Adaptive wavelet collocation method for PDEson the sphere
Mani Mehra
Department of Mathematics and Statistics
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Collaborators
Nicholas Kevlahan (McMaster University)
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Motivation
Adaptive wavelet collocation method on the sphere
Numerical simulation
Conclusions and future directions
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why general manifolds?
(e.g. Sphere)
Application of adaptive wavelet collocation method (AWCM)to the problems of geodesy, climatology, meteorology(Representative examples include forecasting the moisture andcloud water fields in numerical weather prediction).
Many PDEs arise from mean curvature flow, surface diffusionflow and Willmore flow on the sphere.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why general manifolds?
(e.g. Sphere)
Application of adaptive wavelet collocation method (AWCM)to the problems of geodesy, climatology, meteorology(Representative examples include forecasting the moisture andcloud water fields in numerical weather prediction).
Many PDEs arise from mean curvature flow, surface diffusionflow and Willmore flow on the sphere.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why general manifolds?
(e.g. Sphere)
Application of adaptive wavelet collocation method (AWCM)to the problems of geodesy, climatology, meteorology(Representative examples include forecasting the moisture andcloud water fields in numerical weather prediction).
Many PDEs arise from mean curvature flow, surface diffusionflow and Willmore flow on the sphere.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why general manifolds?
(e.g. Sphere)
Application of adaptive wavelet collocation method (AWCM)to the problems of geodesy, climatology, meteorology(Representative examples include forecasting the moisture andcloud water fields in numerical weather prediction).
Many PDEs arise from mean curvature flow, surface diffusionflow and Willmore flow on the sphere.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets?
High rate of data compression.
Fast O(N ) transform.
Dynamic grid adaption to the local irregularities of thesolution.(This situation arises e.g. in the tracking of storms or frontsfor the simulation of global atmospheric dynamics).
Easy to control wavelet properties (e.g. smoothness, boundarycondition,better accuracy near sharp gradients).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets?
High rate of data compression.
Fast O(N ) transform.
Dynamic grid adaption to the local irregularities of thesolution.(This situation arises e.g. in the tracking of storms or frontsfor the simulation of global atmospheric dynamics).
Easy to control wavelet properties (e.g. smoothness, boundarycondition,better accuracy near sharp gradients).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets?
High rate of data compression.
Fast O(N ) transform.
Dynamic grid adaption to the local irregularities of thesolution.(This situation arises e.g. in the tracking of storms or frontsfor the simulation of global atmospheric dynamics).
Easy to control wavelet properties (e.g. smoothness, boundarycondition,better accuracy near sharp gradients).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets?
High rate of data compression.
Fast O(N ) transform.
Dynamic grid adaption to the local irregularities of thesolution.(This situation arises e.g. in the tracking of storms or frontsfor the simulation of global atmospheric dynamics).
Easy to control wavelet properties (e.g. smoothness, boundarycondition,better accuracy near sharp gradients).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets?
High rate of data compression.
Fast O(N ) transform.
Dynamic grid adaption to the local irregularities of thesolution.(This situation arises e.g. in the tracking of storms or frontsfor the simulation of global atmospheric dynamics).
Easy to control wavelet properties (e.g. smoothness, boundarycondition,better accuracy near sharp gradients).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets on manifolds?
(e.g. spherical wavelets)
Spherical triangular grids (quasi uniform triangulations) avoidthe pole problem.Conventional grids
Uniform longitude-latitude grid. Another solution is to avoid the introduction of the metric
term which are unbounded near the poles.
To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods.
Past application of wavelets to turbulence have been mainlyrestricted to flat geometries which severely limiting forgeophysical applications.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets on manifolds?
(e.g. spherical wavelets)
Spherical triangular grids (quasi uniform triangulations) avoidthe pole problem.Conventional grids
Uniform longitude-latitude grid. Another solution is to avoid the introduction of the metric
term which are unbounded near the poles.
To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods.
Past application of wavelets to turbulence have been mainlyrestricted to flat geometries which severely limiting forgeophysical applications.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets on manifolds?
(e.g. spherical wavelets)
Spherical triangular grids (quasi uniform triangulations) avoidthe pole problem.Conventional grids
Uniform longitude-latitude grid. Another solution is to avoid the introduction of the metric
term which are unbounded near the poles.
