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Efficient Numerical Simulation of Advection Diffusion Systems
P. Aaron LottUniversity of Maryland, College Park
Advisors: Howard Elman (CS) & Anil Deane (IPST)
December 19, 2008National Institute of Standards and Technology
Mathematical & Computational Sciences Division Seminar Series
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OUTLINE
• History of machine and algorithmic speedup• Introduction to Advection-Diffusion Systems• Choice of Numerical Discretization• Development of Numerical Solvers• Results• Conclusion/Future Directions
2
Motivation - Efficient Solvers
Faster machines and computational algorithms can dramatically reduce simulation time. (Centuries to milliseconds).
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Motivation - Efficient Solvers
Simulating complex flows doesn’t scale as well.
Image courtesy of D. Donzis
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Motivation - Efficient Solvers
Serial Parallel
FFT Direct nlogn logn
Multigrid Iterative n (logn)^2
GMRES Iterative n n
Lower Bound n logn
Complexity of Modern Linear Solvers
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Model - Steady Advection Diffusion
Inertial and viscous forces occur on disparate scales causing sharp flow features which:
• require fine numerical grid resolution • cause poorly conditioned non-symmetric discrete systems.
These properties make solving the discrete systems computationally expensive.
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Motivation - Efficient Solvers & Discretization
High order methods are accurate & efficient.
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Methods - Spectral Element Discretization
Spectral elements provide:•flexible geometric boundaries•large volume to surface ratio•low storage requirements
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Methods - Spectral Element Discretization
F (!w)u = Mf
F̃ = M̂ ! F̂ (wx) + F̂ (wy) ! M̂
When w is constant in each direction on each element we can use• Fast Diagonal izat ion & Domain Decomposition as a solver.
The discrete system of advection-diffusion equations are of the form:
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Methods - Spectral Element Discretization
Otherwise, we can use this as a Preconditioner for an iterative solver such as GMRES
F (!w)P!1
FPF u = Mf
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Methods -Tensor Products
!What does mean?
Suppose Ak!l and Bm!n
The Kronecker Tensor Product
Matrices of this form have properties that make computations very efficient and save lots of memory!
Ckm!ln = A ! B =
!
"
"
"
#
a11B a12B . . . a1lB
a21B a22B . . . a2lB
.
.
.
.
.
.
.
.
.
ak1B ak2B . . . aklB
$
%
%
%
&
.
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Methods - Fast Diagonalization
Matrix-vector multiplies (A ! B)!u = BUAT
done in O(n3) flops instead of O(n4)
Fast Diagonalization PropertyC = A ! B + B ! A
C!1 = (V ! V )(I ! ! + ! ! I)!1(V T
! VT )
C = (V ! V )(I ! ! + ! ! I)(V T! V
T )
V T AV = !, V T BV = I
Only need an inverse of a diagonal matrix!
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We use Flexible GMRES with a preconditioner based on:
• Local constant wind approximations• Fast Diagonalization• Domain Decomposition
Methods - Solver & Preconditioner
F (!w)P!1
FPF u = Mf
F̃!1
e = (M̂!1/2!M̂
!1/2)(S!T )(!!I +I!V )!1(S!1!T
!1)(M̂!1/2!M̂
!1/2)
P!1
F= R
T0 F̃
!1
0(w̄0)R0 +
N!
e=1
RTe F̃
!1
e (w̄e)Re
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Solver Results - Constant Wind
Solution and contour plots of a steady advection-diffusion flow. Via Domain Decomposition & Fast Diagonalization. Interface solve takes 150 steps to obtain 10^-5 accuracy.
!w = 200(!sin("
6), cos(
"
6))
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Hot plate at wall forms internal boundary layers.
Preconditioner Results - Recirculating Wind
Residual Plot above.•( P + 1 ) [ 1 2 0 N + ( P + 1 ) ] additional flops per step
!w = 200(y(1 ! x2),!x(1 ! y2))
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Coupling Fast Diagonalization & Domain Decomposition provides an efficient solver for the advection-diffusion equation.
Conclusions/Future Directions
•Precondition Interface Solve•Coarse Grid Solve (multilevel DD)•Multiple wind sweeps•Time dependent flows•2D & 3D Navier-Stokes•Apply to study of complex flows
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References
M. Deville, P. Fischer, E. Mund, High-Order Methods for Incompressible Fluid Flow, Cambridge Monographs on Applied and Computational Mathematics, 2002.
H. Elman, D. Silvester, & A. Wathen, Finite Elements and Fast Iterative Solvers with applications in incompressible fluid dynamics, Numerical Mathemat ics and Scient ific Computation, Oxford University Press, New York, 2005.
H. Elman, P.A. Lott Matrix-free preconditioner for the steady advection-diffusion equation with spectral element discretization. In preparation. 2008.
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