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Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
J. Blair Perot
V. Subramanian
Higher-Order Discrete Calculus Methods
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Realistic
Parallel, Moving Mesh, Complex Geometry, …
Slide 1Slide 1
Practical,Practical, CostCost--effective, Physically Accurateeffective, Physically Accurate
���������
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Context
DC Methods are a subset of many other classical approaches
Slide 2Slide 2
Finite Difference Finite Element
Finite Volume Meshless
Mimetic SOMMimetic SOM
StaggeredStaggered
Edge/FaceEdge/Face
Natural Natural NeighborsNeighbors
Discrete Calculus Methods
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Exact Discretization
Solution:Solution:Requires Approximations/ErrorRequires Approximations/Error
Discretization Approximation
Slide 3Slide 3
•• But all approximation errors occur in the constitutive But all approximation errors occur in the constitutive equations (in material properties).equations (in material properties).
•• All numerical errors appear with the modeling errors.All numerical errors appear with the modeling errors.
Discretization:Discretization:Continuous PDE => Finite Dimensional Matrix ProblemContinuous PDE => Finite Dimensional Matrix Problem
This Can Be Done ExactlyThis Can Be Done Exactly
≠
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Example
The Physical (Continuous) System
Physical Equation (Heat Equation)Physical Equation (Heat Equation)
Components of the Physical EquationComponents of the Physical Equation
( )cTk T
t
ρ∂= ∇ ⋅ ∇
∂
0it
∂ + ∇ ⋅ =∂
q Conservation of EnergyConservation of Energy
T= ∇g Definition of GradientDefinition of Gradient
PhysicsPhysics
MathMath
k= −q g Fourier’s LawFourier’s Law
i cTρ= Perfectly Caloric MaterialPerfectly Caloric MaterialMaterial Material ApproximationApproximation
Slide 4Slide 4
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
The Numerical (Discrete) System
Example
Exact Discretization of Physics and CalculusExact Discretization of Physics and Calculus1| | 0n n
fc c ff
idV idV dt dA+ − + ⋅ =�� � � � q n ��� ��
2 1n ned T T⋅ = −� g l
1 0n nc c f
I I Q+ − + =D �� ��
� e ng T= G
Numerically ExactNumerically Exact
Numerical ApproximationsNumerical Approximations
1 efQ M g= −�
2c nI M T=�
�
�
f
e
A
eLfQ k g= − �
�
c c nI cV Tρ=� �
Numerically ApproximateNumerically Approximate
Slide 5Slide 5
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Same Mesh – More Unknowns
Higher Order Discrete Calculus
Higher Order Exact GradientHigher Order Exact Gradient
2 1n nedged T T⋅ = −� g l
�
Applicable to Any Applicable to Any PolyhedronPolyhedron
Slide 6Slide 6
2
2 2 1 1 1( ) ( )
n
e e n n n nedge nd T T Tdl⋅ ⋅ = ⋅ − −� �t x g l t x x
efaceedges
dA Tdl× = �� �n g t
[2]( )
edge
n
eedgee
face
dT
dTdl
dA
� �⋅� �
� �� �� �⋅ ⋅ =� �� �� � �
� �� ��
�
��
�
g l
t x g l G
n g
nT
Td� �
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Higher Order Discrete Calculus
Higher Order Exact DivergenceHigher Order Exact Divergence
••Need one equation for each edge.Need one equation for each edge.
••Integrate over the very thin CVs surrounding the dual faces.Integrate over the very thin CVs surrounding the dual faces.
••Take the ‘thin’ limit carefully Take the ‘thin’ limit carefully –– so thin elements align.so thin elements align.
Slide 7Slide 7
( )0f
ffcnf e
edgefaces
dA dl∂ ⋅
∂ + ⋅ =�� �q n
n q�
�� �
c
dx
f
[ ] 0f ff ledge edgecells faces
dA dl⊥ ⊥
⊥ ⊥∇ ⋅ + =� �� �q q n�
�
oror
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Basis Functions not Required
Reconstruction
HaveHave
Assume qAssume q==--kkg is linear on g is linear on primary mesh cells.primary mesh cells.
Slide 8Slide 8
e
e
f
d
d
dA
� �� �� �
⋅� �� �� �×�
�
�
�
g l
xg l
n g
�
( )f
f
ff
nf
e
dA
dA
d
∂ ⋅∂
� �⋅� �� �� �� �
� ��
�
�
�
q n
q n
q l
�
�
��
�
�
NeedNeed
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Results
Good Cost/Accuracy for ‘Smooth’ Solutions
Slide 9Slide 9
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Conclusions
•• Exact discretizationExact discretization of of PDEsPDEs is possible and is possible and stronglystrongly encouraged.encouraged.
• Excellent numerical/mathematical properties are NOT restricted to FE methods.NOT restricted to FE methods.
• Applicable to ANY polygon mesh (including ANY polygon mesh (including meshlessmeshless), no explicit basis functions, no need to ), no explicit basis functions, no need to build matrices.build matrices.
Slide 10Slide 10
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
JCP Publications
• Subramanian, V., and Perot, J. B. Subramanian, V., and Perot, J. B. Higher Order Mimetic Higher Order Mimetic Methods for Unstructured MeshesMethods for Unstructured Meshes, , J. J. ComputComput. Phys.. Phys., , 219219, 68, 68--85, 200685, 2006
•• Perot, J. B., and Subramanian, V. Perot, J. B., and Subramanian, V. Discrete Calculus Discrete Calculus Methods for DiffusionMethods for Diffusion, , J. J. ComputComput. Phys.. Phys., , 224224 (1), 59(1), 59--81, 2007.81, 2007.