To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods.
Past application of wavelets to turbulence have been mainlyrestricted to flat geometries which severely limiting forgeophysical applications.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets on manifolds?
(e.g. spherical wavelets)
Spherical triangular grids (quasi uniform triangulations) avoidthe pole problem.Conventional grids
Uniform longitude-latitude grid. Another solution is to avoid the introduction of the metric
term which are unbounded near the poles.
To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods.
Past application of wavelets to turbulence have been mainlyrestricted to flat geometries which severely limiting forgeophysical applications.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets on manifolds?
(e.g. spherical wavelets)
Spherical triangular grids (quasi uniform triangulations) avoidthe pole problem.Conventional grids
Uniform longitude-latitude grid. Another solution is to avoid the introduction of the metric
term which are unbounded near the poles.
To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods.
Past application of wavelets to turbulence have been mainlyrestricted to flat geometries which severely limiting forgeophysical applications.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Why wavelets on manifolds?
(e.g. spherical wavelets)
Spherical triangular grids (quasi uniform triangulations) avoidthe pole problem.Conventional grids
Uniform longitude-latitude grid. Another solution is to avoid the introduction of the metric
term which are unbounded near the poles.
To solve PDEs efficiently using adaptivity on general manifoldby wavelet methods.
Past application of wavelets to turbulence have been mainlyrestricted to flat geometries which severely limiting forgeophysical applications.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet multiresolution analysis of L2(S)
DefinitionMRA is characterized by the following axioms
V j V j+1 (subspaces are nested).
j=j= Vj = L2(S).
Each V j has a Riesz basis of scaling function {jk |k Kj}.
Define Wj = {jk(wavelets)|k M
j} to be the complement of Vjin Vj+1, where Vj+1 = Vj + Wj .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet multiresolution analysis of L2(S)
DefinitionMRA is characterized by the following axioms
V j V j+1 (subspaces are nested).
j=j= Vj = L2(S).
Each V j has a Riesz basis of scaling function {jk |k Kj}.
Define Wj = {jk(wavelets)|k M
j} to be the complement of Vjin Vj+1, where Vj+1 = Vj + Wj .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet multiresolution analysis of L2(S)
DefinitionMRA is characterized by the following axioms
V j V j+1 (subspaces are nested).
j=j= Vj = L2(S).
Each V j has a Riesz basis of scaling function {jk |k Kj}.
Define Wj = {jk(wavelets)|k M
j} to be the complement of Vjin Vj+1, where Vj+1 = Vj + Wj .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet multiresolution analysis of L2(S)
DefinitionMRA is characterized by the following axioms
V j V j+1 (subspaces are nested).
j=j= Vj = L2(S).
Each V j has a Riesz basis of scaling function {jk |k Kj}.
Define Wj = {jk(wavelets)|k M
j} to be the complement of Vjin Vj+1, where Vj+1 = Vj + Wj .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet multiresolution analysis of L2(S)
DefinitionMRA is characterized by the following axioms
V j V j+1 (subspaces are nested).
j=j= Vj = L2(S).
Each V j has a Riesz basis of scaling function {jk |k Kj}.
Define Wj = {jk(wavelets)|k M
j} to be the complement of Vjin Vj+1, where Vj+1 = Vj + Wj .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet multiresolution analysis of L2(S)
DefinitionMRA is characterized by the following axioms
V j V j+1 (subspaces are nested).
j=j= Vj = L2(S).
Each V j has a Riesz basis of scaling function {jk |k Kj}.
Define Wj = {jk(wavelets)|k M
j} to be the complement of Vjin Vj+1, where Vj+1 = Vj + Wj .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Construction of spherical wavelets based on sphericaltriangular grids
The set of all vertices S j = {pjk S : pjk = p
j+12k |k K
j} andMj = Kj+1/Kj .
Level 0
Dyadic icosahedral triangulation of the sphereMani Mehra McMaster UniversityAdaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Construction of spherical wavelets based on sphericaltriangular grids
The set of all vertices S j = {pjk S : pjk = p
j+12k |k K
j} andMj = Kj+1/Kj .
Level 1
Dyadic icosahedral triangulation of the sphereMani Mehra McMaster UniversityAdaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Fast wavelet transform
e4 e3
e2e1
f1
f2
v1 v2
m
The members of the neighborhoodsused in wavelet bases (m Mj ,Km = {v1, v2, f1, f2, e1, e2, e3, e4}).
