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

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  • 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

    Simulated Sea Surface Temperature

    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.

    Most of the Ocean’s KE is contained in meso-scale eddies (

  • 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 −1H∇ · u = 0

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

    u · n = 0 on δΩ (2)

    E. L. Foster (BCAM) cG(1)cG(1) for SWE 4 / 13

  • Intro SWE cG(1)cG(1) Tests

    Why finite elements?

    Finite Difference grid of GIOMAS Finite Element mesh of SLIM

    E. L. Foster (BCAM) 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 − PNC1 . 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 − PNC1 . 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 − PNC1 . 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 − PNC1 . 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 − PNC1 . 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 − PNC1 . 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 − PNC1 . 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

    k−1n (un − un−1,v) +Ro−1(ū⊥,v)− Fr−2Θ (η̄,∇ · v) + k−1n (ηn − ηn−1, χ) +H(∇ · ū, χ) + δ1(R1(ū

    n h, η

    n h), R1(v, χ))

    + δ2(R2(ū n h, η

    n h), R2(v, χ))

    (3)

    where

    ūnh = 1

    2 (unh + u

    n−1 h ), η̄

    n h =

    1

    2 (ηnh + η

    n+1 h )

    and

    R1(v, χ)