Intro
SWE
cG(1)cG(1)
Tests
A Stable Equal Order Finite Element Discretization of
the Shallow Water Equations of the Ocean
Erich L Foster
13 January 2014
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
1 / 13
Intro
SWE
cG(1)cG(1)
Tests
Characteristics of the Earth’s Oceans
Absorbs energy from the
Sun and stores it.
Transports heat from the
equator towards the poles.
71% of Eath’s surface is
covered by the oceans.
1000 times the heat capacity
of the atmosphere.
Simulated Sea Surface Temperature
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
Most of the Ocean’s KE is
contained in meso-scale
eddies (<100km).
2 / 13
Intro
SWE
cG(1)cG(1)
Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring several
thousand simulated years for “spin up.”
◦
0.1 resolution or higher needed to capture the bulk of the energy
contained in the meso-scale eddy field.
◦
Large amounts of data, ∼1TB per simulated year for 0.1 grid
resolution.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
3 / 13
Intro
SWE
cG(1)cG(1)
Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring several
thousand simulated years for “spin up.”
◦
0.1 resolution or higher needed to capture the bulk of the energy
contained in the meso-scale eddy field.
◦
Large amounts of data, ∼1TB per simulated year for 0.1 grid
resolution.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
3 / 13
Intro
SWE
cG(1)cG(1)
Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring several
thousand simulated years for “spin up.”
◦
0.1 resolution or higher needed to capture the bulk of the energy
contained in the meso-scale eddy field.
◦
Large amounts of data, ∼1TB per simulated year for 0.1 grid
resolution.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
3 / 13
Intro
SWE
cG(1)cG(1)
Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring several
thousand simulated years for “spin up.”
◦
0.1 resolution or higher needed to capture the bulk of the energy
contained in the meso-scale eddy field.
◦
Large amounts of data, ∼1TB per simulated year for 0.1 grid
resolution.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
3 / 13
Intro
SWE
cG(1)cG(1)
Tests
Challenges
Complex domain, coastlines and undersea mountain ranges.
Small spatial scales, yet long time scales.
Long memory, due to heat capacity and inertia, requiring several
thousand simulated years for “spin up.”
◦
0.1 resolution or higher needed to capture the bulk of the energy
contained in the meso-scale eddy field.
◦
Large amounts of data, ∼1TB per simulated year for 0.1 grid
resolution.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
3 / 13
Intro
SWE
cG(1)cG(1)
Tests
Shallow Water Equations (SWE)
Standard test problem for Ocean Modelling.
Like Navier-Stokes, suffers from spurious computational modes.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
4 / 13
Intro
SWE
cG(1)cG(1)
Tests
Shallow Water Equations (SWE)
Standard test problem for Ocean Modelling.
Like Navier-Stokes, suffers from spurious computational modes.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
4 / 13
Intro
SWE
cG(1)cG(1)
Tests
Shallow Water Equations (SWE)
Standard test problem for Ocean Modelling.
Like Navier-Stokes, suffers from spurious computational modes.
ηt + Θ−1 H∇ · u = 0
ut + (u · ∇) u + Ro−1 u⊥ + F r−2 Θ∇η − Re−1 ∆u = 0
u·n=0
E. L. Foster (BCAM)
on δΩ
cG(1)cG(1) for SWE
on Ω
(1)
(2)
4 / 13
Intro
SWE
cG(1)cG(1)
Tests
Why finite elements?
Finite Difference grid of GIOMAS
E. L. Foster (BCAM)
Finite Element mesh of SLIM
cG(1)cG(1) for SWE
5 / 13
Intro
SWE
cG(1)cG(1)
Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad for
Non-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-Stream
function form.
Use Taylor-Hood or lesser known elements such as P1 − P1N C .
Use a stabilization scheme.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
6 / 13
Intro
SWE
cG(1)cG(1)
Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad for
Non-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-Stream
function form.
Use Taylor-Hood or lesser known elements such as P1 − P1N C .