Analysis(j) :
m Mj : d jm = cj+1m
kKm
sjk,mc
jk ,
m Kj : c jk = cj+1k .
Synthesis(j) :
m Kj : c jk = cjk ,
m Mj : c j+1m = djm +
kKm
sjk,mc
jk .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Fast wavelet transform
e4 e3
e2e1
f1
f2
v1 v2
m
The members of the neighborhoodsused in wavelet bases (m Mj ,Km = {v1, v2, f1, f2, e1, e2, e3, e4}).
Analysis(j) :
m Mj : d jm = cj+1m
kKm
sjk,mc
jk ,
m Kj : c jk = cj+1k .
Synthesis(j) :
m Kj : c jk = cjk ,
m Mj : c j+1m = djm +
kKm
sjk,mc
jk .
For linear basis, sv1 = sv2 = 1/2
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Fast wavelet transform
e4 e3
e2e1
f1
f2
v1 v2
m
The members of the neighborhoodsused in wavelet bases (m Mj ,Km = {v1, v2, f1, f2, e1, e2, e3, e4}).
Analysis(j) :
m Mj : d jm = cj+1m
kKm
sjk,mc
jk ,
m Kj : c jk = cj+1k .
Synthesis(j) :
m Kj : c jk = cjk ,
m Mj : c j+1m = djm +
kKm
sjk,mc
jk .
For Butterfly basis, sv1 = sv2 = 1/2,sf1 = sf2 = 1/8,se1 = se2 = se3 = 1/16
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Fast wavelet transform
e4 e3
e2e1
f1
f2
v1 v2
m
The members of the neighborhoodsused in wavelet bases (m Mj ,Km = {v1, v2, f1, f2, e1, e2, e3, e4}).
Analysis(j) :
m Mj : d jm = cj+1m
kKm
sjk,mc
jk ,
m Mj : c jk = cj+1k + s
jk,md
jm.
Synthesis(j) :
m Mj : c jk = cjk
kKm
sjk,md
jm,
m Mj : c j+1m = djm +
kKm
sjk,mc
jk .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj djm
jm(p)
Test function
1
0
1 10
11
0.5
0
0.5
1
yx
z
Wavelet locations xJk withoutcompression at J = 5, #K5 = 10242
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj
|d jm|
djm
jm(p)
Test function
1
0
1 10
11
0.5
0
0.5
1
yx
zWavelet locations xJk at J = 5, = 105, N() = 995 and ratio
#K5
N() 10
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj
|d jm|
djm
jm(p)
Test function Wavelet locations xJk withoutcompression at J = 6, #K6 = 40962
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj
|d jm|
djm
jm(p)
Test function Wavelet locations xJk st J = 6, = 105, N() = 8175 and ratio
#K6
N() 5
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj
|d jm|
djm
jm(p)
Test function Wavelet locations xJk withoutcompression at J = 7,
#K7 = 163842
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj
|d jm|
djm
jm(p)
Test function Wavelet locations xJk at J = 7, = 105, N() = 20353 and ratio
#K7
N() 8
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj
|d jm|
djm
jm(p)
Test function Wavelet locations xJk withoutcompression at J = 8,
#K8 = 655362
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet compression
uJ(p) =
kK0 cJ0k
J0k (p) +
j=J1j=J0
mMj
|d jm|
djm
jm(p)
Test function Wavelet locations xJk at J = 8, = 105, N() = 64231 and ratio
#K8
N() 10
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet approximation estimates
Approximation error iscontrolled by the waveletthreshold ||uJ(p) uJ(p)|| c1
106
105
104
103
102
101
100
106
105
104
103
102
101
100
||u
J(p
)u
J (p
)||
LinearLinear liftedButterflyButterfly lifted
O()
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Wavelet approximation estimates
controls the total active gridpoints N() by the followingrelation N() c2
nk
Therefore, approximation erroris controlled by active gridpoints
||uJ(p) uJ(p)|| c3N() k
n
102
103
104
105
106
106
105
104
103
102
101
100
N()
||u
J(p
)u
J (p
)||
LinearLinear liftedButterflyButterfly lifted
O(N()1)
O(N()2)
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Calculation of Laplace-Beltrami operator on adaptive grid
Su(pji ) =
1
AS(pji)
kN(i)cot i,k+cot i,k
2 [u(pjk) u(p
ji )]
where AS(pji ) is the area
of one-ringneighborhood regiongiven by AS(p
ji ) =
18
kN(i)(coti ,k +
cot i ,k)pjk p
ji
2.
