     # Higher-Order Discrete Calculus Higher-Order Discrete Calculus Methods. Higher-Order Discrete Calculus

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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

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•• 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

( )cT k T

t

ρ∂ = ∇ ⋅ ∇

0 i t

∂ + ∇ ⋅ = ∂

q Conservation of EnergyConservation of Energy

PhysicsPhysics

MathMath

k= −q g Fourier’s LawFourier’s Law

i cTρ= Perfectly Caloric MaterialPerfectly Caloric Material Material 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 Calculus 1| | 0n n

fc c f f

idV idV dt dA+ − + ⋅ =�� � � � q n ��� � �

2 1n ne d T T⋅ = −� g l

1 0n nc c fI I Q + − + =D �� ��

� e ng T= G

Numerically ExactNumerically Exact

Numerical ApproximationsNumerical Approximations

1 ef Q M g= −�

2c nI M T=�

f

e

A

eLf Q 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

2 1n nedge d 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 n d T T Tdl⋅ ⋅ = ⋅ − −� �t x g l t x x

eface edges

dA Tdl× = �� �n g t

( )

edge

n

eedge e

face

d T

d Tdl

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

f fcnf e

edge faces

dA dl ∂ ⋅

∂ + ⋅ =�� � q n

n q� �� �

c

dx

f

[ ] 0f ff l edge edge cells 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

• 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 f

f

Q Q

�� � � � D

Slide 6Slide 6

Curl Operator Curl Operator CC e e e

s s

�� � � � C

Rotation Operator Rotation Operator RR f f f

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

0 i t

∂ + ∇ = ∂

q�

T= ∇g

PhysicallyPhysically

ExactExact

k= −q g

i cTρ= PhysicallyPhysically

ApproximateApproximate

NumericallyNumerically

ApproximateApproximate

e f f

e

g Q k A

L = − �

c c nI cV Tρ= �

( )cT k T

t

ρ∂ = ∇ ∇

∂ �

( )I fc n n

AcV T k 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--

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