A comparison between

a direct and a multigrid sparse linear solvers

for highly heterogeneous flux computations

A. Beaudoin, J.-R. De Dreuzy and J. ErhelA. Beaudoin, J.-R. De Dreuzy and J. Erhel ECCOMAS CFD 06, Egmond aan Zee, the Netherlands, September 2006

2D Heterogeneous 2D Heterogeneous permeability fieldpermeability fieldStochastic model Y = ln(K)Stochastic model Y = ln(K)with correlation functionwith correlation function

2( ) expY YY

C

rr

31 Y

Physical flow modelPhysical flow model

Q = - K*grad (h)

div (Q) = 0

Fix

ed

head

Fix

ed

head

Nul flux

Nul flux

Examples of simulationsExamples of simulationsσσ=0.5 and =0.5 and σσ=3=3

Numerical method for 2D heterogeneous Numerical method for 2D heterogeneous porous mediumporous medium

Finite Volume Method with a regular mesh

Large sparse structured matrix of order N with 5 entries per row

Sparse direct solverSparse direct solver

memory size and CPU time with memory size and CPU time with PSPASESPSPASES

Theory : NZ(L) = O(N logN) Theory : Time = O(N1.5)

Multigrid sparse solverMultigrid sparse solver

convergence and CPU time with convergence and CPU time with HYPRE/SMGHYPRE/SMG

Parallel architectureParallel architecturedistributed memorydistributed memory

2 nodes of 32 bi – processors 2 nodes of 32 bi – processors (Proc AMD Opteron 2Ghz with 2Go (Proc AMD Opteron 2Ghz with 2Go

of RAM)of RAM)

Parallel architectureParallel architecture

Direct and multigrid solversDirect and multigrid solvers

Parallel CPU times for various sizesParallel CPU times for various sizes

Direct and multigrid solversDirect and multigrid solvers

Speed-ups for various sizesSpeed-ups for various sizes

Direct solverDirect solver

Scalability analysis with PSPASES : Scalability analysis with PSPASES : isoefficiencyisoefficiency

PTp

TE S

PTp

NR

P N Tp R

2 0.26 106 5.60 1.20 106

8 1.05 106 11.33 1.18 106

32 4.19 106 25.70 1,04 106

4 0.26 106 2.92 1.15 106

16 1.05 106 6.06 1.11 106

64 4.19 106 13.08 1,05 106

5.1

Multigrid solverMultigrid solver

Impact of permeability standard deviation Impact of permeability standard deviation and system sizeand system size

Convergence and CPU timeConvergence and CPU time

Multigrid solverMultigrid solver

Impact of permeability standard deviation Impact of permeability standard deviation and system sizeand system size

Convergence and CPU timeConvergence and CPU time

Direct and multigrid solversDirect and multigrid solvers

Impact of permeability standard Impact of permeability standard deviationdeviation

Direct and multigrid solversDirect and multigrid solvers

SummarySummary

• PSPASES is more efficient for small matrices• PSPASES is scalable and is more efficient with many processors• HYPRE requires less memory• HYPRE is more efficient for large matrices • HYPRE is very sensitive to the permeability variance

• Another method for large matrices and large variance ?