pij
pkj
i,k
i,kp
k1j p
k+1j
AM
(pij)
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Calculation of Laplace-Beltrami operator on adaptive grid
Su(pji ) =
1
AS(pji)
kN(i)cot i,k+cot i,k
2 [u(pjk) u(p
ji )]
Convergence result for theLaplace-Beltrami operator ofuJ(p) for the test function
||SuJ(p) Su
J(p)|| c4
106
105
104
103
102
101
100
101
106
105
104
103
102
101
100
/||uJ(p)||
||
Su
J(p
)
Su
J (p
)||
/||
Su
J(p
)||
Butterfly
Butterfly lifted
O()
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Calculation of Laplace-Beltrami operator on adaptive grid
Su(pji ) =
1
AS(pji)
kN(i)cot i,k+cot i,k
2 [u(pjk) u(p
ji )]
Convergence result for theLaplace-Beltrami operator ofuJ(p) for the test function
||SuJ(p) Su
J(p)|| c4
c5N() k2
n
103
104
105
106
106
105
104
103
102
101
100
N()
||
Su
J(p
)
Su
J (p
)||
/||
Su
J(p
)||
ButterflyButterfly lifted
O(N()3/2)
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Calculation of other differential operators on adaptive grid
The flux term present in the spherical advection equation canbe expressed in the form of flux divergence and Jacobianoperators, .(Vu) = .(u) J(u, )
JS(u(pji ), (p
ji )) =
1
6AS(pji)
kN(i)(u(pjk) + u(p
ji ))((p
jk ) (p
ji ))
S .(u(pji ),S(p
ji )) =
1
2AS(pji )
kN(i)
coti ,k + cot i ,k2
(u(pjk) + u(pji ))((p
jk ) (p
ji )).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Calculation of other differential operators on adaptive grid
The flux term present in the spherical advection equation canbe expressed in the form of flux divergence and Jacobianoperators, .(Vu) = .(u) J(u, )
JS(u(pji ), (p
ji )) =
1
6AS(pji)
kN(i)(u(pjk) + u(p
ji ))((p
jk ) (p
ji ))
S .(u(pji ),S(p
ji )) =
1
2AS(pji )
kN(i)
coti ,k + cot i ,k2
(u(pjk) + u(pji ))((p
jk ) (p
ji )).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Calculation of other differential operators on adaptive grid
The flux term present in the spherical advection equation canbe expressed in the form of flux divergence and Jacobianoperators, .(Vu) = .(u) J(u, )
JS(u(pji ), (p
ji )) =
1
6AS(pji)
kN(i)(u(pjk) + u(p
ji ))((p
jk ) (p
ji ))
S .(u(pji ),S(p
ji )) =
1
2AS(pji )
kN(i)
coti ,k + cot i ,k2
(u(pjk) + u(pji ))((p
jk ) (p
ji )).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Calculation of other differential operators on adaptive grid
The flux term present in the spherical advection equation canbe expressed in the form of flux divergence and Jacobianoperators, .(Vu) = .(u) J(u, )
JS(u(pji ), (p
ji )) =
1
6AS(pji)
kN(i)(u(pjk) + u(p
ji ))((p
jk ) (p
ji ))
S .(u(pji ),S(p
ji )) =
1
2AS(pji )
kN(i)
coti ,k + cot i ,k2
(u(pjk) + u(pji ))((p
jk ) (p
ji )).
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Application of spherical wavelets to compress turbulence
Turbulence can be divided into two orthogonal parts: aorganized (coherent vortices), inhomogeneous, non-Gaussiancomponent and random noise (incoherent), homogeneous andGaussian component.
The coherent vortices must be resolved, but the noise may bemodeled (or neglected entirely).
The coherent vortices can be extracted using nonlinearwavelet filtering.
This is the concept of Coherent Vortex Simulation which wasdeveloped by Marie Farge, Kai Schneider and Nicholas Kevlahan inflat geometries.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Application of spherical wavelets to compress turbulence
Turbulence can be divided into two orthogonal parts: aorganized (coherent vortices), inhomogeneous, non-Gaussiancomponent and random noise (incoherent), homogeneous andGaussian component.
