Multigrid MethodsThe Implementation

Wei E

CSE@Technische Universität München.

Ferien Akademie 19th Sep. 2005

Content

IntroductionAlgorithmResults & PerformanceA Failing ExampleConclusion

Content

IntroductionAlgorithmResults & PerformanceA Failing ExampleConclusion

Introduction

Fluid DynamicsSolving PDEsComputational Fluid DynamicsNumerical Solution

The Navier-Stokes Equations (1)

Non-stationary incompressible viscous fluids

2D Cartesian coordinates system of partial differential equations two momentum equations + continuity

equation

The Navier-Stokes Equations (2)

xgy

uv

x

u

y

u

x

u

x

p

t

u

)()(

)(Re

1 2

2

2

2

2

ygx

uv

y

v

y

v

x

v

x

p

t

v

)()(

)(Re

1 2

2

2

2

2

0

y

v

x

u

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.

0,10,10),,( yxyxfuuu yyxx

Discretization (1)

11,11,0

,22

00

,,2

1,,1,

2

,1,,1

njmivvvv

fvh

vvv

h

vvv

mjjini

jijiy

jijiji

x

jijiji

Finite Difference Scheme:

Discretization (2)

The corresponding matrix representation is:

wherefAv

1

1

1

1

2

.

.

.

,

.

.

.

,

..

...1

mm f

f

f

v

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BaI

aI

aIBaI

aIB

hA ,

21

1..

...

121

12

1

2

2

2

2

h

h

h

hB

Content

IntroductionAlgorithmResults & PerformanceA Failing ExampleConclusion

Multigrid Implementation

The two classical schemes:

V-Cycle

Full Multigrid (FMG)

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

Grid[j].v <- relax(Grid[j].v, Grid[j].f, num_sweeps_down);

Grid[j+1].f <- restrict(Grid[j].f- calculate_rhs(Grid[j].v));

endfor

Grid[coarsest].v = direct_solve(Grid[coarsest].v, Grid[coarsest].f);

for j = coarsest – 1 to 0

Grid[j].v <- Grid[j].v + interpolate(Grid[j+1].v);

Grid[j].v <- relax(Grid[j].v, Grid[j].f, num_sweeps_up);

endfor

V-Cycle: Comments (1)

a) Non-recursive structure; b) Gauss-Seidel is used as the relaxation met

hod;c) calculate_rhs() is a function to calculate th

e 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);

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 it’s four neighbors.

restriction interpolation

FMG: The Algorithm

Once we have the V-cycle, FMG would be rather easy to implement:

for j = 0 to coarsest - 1 Grid[coarsest-j+1].v <- Grid[coarsest-j+1].v + interpolate_fine(Grid[coarsest-j].v); Grid[coarsest-j+1].v <- V-cycle(Grid[coarsest-j+1].v, Grid[coarsest-j+1].f);

endfor

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:

1331

3993

3993

1331

16

1

Content

IntroductionAlgorithmResults & PerformanceA Failing ExampleConclusion

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.

Simulation result (1)

The Hidden Step:

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.

Simulation result (2)

Von Karman Vortex

Performance (1)

Size

Method

128*32 256*64 128*128 256*128

SOR 18 sec 289 sec 700 sec 1635 sec

V-Cycle 10 sec 76 sec 80 sec 172 sec

FMG 7 sec 33 sec 35 sec 75 sec

Testing platform: P4 2.4GHz, 1GB Memory, SUSE Linux 9.3

Performance (2)

Performance Chart

0

500

1000

1500

2000

128*32 256*64 128*128 256*128Probl em Si ze

Time

(se

c)

SORV-cycl eFMG

Content

IntroductionAlgorithmResults & PerformanceA Failing ExampleConclusion

A Failing Example

Throttle :

Simulation Result

The Reason

Content

IntroductionAlgorithmResults & PerformanceA Failing ExampleConclusion

Conclusion

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

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.

Thank You!