Use a stabilization scheme.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
6 / 13
Intro
SWE
cG(1)cG(1)
Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad for
Non-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-Stream
function form.
Use Taylor-Hood or lesser known elements such as P1 − P1N C .
Use a stabilization scheme.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
6 / 13
Intro
SWE
cG(1)cG(1)
Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad for
Non-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-Stream
function form.
Use Taylor-Hood or lesser known elements such as P1 − P1N C .
Use a stabilization scheme.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
6 / 13
Intro
SWE
cG(1)cG(1)
Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad for
Non-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-Stream
function form.
Use Taylor-Hood or lesser known elements such as P1 − P1N C .
Use a stabilization scheme.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
6 / 13
Intro
SWE
cG(1)cG(1)
Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad for
Non-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-Stream
function form.
Use Taylor-Hood or lesser known elements such as P1 − P1N C .
Use a stabilization scheme.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
6 / 13
Intro
SWE
cG(1)cG(1)
Tests
Some Known Issues with Finite Elements
Mathematically sophisticated (Good for Mathematicians bad for
Non-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-Stream
function form.
Use Taylor-Hood or lesser known elements such as P1 − P1N C .
Use a stabilization scheme.
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
6 / 13
Intro
SWE
cG(1)cG(1)
Tests
cG(1)cG(1) Finite Element
Spatial Discretization
Trial Functions - Piecewise linear
Test Functions - Piecewise linear
Temporal Discretization
Trial Functions - Piecewise linear
Test Functions - Piecewise constant
Weighted least squares stabilization
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
7 / 13
Intro
SWE
cG(1)cG(1)
Tests
Discretization of SWE
kn−1 (un
− un−1 , v) + Ro−1 (ū⊥ , v) − F r−2 Θ (η̄, ∇ · v)
+ kn−1 (ηn − ηn−1 , χ) + H(∇ · ū, χ)
+ δ1 (R1 (ūnh , ηhn ), R1 (v, χ))
(3)
+ δ2 (R2 (ūnh , ηhn ), R2 (v, χ))
where
1
ūnh = (unh + un−1
h ),
2
1
η̄hn = (ηhn + ηhn+1 )
2
and
R1 (v, χ) = (ū · ∇) v + Ro−1 v⊥ + F r−2 Θ∇χ
R2 (v, χ) = Θ−1 ∇ · v
are the linearized strong residuals while
δ1 =
Ro F r2 Θ−1 −2
−1/2
(kn + |un |2 h−2
,
n )
2
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
δ2 =
Θ −2
−1/2
(k + |η n |2 h−2
.
n )
2 n
8 / 13
Intro
SWE
cG(1)cG(1)
Tests
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−1 H∇ · u = 0
ut + Ro−1 u⊥ + F r−2 Θ∇η = 0
u·n=0
on δΩ
on Ω
(4)
(5)
Ro = 0.1
F r = 0.1
Θ=1
H = 1.63
Initial Condition:
u0 = 0
2
2
2
η0 = A e−(x0 +x1 )/(2∗σ ) ,
(6)
A = 1.0, σ = 5 × 10−2
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
9 / 13
Intro
SWE
cG(1)cG(1)
Tests
Simulated Gaussian Drop for Linear Inviscid SWE, Height
Left:P1 − P1 , Right: cG(1)cG(1)
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
10 / 13
Intro
SWE
cG(1)cG(1)
Tests
Flow Around an Island
Compare the standard P1 − P1 finite element pair to cG(1)cG(1)
Re = 1 000
Ro = 0.1
F r = 0.1
Θ=1
H = 1.63
η = 1 at inflow and η = 0 at outflow.
(u0 , η0 ) = (0, 0)
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
11 / 13
Intro
SWE
cG(1)cG(1)
Tests
Simulated flow around an Island for SWE, Velocity
Top:P1 − P1 , Bottom: cG(1)cG(1)
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
12 / 13
Intro
SWE
cG(1)cG(1)
Tests
Questions?
E. L. Foster (BCAM)
cG(1)cG(1) for SWE
13 / 13