The coherent vortices must be resolved, but the noise may bemodeled (or neglected entirely).
The coherent vortices can be extracted using nonlinearwavelet filtering.
This is the concept of Coherent Vortex Simulation which wasdeveloped by Marie Farge, Kai Schneider and Nicholas Kevlahan inflat geometries.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Application of spherical wavelets to compress turbulence
Turbulence can be divided into two orthogonal parts: aorganized (coherent vortices), inhomogeneous, non-Gaussiancomponent and random noise (incoherent), homogeneous andGaussian component.
The coherent vortices must be resolved, but the noise may bemodeled (or neglected entirely).
The coherent vortices can be extracted using nonlinearwavelet filtering.
This is the concept of Coherent Vortex Simulation which wasdeveloped by Marie Farge, Kai Schneider and Nicholas Kevlahan inflat geometries.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Application of spherical wavelets to compress turbulence
Turbulence can be divided into two orthogonal parts: aorganized (coherent vortices), inhomogeneous, non-Gaussiancomponent and random noise (incoherent), homogeneous andGaussian component.
The coherent vortices must be resolved, but the noise may bemodeled (or neglected entirely).
The coherent vortices can be extracted using nonlinearwavelet filtering.
This is the concept of Coherent Vortex Simulation which wasdeveloped by Marie Farge, Kai Schneider and Nicholas Kevlahan inflat geometries.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Application of spherical wavelets to compress turbulence
Vorticity function. Reconstructed with 40%significant wavelets.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Application of spherical wavelets to compress turbulence
E (n, 0) = An/2
(n+n0)
100
101
102
107
106
105
104
103
102
k
E(k
)
E(k)E(k) u
Energy spectrum reconstructedwith 40% significant wavelets.
100
101
102
107
106
105
104
103
102
kE
(k)
E(k)E(k) u
Energy spectrum reconstructedwith 2% significant wavelets
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Application of spherical wavelets to compress turbulence
102
103
104
105
106
107
107
106
105
104
103
102
101
N()
LinearLinear liftedButterflyButterfly lifted
Relation between and N() for thevorticity function.
20 40 60 80 10010
6
104
102
100
102
% of coefficients used
% o
f err
or
LinearLinear liftedButterflyButterfly lifted
Rate of relative error as a function ofrate of significant coefficients
=N()100N(=0) .
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Diffusion equation
ut = Su + f
where f is localized source chosen such a way that the solution ofdiffusion equation is given by
u(, , t) = 2e
(0)2+(0)
2
(t+1)
Such equations arise in the application of Willmore flow, surfacediffusion flow etc.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Diffusion Equation
Initial condition at t = 0 Adaptive grid at t = 0
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Diffusion Equation
Solution using AWCM att = 0.5
Adaptive grid at t = 0.5
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Diffusion Equation
The potential of adaptivealgorithm is measured bycompression coefficients
C =N( = 0)
N()
0 0.1 0.2 0.3 0.4 0.58
10
12
14
16
18
20
22
24
t
CThe compression coefficient C
as a function of time, = 105.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Spherical advection equation
u
t+ V .Su = 0,
The nondivergent driving velocity field V = (v1, v2) for all times isgiven by
v1 = u0
(
cos() cos() + sin() cos(+3
2) sin()
)
,
v2 = u0 sin(+3
2) sin().
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Spherical advection equation
u
t+ V .Su = 0,
The initial cosine bell test pattern to be advected is given by
u(, ) =
{
u02
(
1 + cos(rR
)
if r < R
0 if r R ,
where u0 = 1000m, radius R = a/3 and r = a arccos[sin(c) sin() + cos(c) cos() cos( c), which is the greatcircle distance between (, ) and the center (c , c) = (0, 0).Such equations arise typically in the context of numerical weatherforecast or in climatological studies, and they provide some of themost challenging and CPU time consuming problems in moderncomputational fluid dynamics.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Spherical advection equation
1 0.5 0 0.5 10.2
0
0.2
0.4
0.6
0.8
1
Analytic sol.AWCM
Solid body rotation of cosine bell using AWCM for = 105.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Spherical advection equation
Solution using AWCM Adapted grid for the solution.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Spherical advection equation
0 0.05 0.1 0.150
5
10
15
20
25
30
35
t
C
The compression coefficient Cas a function of time,
= 105.
