Multigrid solution methods for nonlinear time-dependent ...
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Multigrid solution methods for nonlinear time-dependentsystems
Feng Wei Yang
Department of MathematicsUniversity of Sussex
4 December 2014
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Objectives
To solve complex non-linear parabolic systems by applying:
2nd order central Finite Difference Method (FDM)
2nd order Backward Differentiation Formula (BDF2)
Nonlinear multigrid method with Full Approximation Scheme (FAS)
Adaptive Mesh Refinement (AMR)
Adaptive time-stepping
Parallel technique
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Outline
Multigrid methods
Linear multigridNonlinear multigrid
Thin film models from Gaskell et al.
Adaptive multigrid solver
Cahn-Hilliard-Hele-Shaw system of equations from Wise
Tumour modelling
Tumour model from Wise et al.
2nd order convergence rate
3-D results
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Jacobi/Gauss-Seidel iterative methods
Well-known methodsRequire diagonally-dominant matricesTypically have complexity of O(n2) for general sparse matrices...Smoothing property
Low frequency of error High frequency of error
S.H. Lui Numerical Analysis of Partial Differential Equations, 2011
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Convergence of a typical Jacobi iterative method
source: nkl.cc.u-tokyo.ac.jp
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Multigrid v-cycle
Finest grid
Coarsest grid Grid level 1
Grid level 2
Grid level 3
Grid level 4
xy
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Linear multigrid
A linear problem:Au = b, (1)
exact error can be obtained as
E = u − v , (2)
residual can be calculated as:
r = b − Av . (3)
Error equation:
AE = A(u − v)
= Au − Av
= b − Av
= r .
(4)
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Linear multigrid
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Nonlinear multigrid
The Error Equation (4) does not exist in a nonlinear case
Full Approximate Scheme (FAS)
For problem on coarser grids, a modified RHS is included
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Nonlinear multigrid
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Droplet spreading model
∂h
∂t= ∂
∂x
[h3
3
(∂p∂x −
Boε sinα
)]+ ∂
∂y
[h3
3
(∂p∂y
)]p = −4(h)− Π(h) + Boh cosα
with Neumann boundary conditions:
∂nh = 0 ∂np = 0 on ∂Ω
Gaskell et al. Int. J. Numer. Meth. Fluids, 45:1161-1186, 2004
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Our solver
Cell-centred 2nd order finite difference method
PARAMESH library for mesh generation and AMR
Fully implicit BDF2 method with adaptive time-stepping
MLAT variation of FAS multigrid at each time-step
Newton-block 2× 2 Red-Black (weighted) Gauss-Seidel smoother
Full weighting restriction and bilinear interpolation
Parallelism achieved using ARC2
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Validation
0 0.2 0.4 0.6 0.8 1x 10−5
0
1
2
3
4
5
t
h 0(t)
32x3264x64128x128256x256512x5121024x1024
Results from Gaskell et al. on the left and our results on the right.
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Multigrid linear complexity
104 105 106 107
100
101
102
No. grid points on the finest grid.
Ave
rage
CP
U ti
me
per t
ime
step
(sec
onds
).
CPU time required Line with slope of 1
A log-log plot demonstrating the linear complexity of multigrid.
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Multigrid performance
Results from Gaskell et al. on the left and our results on the right.
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AMR
AMR with initial condition on the left and final solution on the right.
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Adaptive time-stepping
0 0.2 0.4 0.6 0.8 1x 10−5
1
2
3
4
5
6
7
8
9
10
11x 10−7
Time
Tim
e st
ep s
ize
adaptive time−stepping 1024x1024
Evolution of δt during T = [0, 1× 10−5].
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Adaptive multigrid solver
Cases No.leaf nodes
Uniform 10242 1,048,576AMR 168,480
Cases No. time CPU timesteps (seconds)
Fixed δt 1000 16721.3ATS 45 574.4
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Multigrid in parallel
No. cores 1 2 4 8 16 32 64
CPU time(seconds) 3282.6 1687.1 843.5 774.1 490.6 348.6 368.3
Table: A grid hierarchy 16× 16− 1024× 1024, and mesh block size is 8× 8.
No. cores 1 2 4 8 16 32 64
CPU time(seconds) 3264.1 1625.1 840.1 687.3 439.5 351.2 301.7
Table: Another grid hierarchy of 32× 32− 1024× 1024, and mesh block size of8× 8.
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Fully-developed flow
∂h
∂t= ∂
∂x
[h3
3
(∂p∂x − 2
)]+ ∂
∂y
[h3
3
(∂p∂y
)]p = −64(h + s) + 2 3
√6N(h + s)
with mixed boundary conditions:
h(x = 0, y) = g ,∂p
∂x|x=0 = 0,
∂h
∂x|x=1 =
∂p
∂x|x=1 = 0,
∂p
∂y|y=0 =
∂p
∂y|y=1 =
∂h
∂y|y=0 =
∂h
∂y|y=1 = 0.
