Multigrid Methods The Implementation

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Multigrid MethodsThe Implementation

• Wei E
• CSE_at_Technische Universität München.
• Ferien Akademie 19th Sep. 2005

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Content

• Introduction
• Algorithm
• Results Performance
• A Failing Example
• Conclusion

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Content

• Introduction
• Algorithm
• Results Performance
• A Failing Example
• Conclusion

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Introduction

• Fluid Dynamics
• Solving PDEs
• Computational Fluid Dynamics
• Numerical Solution

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The Navier-Stokes Equations (1)

• Non-stationary incompressible viscous fluids
• 2D Cartesian coordinates
• system of partial differential equations
• two momentum equations continuity equation

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The Navier-Stokes Equations (2)
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Poisson Equation

• Where f(x,y) is the right-hand side calculated by
  the quantities in the previous time step
• the unknown u is to be solved in the current time
  step.

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Discretization (1)

• Finite Difference Scheme

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Discretization (2)

• The corresponding matrix representation is
• where

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Content

• Introduction
• Algorithm
• Results Performance
• A Failing Example
• Conclusion

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Multigrid Implementation

• The two classical schemes
• V-Cycle
• Full Multigrid (FMG)

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V-Cycle The Algorithm

• grid double Ddim_array f // the right hand
  side
• double Ddim_array v // the current
  approximation
• Grid array of structure grid.
• for j 0 to coarsest - 1
• Gridj.v lt- relax(Gridj.v, Gridj.f,
  num_sweeps_down)
• Gridj1.f lt- restrict(Gridj.f-
  calculate_rhs(Gridj.v))
• endfor
• Gridcoarsest.v direct_solve(Gridcoarsest.v,
  Gridcoarsest.f)
• for j coarsest 1 to 0
• Gridj.v lt- Gridj.v interpolate(Gridj1.
  v)
• Gridj.v lt- relax(Gridj.v, Gridj.f,
  num_sweeps_up)
• endfor

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V-Cycle Comments (1)

• a) Non-recursive structure
• b) Gauss-Seidel is used as the relaxation method
• c) calculate_rhs() is a function to calculate the
  right-hand side based on current approximation
• d) Several methods can be used to solve the
  small-size problem, we choose SOR (successive
  over relaxation)

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V-Cycle Comments (2)

• e) For restriction, we take the mean value of the
  four neighbors as the result
• f) For interpolation, we use the similar method
  as restriction spreading the value into its
  four neighbors.
• restriction interpolation

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FMG The Algorithm

• Once we have the V-cycle, FMG would be rather
  easy to implement
• for j 0 to coarsest - 1
• Gridcoarsest-j1.v lt- Gridcoarsest-j1.v
  interpolate_fine(Gridcoarsest-j.v)
• Gridcoarsest-j1.v lt- V-cycle(Gridcoarsest-j
  1.v, Gridcoarsest-j1.f)
• endfor

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FMG Comments

• a) Initialization for all the approximations and
  right-hand sides should be made before executing
  the FMG main loop
• b) interpolate_fine() stands for a higher order
  interpolator. In practice, we use the
  interpolation matrix

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Content

• Introduction
• Algorithm
• Results Performance
• A Failing Example
• Conclusion

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Model Problem (1) Hidden Step

• Fluid flows with a constant velocity through a
  channel with a hidden obstacle on one side.
  No-slip conditions are imposed at the upper and
  lower walls.

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Simulation result (1)

• The Hidden Step

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Model Problem (2) Karman Vortex

• The flow in a channel can meet a tilted plate. At
  the left boundary, the fluid inflow has a
  constant velocity profile, while at the upper and
  lower boundaries no-slip conditions are imposed.

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Simulation result (2)

• Von Karman Vortex

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Performance (1)
Testing platform P4 2.4GHz, 1GB Memory, SUSE
Linux 9.3
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Performance (2)
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Content

• Introduction
• Algorithm
• Results Performance
• A Failing Example
• Conclusion

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A Failing Example

• Throttle

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Simulation Result
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The Reason
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Content

• Introduction
• Algorithm
• Results Performance
• A Failing Example
• Conclusion

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Conclusion

• An Optimal (i.e., O(N)) Solver.
• Highly Modular Program Structure
• Advanced Debugging Technique

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Reference

• 1 Practical Course Scientific Computing and
  Visualization Worksheet, Lehrstuhl für Informatik
  mit Schwerpunkt Wissenschaftliches Rechnen,
  TU-Muenchen, 2005.
• 2 Krzysztof J. Fidkowski, A High-Order
  Discontinuous Galerkin Multigrid Solver for
  Aerodynamic Applications, Master Thesis in
  Aerospace Engineering at the MASSACHUSETTS
  INSTITUTE OF TECHNOLOGY
• 3 S. McCormick, B. Briggs, and V. Henson, "A
  Multigrid Tutorial, second edition,
  SIAM,Philadelphia, June 2000.

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Thank You!