0 0.05 0.1 0.1510
8
106
104
102
100
t
||u(p
,t)
uex(
p,t)|
| Time evolution of the
pointwise L error of thesolution.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Conclusions
Adaptive wavelet collocation method:
Used for time evolution problems.
Dynamic adaptivity is necessary for atmospheric modeling.
Fast O(N ) wavelet transform and O(N ) hierarchical finitedifference schemes over triangulated surface for thedifferential operators is used.
Verified convergence result predicted in theory.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Conclusions
Adaptive wavelet collocation method:
Used for time evolution problems.
Dynamic adaptivity is necessary for atmospheric modeling.
Fast O(N ) wavelet transform and O(N ) hierarchical finitedifference schemes over triangulated surface for thedifferential operators is used.
Verified convergence result predicted in theory.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Conclusions
Adaptive wavelet collocation method:
Used for time evolution problems.
Dynamic adaptivity is necessary for atmospheric modeling.
Fast O(N ) wavelet transform and O(N ) hierarchical finitedifference schemes over triangulated surface for thedifferential operators is used.
Verified convergence result predicted in theory.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Conclusions
Adaptive wavelet collocation method:
Used for time evolution problems.
Dynamic adaptivity is necessary for atmospheric modeling.
Fast O(N ) wavelet transform and O(N ) hierarchical finitedifference schemes over triangulated surface for thedifferential operators is used.
Verified convergence result predicted in theory.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Conclusions
Adaptive wavelet collocation method:
Used for time evolution problems.
Dynamic adaptivity is necessary for atmospheric modeling.
Fast O(N ) wavelet transform and O(N ) hierarchical finitedifference schemes over triangulated surface for thedifferential operators is used.
Verified convergence result predicted in theory.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Future directions
Implementation of wavelet bases based on optimal sphericaltriangulation.
Ref.:Discrete Laplace-Beltrami operator on sphere and optimalspherical triangulation, Int. J. Comp. Geometry Appl. 16 (1)(2006) 75-93.
Incorporation of appropriate time integration method inspherical domain which takes advantage of wavelet multileveldecomposition.
Ref.:An adaptive multilevel wavelet collocation method for ellipticproblems, J. Comp. Phys. 206 (2005) 412-431.
Application of AWCM to more realistic models.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Future directions
Implementation of wavelet bases based on optimal sphericaltriangulation.
Ref.:Discrete Laplace-Beltrami operator on sphere and optimalspherical triangulation, Int. J. Comp. Geometry Appl. 16 (1)(2006) 75-93.
Incorporation of appropriate time integration method inspherical domain which takes advantage of wavelet multileveldecomposition.
Ref.:An adaptive multilevel wavelet collocation method for ellipticproblems, J. Comp. Phys. 206 (2005) 412-431.
Application of AWCM to more realistic models.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Future directions
Implementation of wavelet bases based on optimal sphericaltriangulation.
Ref.:Discrete Laplace-Beltrami operator on sphere and optimalspherical triangulation, Int. J. Comp. Geometry Appl. 16 (1)(2006) 75-93.
Incorporation of appropriate time integration method inspherical domain which takes advantage of wavelet multileveldecomposition.
Ref.:An adaptive multilevel wavelet collocation method for ellipticproblems, J. Comp. Phys. 206 (2005) 412-431.
Application of AWCM to more realistic models.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
o
Motivation Adaptive wavelet collocation method on the sphere Numerical simulation Conclusions and future directions
Future directions
Implementation of wavelet bases based on optimal sphericaltriangulation.
Ref.:Discrete Laplace-Beltrami operator on sphere and optimalspherical triangulation, Int. J. Comp. Geometry Appl. 16 (1)(2006) 75-93.
Incorporation of appropriate time integration method inspherical domain which takes advantage of wavelet multileveldecomposition.
Ref.:An adaptive multilevel wavelet collocation method for ellipticproblems, J. Comp. Phys. 206 (2005) 412-431.
Application of AWCM to more realistic models.
Mani Mehra McMaster University
Adaptive wavelet collocation method for PDEs on the sphere
MotivationAdaptive wavelet collocation method on the sphereNumerical simulationConclusions and future directions