• Subramanian, V., and Perot, J. B. Subramanian, V., and Perot, J. B. A Discrete Calculus A Discrete Calculus Analysis of the KellerAnalysis of the Keller--Box Scheme and a Box Scheme and a Generalization of the Method to Arbitrary MeshesGeneralization of the Method to Arbitrary Meshes, , Accepted to Accepted to J. J. ComputComput. Phys.. Phys., 2007., 2007.
Slide 11Slide 11
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Discrete Calculus Operators
Gradient Operator Gradient Operator GG ( )2 1n n nT T T− �G� � �
Divergence Operator Divergence Operator DD f ff
Q Q �
�� � �� D
Slide 6Slide 6
Curl Operator Curl Operator CC e ee
s s �
�� � �� C
Rotation Operator Rotation Operator RR f ff
U U �
�� � �� R
T= −G D
T=C R
0 constantφ φ∇ = � =
{ } { }0n nT T c= � =G� �
( ) 0∇ ∇× =�
0=DC
( ) 0∇×∇ =
0=CG
Discrete Calculus Operators mimic Continuous Operators
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
The Discrete Calculus Approach
Continuous vs. Discrete System
PhysicsPhysics NumericsNumerics
0it
∂ + ∇ =∂
q�
T= ∇g
PhysicallyPhysically
ExactExact
k= −q g
i cTρ=PhysicallyPhysically
ApproximateApproximate
NumericallyNumerically
ApproximateApproximate
ef f
e
gQ k A
L= − �
�
c c nI cV Tρ=�
( )cTk T
t
ρ∂= ∇ ∇
∂�
( )I fc nn
AcV Tk T
t L
ρ∂ �= −� ∂ � �
D G�
�
e ng GT=� �
�
NumericallyNumerically
ExactExact
1 0+ − + =Dn nc c fI I Q
Slide 7Slide 7
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Discrete Calculus Methods
DC Approach is a methodology – not a particular method!
Slide 9Slide 9
FaceFace--Based MethodBased Method,f fT Q(KB)(KB)
Higher Order MethodHigher Order Method,n eT T
CellCell--Based MethodBased Method,n fT Q�(Mixed)(Mixed)
NodeNode--Based MethodBased MethodnT
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Physical Accuracy
Linearly Complete as well as Physically Realistic
Discontinuous Diffusion: Heat Flow at an AngleDiscontinuous Diffusion: Heat Flow at an Angle4 0 0.51 0.5 1
xk
x
< <�= � < <�
1 0 0.54 0.5 0.5 1exact
x y xT
x y x
+ + ≤ ≤�= � + − ≤ ≤�
Slide 10Slide 10
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Cost of DC Methods
More Cost-Effective than Traditional Methods
Slide 12Slide 12
For any accuracy For any accuracy level, DC is more level, DC is more costcost--effective than effective than Finite VolumeFinite Volume
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Implications
Methods that capture physics well
Slide 3Slide 3
Exact Discretization:Exact Discretization:•• Discrete Operators (Div, Grad, Curl) Discrete Operators (Div, Grad, Curl)
behave just like the continuous operators.behave just like the continuous operators.•• Mimetic.Mimetic.•• Discrete de Discrete de RhamRham Complex (algebraic topology).Complex (algebraic topology).•• , etc, etc•• Physics is always captured Physics is always captured
(conservation, wave propagation, max principal, …).(conservation, wave propagation, max principal, …).•• No spurious modes.No spurious modes.•• No surprises. No surprises.
( ) 0∇ ×∇ =
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Total cost for a desired accuracy level
High Matrix Condition Numbers Adversely Impacts the Cost
Slide 8 / 16Slide 8 / 16
FaceFace--based DC based DC converges more converges more slowly than Finite slowly than Finite VolumeVolume
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
Discrete Calculus Operators
Discrete Gradient OperatorDiscrete Gradient Operator ( )n e→� �
1 2 1e n ng T T= −� � �
2 2 2bc
e n fg T T= − +� �
3 2 3bc
e n fg T T= − +� �
4 1 4bc
e n fg T T= − +� �
5 1 5bc
e n fg T T= − +� �
�
1 10 10 11 01 0
G
−� �� �−� �� �= −� �−� �� �−�
�
Discrete Divergence OperatorDiscrete Divergence Operator ( )f c→
1 4 51f f f fcDQ Q Q Q= + +
1 2 32f f f fcDQ Q Q Q= − + +
�1 0 0 1 11 1 1 0 0
D� �
= � �−�
TG D= −�
Higher-Order Discrete Calculus
22nd Dundee
B. Perot 06/28/2007
More Discrete Calculus Operators
1 1 0 01 0 0 10 1 0 11 0 1 0
0 1 1 0
C
−� �� �−� �� �= −� �−� �� �−�
( )e f→ ��Discrete Rotation OperatorDiscrete Rotation Operator
Discrete Curl OperatorDiscrete Curl Operator ( )e f→
1 1 0 1 01 0 1 0 1
0 0 0 1 10 1 1 0 0
C
−� �� �− −� �=� �−� �−�
�
TC C= �