements

Mathematically sophisticated (Good for Mathematicians bad forNon-Mathematicians).

Complicated to program.

Use packages such as FEniCS, FreeFEM, OpenFOAM, etc.

Spurious computational modes for certain finite element pairs.(similar problem with finite differences)

Use a different formulation of the problem, e.g. Vorticity-Streamfunction form.Use Taylor-Hood or lesser known elements such as P1 − PNC

1 .Use a stabilization scheme.

cG(1)cG(1) Finite Element

Spatial Discretization

Trial Functions - Piecewise linearTest Functions - Piecewise linear

Temporal Discretization

Trial Functions - Piecewise linearTest Functions - Piecewise constant

Weighted least squares stabilization

Discretization of SWE

k−1n (un − un−1,v) +Ro−1(u⊥,v)− Fr−2Θ (η,∇ · v)

+ k−1n (ηn − ηn−1, χ) +H(∇ · u, χ)

+ δ1(R1(unh, η

nh), R1(v, χ))

+ δ2(R2(unh, η

nh), R2(v, χ))

unh =1

2(unh + un−1h ), ηnh =

2(ηnh + ηn+1

R1(v, χ) = (u · ∇) v +Ro−1v⊥ + Fr−2Θ∇χR2(v, χ) = Θ−1∇ · v

are the linearized strong residuals while

δ1 =RoFr2 Θ−1

2(k−2n + |un|2h−2n )−1/2, δ2 =

2(k−2n + |ηn|2h−2n )−1/2.

Linear Inviscid SWE

Compare the standard P1 − P1 finite element pair to cG(1)cG(1)applied to the Linear Inviscid SWE, i.e.

ηt + Θ−1H∇ · u = 0

ut +Ro−1u⊥ + Fr−2Θ∇η = 0on Ω (4)

u · n = 0 on δΩ (5)

Ro = 0.1Fr = 0.1Θ = 1H = 1.63Initial Condition:

u0 = 0

η0 = Ae−(x20+x

21)/(2∗σ2),

A = 1.0, σ = 5× 10−2

Simulated Gaussian Drop for Linear Inviscid SWE, HeightLeft:P1 − P1, Right: cG(1)cG(1)

Flow Around an Island

Compare the standard P1 − P1 finite element pair to cG(1)cG(1)

Re = 1 000

Ro = 0.1

Fr = 0.1

Θ = 1

H = 1.63

η = 1 at inflow and η = 0 at outflow.

(u0, η0) = (0, 0)

Simulated flow around an Island for SWE, VelocityTop:P1 − P1, Bottom: cG(1)cG(1)

Questions?

A Stable Equal Order Finite Element Discretization of the ...A Stable Equal Order Finite Element...

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