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Novel non-hydrostatic numerical schemes based on discontinuous Galerkin finite element method Matt Ueckermann, Pierre Lermusiaux 16 September, 2009 1 of 29 Outline Motivation Hybrid Discontinuous Galerkin (HDG) Methods Formulation

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16 September, 2009 1 of 29

Novel non-hydrostatic numerical schemes based

  • n discontinuous Galerkin finite element method

Matt Ueckermann, Pierre Lermusiaux

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16 September, 2009 2 of 29

Outline

  • Motivation
  • Hybrid Discontinuous Galerkin (HDG) Methods
  • Formulation
  • Similarity to other methods (LDG, IP, CG)
  • Post-processing
  • Simple Test cases: Convergence studies
  • Projection method
  • Results for analytical stokes problem
  • Results for Lock-exchange problem
  • Comments on GPU implementation of HDG algorithm
  • Future work
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Motivation

  • Major complaint of DG methods:
  • Too many degrees of freedom (DOFs)!

Duplication at corners, but no interior DOFs! Duplication at edges No duplication of DOF Hybrid Discontinuous Galerkin Discontinuous Galerkin Continuous Galerkin

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Hybrid Discontinuous Galerkin - Formulation

  • Basic idea: if boundary value of element is known, the

element can be independently solved.

  • 1. Nguyen, N. C., Peraire, J., and Cockburn, B. (2009). An implicit high-order hybridizable

discontinuous galerkin method for linear convection-difgusion equations. Journal of Computational Physics, 228(9):3232-3254.

  • 2. Cockburn, B., Gopalakrishnan, J., and Lazarov, R. (2009). Unified hybridization of

discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic

  • problems. Siam Journal on Numerical Analysis, 47(2):1319-1365.

Boundary conditions known Solve Independently

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Hybrid Discontinuous Galerkin - Formulation

  • Consider:
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Hybrid Discontinuous Galerkin - Formulation

  • Finite Element (FE) spaces:
  • FE formulation
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Hybrid Discontinuous Galerkin - Formulation

  • Finite Element (FE) spaces:
  • FE formulation
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Hybrid Discontinuous Galerkin - Formulation

  • Finite Element (FE) spaces:
  • FE formulation

Couples u and q solutions. Inefficient?

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Hybrid Discontinuous Galerkin - Formulation

  • Additional FE space
  • Flux Definition(1)
  • Final Equations
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Hybrid Discontinuous Galerkin - Formulation

  • Algebraic Equations

Block Diagonal! Element-local inversion

  • Global system of unknowns
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Hybrid Discontinuous Galerkin - Formulation

  • Algebraic Equations -- Simple implementation
  • May also implement this by constructing the K matrix directly.
  • Considerably faster
  • Much more difficult to implement
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Hybrid Discontinuous Galerkin - Comparison

  • HDG in the standard form:

Compared to LDG/IP:

  • HDG: Sparser matrix, Fewer globally coupled degrees of freedom,
  • Always for 2D(2)
  • For higher than 2nd order basis in 3D(2)
  • What about expense of local solvers?

Compared to CG/Mixed-method

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Hybrid Discontinuous Galerkin - Post-Processing

  • HDG convergence for
  • Both u and q converge at order p+1
  • Allows for local post-processing of solution
  • Only required at final time
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Test Case Examples - Steady Diffusive Problem

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Test Case Examples - Steady Diffusive Problem

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Test Case Examples - Unsteady ADR

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Projection Method

  • Solution scheme: Incremental pressure correction scheme(3)
  • Uses Pressure Poisson Equation (PPE)
  • HDG provides accurate derivative terms
  • 3. Guermond, J.L., Minev, P., and Shen, Jie. (2006). An overview of projection methods for

incompressible flows. Comput. Methods Appl. Mech. Engrg, 195:6011-6045.

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Solution to Stokes equations, tau = 1

  • Tau = 1
  • Pressure postprocessing has

significant effect

Postprocessed

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Solution to Stokes equations, tau = 1

  • Tau = 1
  • Velocity

postprocessing has significant effect

  • Reduce error by one
  • rder
  • Divergence field very

noisy

Postprocessed

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Solution to Stokes equations, tau = 10,000

  • Tau = 10,000
  • Pressure postprocessing has no

effect

Postprocessed

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Solution to Stokes equations, tau = 10,000

  • Tau = 10,000
  • Velocity

postprocessing reduces error

Postprocessed

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Stokes: Pressure error comparison

  • Tau = 10,000 pressure error

compares well with literature

  • Tau = 1 pressure error very noisy
  • Numerical boundary Pressure

boundary layer

Tau = 10,000 Guermond (2006) Tau = 1

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Stokes: Convergence

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Stokes: Convergence

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Lock Exchange Problem

  • 37,000 DOF, 14,000 HDG unknowns
  • 13.5 hrs
  • 1320 Elements
  • p=6
  • Gr = 1.25x106, Sc=0.71
  • 4. Hartel, C., Meinburg, E., and Freider, N. (2000). Analysis and direct numerical simulations of the flow at a gravity-

current head. Part 1. Flow topology and front speed for slip and no-slip boundaries. J. Fluid. Mech, 418:189-212.

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Lock Exchange Problem

Time = 10 Time = 5

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Implementation issues: GPU’s

  • Local HDG solver has three steps
  • 1. Build matrix and right-hand side vector: Local operation
  • 2. Solve global system of equations: Global operation
  • 3. Reconstruct solution: Local operation
  • Local steps (1, 3) very well-suited to parallel architectures

such as GPU

  • Local read patterns very well-suited to coalesced memory

access

  • Write pattern less localized (to form matrix)
  • Negates the problem with “expensive” local solutions
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Future work

  • Investigate methods to reduce tau in PPE
  • Test different algorithms for solving Boussinesq Equations
  • Can we do better?
  • Some ideas to further reduce global DOF
  • Investigate adaptive/multiscale strategies
  • Octree-structured (2p+1 accurate post-processed solution)
  • Unstructured
  • Nesting
  • 2D Non-hydrostatic dynamical studies
  • Implement efficient 3D code
  • Exploit HDG parallelism: domain decomposition
  • MPI?
  • GPU?
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Thank You!

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Notation

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Stokes: Convergence tempora

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Lock exchange: vorticity

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Lock Exchange Pressure

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Lock Exchange: Vorticity and Pressure

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Lock Exchange Velocity

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Preconditioners with HDG

  • Have compared LDG descretization with HDG
  • Solved matrixes using various preconditioners
  • HDG at least 3x faster than LDG for p=4 using similar preconditioners

LDG, solve system once HDG, solve system 3 times

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Test Case Examples - Steady advection-diffusion