SLIDE 1 Efficient Numerical Simulation of Advection Diffusion Systems
University of Maryland, College Park Advisors: Howard Elman (CS) & Anil Deane (IPST) December 19, 2008 National Institute of Standards and Technology Mathematical & Computational Sciences Division Seminar Series
SLIDE 2 OUTLINE
- History of machine and algorithmic speedup
- Introduction to Advection-Diffusion Systems
- Choice of Numerical Discretization
- Development of Numerical Solvers
- Results
- Conclusion/Future Directions
SLIDE 3
Motivation - Efficient Solvers
Faster machines and computational algorithms can dramatically reduce simulation time. (Centuries to milliseconds).
SLIDE 4 Motivation - Efficient Solvers
Simulating complex flows doesn’t scale as well.
Image courtesy of D. Donzis
SLIDE 5
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
SLIDE 6 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.
SLIDE 7
Motivation - Efficient Solvers & Discretization
High order methods are accurate & efficient.
SLIDE 8 Methods - Spectral Element Discretization
Spectral elements provide:
- flexible geometric boundaries
- large volume to surface ratio
- low storage requirements
SLIDE 9 Methods - Spectral Element Discretization
When w is constant in each direction on each element we can use
Diagonalization & Domain Decomposition as a solver. The discrete system of advection-diffusion equations are of the form:
SLIDE 10
Methods - Spectral Element Discretization
Otherwise, we can use this as a Preconditioner for an iterative solver such as GMRES
SLIDE 11
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 .
SLIDE 12
Methods - Fast Diagonalization
Matrix-vector multiplies (A ⊗ B) u = BUAT done in O(n3)flops instead of O(n4) Fast Diagonalization Property C = A ⊗ B + B ⊗ A C−1 = (V ⊗ V )(I ⊗ Λ + Λ ⊗ I)−1(V T ⊗ V T ) C = (V ⊗ V )(I ⊗ Λ + Λ ⊗ I)(V T ⊗ V T ) V T AV = Λ, V T BV = I Only need an inverse of a diagonal matrix!
SLIDE 13 We use Flexible GMRES with a preconditioner based on:
- Local constant wind approximations
- Fast Diagonalization
- Domain Decomposition
Methods - Solver & Preconditioner
F( w)P −1
F PF u = Mf ˜ F −1
e
= ( ˆ M −1/2 ⊗ ˆ M −1/2)(S ⊗T)(Λ⊗I +I ⊗V )−1(S−1 ⊗T −1)( ˆ M −1/2 ⊗ ˆ M −1/2)
P −1
F
= RT
0 ˜
F −1 ( ¯ w0)R0 +
N
RT
e ˜
F −1
e
( ¯ we)Re
SLIDE 14 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.
6 ), cos(π 6 ))
SLIDE 15 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))
SLIDE 16 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
SLIDE 17 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 Mathematics and Scientific 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.
