A finite volume method for solving parabolic …grids and nite volume methods for PDEs in circular...
Transcript of A finite volume method for solving parabolic …grids and nite volume methods for PDEs in circular...
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A finite volume method for solving parabolicequations on curved surfaces
Donna CalhounCommissariat a l’Energie Atomique
DEN/SMFE/DM2S, Centre de Saclay, France
Christiane HelzelRuhr-University Bochum, Germany
Workshop on Numerical Methods for PDEs on SurfacesFreiburg, Germany, Sept. 14-17, 2009
Solving parabolic equations on surfaces
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Problem
Solve advection-reaction-diffusion equations
qt +∇ · f(q) = D∇2q + G(q)
using a finite-volume scheme on logically Cartesian smooth surfacemeshes.
I The operators ∇· and ∇2 are the surface divergence andsurface Laplacian, respectively, and
I q is a vector valued function, f (q) is a flux function, and D isa diagonal matrix of constant diffusion coefficients
Solving parabolic equations on surfaces
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Applications
I Diffusion on cell surfaces
I Biological pattern formation on realistic shapes (Turingpatterns, chemotaxis, and so on)
I Phase-field modeling on curvilinear grids (dendritic growthproblems)
I Navier-Stokes equations on the sphere for atmosphericapplications
Solving parabolic equations on surfaces
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Disk and sphere grids
I Single logically Cartesian grid → disk
I Nearly uniform cell sizes
Solving parabolic equations on surfaces
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Disk and sphere grids
I Single logically Cartesian grid → sphere
I Nearly uniform cell sizes
Solving parabolic equations on surfaces
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Other grids
“Super-shape”
Solving parabolic equations on surfaces
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Fractional step approach
To solveqt +∇ · f(q) = D∇2q + G(q)
we alternate between these two steps :
(1) qt +∇ · f(q) = 0
(2) qt = D∇2q + G(q)
Take a full time step ∆t of each step. Treat each sub-problemindependently.
The focus of this talk is on describing a finite-volume scheme forsolving the parabolic step.
Solving parabolic equations on surfaces
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Assumptions and requirements
Parabolic surface problem :
qt = ∇2q + G(q)
Parabolic scheme should couple well with our finite-volumehyperbolic solvers.
I We assume that our surfaces can be described parametrically,
I We do not want to involve analytic metric terms, and
I Scheme should use cell-centered values.
We need a finite-volume discretization of the Laplace-Beltramioperator on smooth quadrilateral surface meshes
Solving parabolic equations on surfaces
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Previous work
I Finite element methods for triangular surface meshes (Dzuik,Elliot, Polthier, Pinkall, Desbrun, Meyer, and others),
I Finite-volume schemes for diffusion equations on unstructuredgrids in Euclidean space (Hermeline, Eymard, Gallouet,Herbin, LePotier, Hubert, Boyer, Shaskov, Omnes, Z. Sheng,G. Yuan, and so on)
I Approximating curvature by discretizing the Laplace-Beltramioperator on quadrilateral meshes (G. Xu)
Solving parabolic equations on surfaces
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Laplace-Beltrami operator
∇2 q =1√a
{∂
∂ξ
√a
(a11∂q
∂ξ+ a21∂q
∂η
)+
∂
∂η
√a
(a21∂q
∂ξ+ a22∂q
∂η
)}
with mapping
T (ξ, η) = [X (ξ, η),Y (ξ, η),Z (ξ, η)]T
and conjugate metric tensor
(a11 a12
a21 a22
)=
(a11 a12
a21 a22
)−1
=
(Tξ · Tξ Tξ · TηTη · Tξ Tη · Tη
)−1
where a ≡ a11a22 − a12a21
Solving parabolic equations on surfaces
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Computing fluxes at cell edges
θ
qi−1,j
qi,j
ti−1/2,j
ti−1/2,j
qi,j+1
qi,j
Flux :
∫
edge
dq
dnds ≈
√a
(a11 ∂q
∂ξ+ a12 ∂q
∂η
)∆η
Solving parabolic equations on surfaces
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Computing fluxes at cell edges
T (ξ, η) = [X (ξ, η),Y (ξ, η),Z (ξ, η)]T
Flux :
∫
edge
dq
dnds ≈
√a
(a11 ∂q
∂ξ+ a12 ∂q
∂η
)∆η
a11 = Tξ · Tξ ≈ t · t = |t|2
a12 = a21 = Tξ · Tη ≈ t · t = |t||t| cos(θ)
a22 = Tη · Tη ≈ t · t = |t|2√
a = |Tξ × Tη| ≈ |t × t| = |t||t| sin(θ)
a11 = a22/a, a12 = a21 = −a12/a, a22 = a11/a
Solving parabolic equations on surfaces
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Computing edge-based fluxes
∫ bxi,j+1
bxi,j
dq
dnds ≈ |t|
|t|csc(θ)∆q − cot(θ) ∆q
θ
qi−1,j
qi,j
ti−1/2,j
ti−1/2,j
qi,j+1
qi,j
Solving parabolic equations on surfaces
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Discrete Laplace-Beltrami operator
∇2q ≈ L(q) ≡ 1
Area
4∑
k=1
|tk ||tk |
csc(θk)∆kq − cot(θk) ∆k q
I ∆kq is the difference in cell centered values of q
I ∆k q is the difference of nodal values of q, and
I θk is the angle between tk and tk .