Gaskell et al. J. Fluids Mech, 509:253-280, 2004
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Sketch of the fully-developed flow
α
Free surface
Inclined plane
g
Z Y
X
H(X,Y)
flow directionUpstream x=0
S(X,Y)
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Our results with a trench topography
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Cahn-Hilliard-Hele-Shaw system of equations
∂φ
∂t= 4µ−∇ · (φuuu) ,
µ = φ3 − φ− ε24φ,uuu = −∇p − γφ∇µ,
∇ · uuu = 0,
with Neumann boundary conditions:
∂nφ = ∂nµ = ∂np = 0 on ∂Ω,
Wise J. Sci. Comput., 44:38-68, 2010
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Cahn-Hilliard-Hele-Shaw system of equations
∂φ
∂t= ∇ · (M(φ)∇µ)−∇ · (φ∇p),
µ = φ3 − φ− ε24φ,−4p = γ∇ · (φ∇µ),
with Neumann boundary conditions:
∂nφ = ∂nµ = ∂np = 0 on ∂Ω,
Wise J. Sci. Comput., 44:38-68, 2010
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2nd order convergence rate
For variable φ
Levels Time steps Infinity norm Ratio Two norm Ratio
3 40 - - - -4 80 1.074× 10−3 - 3.885× 10−4 -5 160 2.718× 10−4 3.95 9.781× 10−5 3.976 320 6.905× 10−5 3.93 2.468× 10−5 3.96
Table: 2nd order convergence rate seen from CHHS system of equations.
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Tumour modelling - avascular tumour growth
Starts with a small cluster ofcells
Nutrient supply throughdiffusion
Internal adhesion force
Three layers of cells:
Proliferative cellsDormant cellsDead cells (necrosis)
Volume loss in necrotic coresource: www.bioinfo.de
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Tumour modelling - vascular tumour growth
TAF chemical factor
Inducing blood vessel(angiogenesis)
Exponential growth rate
Develop secondaries throughmetastasis
source: www.maths.dundee.ac.uk
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Tumour modelling - micro-environments
J. S. Lowengrub, H. B. Frieboes, Y-L. Chuang, F. Jin, X. Li P. Macklin, S. M. Wise,
Nonlinearity 23:R1-R91, 2010
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Tumour model from Wise et al.
∂tφT = M∇ · (φT∇µ) + ST (φT , φD , n)−∇ · (uSφT )
µ = f ′(φT )− ε2∆φT
∂tφD = M∇ · (φD∇µ) + SD(φT , φD , n)−∇ · (uSφD)
uS = −(∇p − γ
εµ∇φT )
∇ · uS = ST (φT , φD , n)
0 = ∆n + TC (φT , n)− n(φT − φD)
with mixed boundary conditions:
µ = p = 0 n = 1 ∂nφT = ∂nφD = 0 on ∂Ω,
S. M. Wise, J. S. Lowengrub, V. Cristini,
Math. Comput. Modelling, 53: 1-20, 2011.
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Tumour model from Wise et al.
∂tφT = M∇ · (φT∇µ) + ST (φT , φD , n)−∇ · (uSφT )
µ = f ′(φT )− ε2∆φT
∂tφD = M∇ · (φD∇µ) + SD(φT , φD , n)−∇ · (uSφD)
[uS = −(∇p − γ
εµ∇φT )]
−∆p = ST (φT , φD , n)−∇ · (γεµ∇φT )
0 = ∆n + TC (φT , n)− n(φT − φD)
with mixed boundary conditions:
µ = p = 0 n = 1 ∂nφT = ∂nφD = 0 on ∂Ω,
S. M. Wise, J. S. Lowengrub, V. Cristini,
Math. Comput. Modelling, 53: 1-20, 2011.
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Validation
t=100
t=200
Validation between results of Wise et al. and ours.
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2nd order convergence rate
For variable φT
Levels Time steps Infinity norm Ratio Two norm Ratio
5(1282) 1250 - - - -6(2562) 2500 9.118× 10−2 - 7.836× 10−3 -7(5122) 5000 1.322× 10−2 6.90 1.131× 10−3 6.93
8(10242) 10000 2.579× 10−3 5.13 2.367× 10−4 4.789(20482) 20000 6.415× 10−4 4.02 5.833× 10−5 4.06
2nd order convergence rate seen from the model of tumour growth.
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3-D results
t=50 t=100
t=150 t=200
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3-D results
t=200Cutting throughx plane
Cutting throughy plane
Cutting throughz plane
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The End
P. C. Bollada, C. E. Goodyer, P. K. Jimack, A. M. Mullis, F. Yang
J. Comput. Phys., submitted 2014
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