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Multigrid MethodsThe Implementation
Wei ECSE@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)
xgyuv
xu
yu
xu
xp
tu
)()()(
Re1 2
2
2
2
2
ygxuv
yv
yv
xv
xp
tv
)()()(
Re1 2
2
2
2
2
0
yv
xu
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
,,21,,1,
2,1,,1
njmivvvv
fvh
vvvh
vvv
mjjini
jijiy
jijiji
x
jijiji
Finite Difference Scheme:
Discretization (2)
The corresponding matrix representation is:
where fAv
1
1
1
1
2
.
.
.,
.
.
.,
..
...1
mm f
f
f
v
v
v
BaIaI
aIBaIaIB
hA ,
211..
...121
12
1
2
2
2
2
h
hh
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:
1331399339931331
161
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!