Solving parabolic equations on surfaces
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Obtaining node values
I In regions where the mesh is smooth, node values may beobtained by an arithmetic average of the cell-centered values.
I Along diagonal “seams”, we average using only cell centeredvalues on the diagonal.
Solving parabolic equations on surfaces
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Physical boundaries for open surfaces
Impose boundary conditions to obtain edge values at boundary :
a q + bdq
dn= c
x1,j+1
x1,j+1 x2,j
x2,j+1
x2,j
x1,j
x1,j
x0,j
Obtain tridiagonal system for node values at the boundary.
Solving parabolic equations on surfaces
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Equator conditions for the sphere
Match fluxes at the equator and obtain a tridiagonal system forthe node values at the equator
Solving parabolic equations on surfaces
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Properties of the discrete operator
I 9-point stencil involving only cell-centers
I Requires only physical location of mesh cell centers and nodes
I No surface normals are required, since discretization isintrinsic to the surface.
I Orthogonal and non-orthogonal grids both treated.
I On smooth or piecewise-smooth mappings, numericalconvergence tests show second order accuracy.
Solving parabolic equations on surfaces
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Accuracy
Solving parabolic equations on surfaces
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Superconvergence property
Discretization is not consistent∥∥∥∥L(q)− 1
Area
∫∇2q dS
∥∥∥∥ ∼ O(1)
so convergence of solutions to PDEs involving L(q) relies on asuperconvergence property often seen in FV schemes.
I This operator of little use in estimating curvatures of surfacesmeshes
Solving parabolic equations on surfaces
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Connection to other schemes
∇2q ≈ L(q) ≡ 1
Area
4∑
k=1
|tk ||tk |
csc(θk)∆kq − cot(θk) ∆k q
I L(q) reduces to familar stencils on Cartesian and polar grids,
I On a subset of flat Delaunay surface triangulations, L(q)reduces to the “cotan” formula
I Closely related to “diamond-cell” and “Discrete Duality FiniteVolume” (DDFV) schemes for discretizing diffusion terms onflat unstructured, polygonal meshes (Coudiere, Hermeline,Omnes, Komolevo, Herbin, Eymard, Gallouet...)
Solving parabolic equations on surfaces
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Connection to the cotan formula
x6x0
x1
x2
x3
x3
x4
x2
x4
x1
x5
x5
x6
∫
[x1,x2]
∂q
∂ndL ≈ |x1 − x2|
|x0 − x2|(q(x2)− q(x0)) (1)
Solving parabolic equations on surfaces
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Connection to the cotan formula
x2
x1
βαα
x3β
x0
|x1 − x2||x0 − x2|
=1
2(cotα0,2 + cotβ0,2) (2)
Solving parabolic equations on surfaces
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Connection to the cotan formula
x6x0
x1
x2
x3
x3
x4
x2
x4
x1
x5
x5
x6
∫
D0
∇2q dA ≈6∑
j=1
1
2(cot(α0,j) + cot(β0,j)) (q(xj)− q(x0))
Solving parabolic equations on surfaces
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Advection-Reaction-diffusion equations
qt +∇(u q) = ∇2q + f (q)
a q + bdq
dn= c
To handle time dependency,
I Runge-Kutta-Chebyschev (RKC) solver for explicit timestepping of diffusion term (Sommeijer, Shampine, Verwer,1997).
I Wave-propagation algorithms for advection terms (SeeClawpack, R. J. LeVeque).
Solving parabolic equations on surfaces
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Chemotaxis in a petri-dish
∂u
∂t= du∇2u − α∇ ·
((∇v
(1 + v)2
)u
)+ ρu(δ − u)
∂v
∂t= ∇2v + βu2 − uv .
Solving parabolic equations on surfaces
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Turing patterns
∂u
∂t= Dδ∇2u + αu
(1− τ1v2
)+ v(1− τ2u)
∂v
∂t= δ∇2v + βv
(1 +
ατ1β
uv
)+ u(γ + τ2v)
Solving parabolic equations on surfaces
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Turing patterns
Solving parabolic equations on surfaces
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Flow by mean curvature
Allen-Cahn equation
ut = D2∇2 + (u − u3)
Solving parabolic equations on surfaces
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Spiral waves
Spiral waves using the Barkley model
ut = ∇2u +1
εu(1− u)(u − v + b
a)
vt = u − v , ε = 0.02, a = 0.75, b = 0.02
Solving parabolic equations on surfaces
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More?
I D. Calhoun, C. Helzel, R. J. LeVeque, ”Logically rectangulargrids and finite volume methods for PDEs in circular andspherical domains”. SIAM Review, 50-4 (2008).
I D. Calhoun, C. Helzel, ”A finite volume method for solvingparabolic equations on logically Cartesian curved surfacemeshes”, (to appear, SISC).http://www.amath.washington.edu/∼calhoun/Surfaces
Code is available!
Solving parabolic equations on surfaces